Fractal Functions, Fractal Surfaces, and Wavelets, Second Edition [2ed.] 012804408X, 978-0-12-804408-7, 9780128044704, 0128044705

Table of contents :
Content: Part I: Foundations 1. Mathematical preliminaries 2. Construction of fractal sets 3. Dimension theory 4. Dynamical systems and dimension Part II: Fractal Functions and Fractal Surfaces 5. Construction of fractal functions 6. Fractels and self-referential functions 7. Dimension of fractal functions 8. Fractal functions and wavelets 9. Fractal surfaces 10. Fractal surfaces and wavelets in â n

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Fractal Functions, Fractal Surfaces, and Wavelets

Fractal Functions, Fractal Surfaces, and Wavelets

Second Edition

Peter R. Massopust Centre of Mathematics Technische Universität München München, Germany

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2016, 1994 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-804408-7 For information on all Academic Press publications visit our website at https://www.elsevier.com/

Publisher: Nikki Levy Acquisition Editor: Graham Nisbet Editorial Project Manager: Susan Ikeda Production Project Manager: Paul Prasad Chandramohan Cover Designer: Matthew Limbert Typeset by SPi Global, India

This book is dedicated to my wife Maritza, my son Jonathan, and my daughter Sarah.

About the author Peter R. Massopust is a Privatdozent in the Center of Mathematics at the Technical University of Munich, Germany. He received his PhD degree in Mathematics from Georgia Institute of Technology in Atlanta, Georgia, United States, and his habilitation from the Technical University of Munich. He worked at several universities in the United States, at Sandia National Laboratories in Albuquerque (United States), and as a senior research scientist in industry before returning to the academic environment. He has written more than 60 peer-reviewed articles in the mathematical areas of Fourier analysis, approximation theory, fractals, splines, and harmonic analysis and wrote more than 20 technical reports while working in the nonacademic environment. He has authored or coauthored two textbooks and two monographs and has coedited two contemporary mathematics volumes and several special issues for peer-reviewed journals. He is on the editorial board of several mathematics journals, and has given more than 100 invited presentations at national and international conferences, workshops, and seminars.

Preface to first edition This monograph gives an introduction to the theory of fractal functions and fractal surfaces with an application to wavelet theory. The study of fractal functions goes back to Weierstraß’s nowhere differentiable function and beyond. However, it was not until the publication of B. Mandelbrot’s book (cf. [1]) in which the concept of a fractal set was introduced and common characteristics of these sets, such as nonintegral dimension and geometric self-similarity, identified that the theory of functions with fractal graphs developed into an area of its own. Seemingly different types of nowhere differentiable functions, such as those investigated by Besicovitch, Ursell, Knopp, and Kiesswetter, to only mention a few, were unified under the fractal point of view and this unification led to new mathematical methods and applications in areas that include: dimension theory, dynamical systems and chaotic dynamics, image analysis, and wavelet theory. The objective of this monograph is to provide essential results from the theory of fractal functions and surfaces for those interested in this fascinating area, to present new and exciting applications and to indicate in which interesting directions the theory can be extended. The book is essentially self-contained and covers the basic theory and the different types of fractal constructions as well as some specialized and advanced topics such as dimension calculations and function space theory. The material is presented in the following way. The first part of the book contains basically background material and consists of four chapters. The first chapter introduces the relevant notation and terminology and gives a brief review of some of the basic concepts from classical analysis, abstract algebra, and probability theory that are necessary for the remainder of the book. The reader who is not quite familiar with some of the material presented in this first chapter is referred to a list of references in the bibliography where most of these concepts are defined and motivated. However, efforts were made to keep the mathematical requirements at a level where a graduate student with a solid background in the above-mentioned areas will be able to work through most of the book. The second chapter introduces same basic constructions of fractal sets. The first is based upon the approach by Hutchinson [2] and Barnsley and Demko [3] using what is now called an iterated function system. This method is then generalized and compared to Dekking’s [4] construction of so-called recurrent sets associated with certain semigroup endomorphisms and Bandt’s approach [5, 6] via topological Markov chains. Finally, a graph-directed fractal construction due to Mauldin and Williams [7] is presented. The emphasis is on iterated function systems and their generalizations; however. In this chapter the foundations for the rigorous treatment of univariate and multivariate fractal functions are laid. Next, the concept of dimension of a set is introduced. This is done by first reviewing the different notions of dimension that are used to characterize and describe sets. The

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Preface to first edition

last two sections in this chapter are devoted to the presentation of dimension results for self-affine fractal sets. A short chapter dealing with the fascinating theory of dynamical systems follows. The emphasis is on the geometric aspects of the theory and it is shown how they can be used to describe attractors of iterated function systems. In the second part of this book, univariate and multivariate fractal functions are discussed. Chapter 5 introduces fractal functions as the fixed points of a ReadBajraktarevi´c operator. This approach differs from that undertaken by Barnsley [8] who introduced fractal functions for interpolation and approximation purposes. It is also shown how Dekking’s approach to fractals can be used to define fractal functions and the iterative interpolation process of Dubuc and his co-workers is presented. The remainder of the chapter deals with different classes of fractal functions and discusses several of their properties. Chapter 6 is devoted to dimension calculations. Formulae for the box dimension of the graphs of most of the fractal functions introduced in the previous chapter are presented here. The second part of the chapter deals with an interesting relationship between certain classes of smoothness spaces and the box dimension of the graphs of affine fractal functions. In Chapter 7 the basic concepts and notions from wavelet theory are introduced, and it is demonstrated how a certain class of fractal functions generated by iterated function systems can be used to generate a multiresolution analysis of L2 (R). This class of fractal functions then provides a new construction of continuous, compactly supported, and orthogonal scaling functions and wavelets. The next chapter introduces multivariate fractal functions. The graphs of these functions are called fractal surfaces. Properties of fractal surfaces are then discussed and formulae for the box dimension derived. In order to construct multiresolution analyses based on the fractal surfaces defined in Chapter 8, the theory of Coxeter groups needs to be employed. This is done in Chapter 9, after some rudimentary concepts from this theory are introduced. Because of the limited scope of this monograph, certain topics could not be covered. These include, among others, a more in-depth presentation of the geometric theory of dynamical systems and the role fractals play in this theory. Furthermore, some of the work of T. Lindstrøm on nonstandard analysis, iterated function systems, fractals, and especially Brownian motion on fractals, turned out to be beyond the limits of the present book. The interesting work of J. Harrison dealing with geometric integration theory and fractals could also not be described. However, references pertaining to these as well as other topics are listed in the bibliography. The bibliography also contains research papers and books not explicitly used or mentioned in this monograph. They were included to give the reader a more well-rounded perspective of the subject. This book grew out of the work of many mathematicians from several areas of mathematics, and the author has greatly benefited from numerous conversations and discussions with my colleagues. Special thanks go to Doug Hardin and Jeff Geronimo, who have influenced and shaped some of my thoughts and ideas. In particular, I am grateful to Doug Hardin for allowing me to use his Mathematica packages to make

Preface to first edition

xiii

some of the figures in this monograph. I also wish to thank Patrick Van Fleet for introducing me to the theory of Dirichlet splines and special functions. Working with Academic Press was a pleasure. I would like to especially express my gratitude to Christina Wipf, who gave me the idea of writing this monograph, and to Peter Renz, who guided me through the final stages. Last but not least I wish to thank my wife Maritza and my family for their continuous support and encouragement during the preparation of this monograph. Peter R. Massopust Houston, Texas, USA December 1993

Preface to second edition Since the publication of the original monograph in 1995, the fields of fractals and wavelets have experienced phenomenal growth. New concepts and ideas have been introduced and are now at the heart of each theory. Given these circumstances, it was natural to change the perspective of the original monograph and to incorporate some of the new ideas and to remove those that are no longer at the core of either theory. Some of the new concepts introduced in the second edition are fractal transformations between fractal sets, local iterated function systems, local fractal functions and surfaces, fractels (ie, fractal elements), and self-referential functions. The overall structure of the original edition remains intact. Some mathematical preliminaries were moved from the main text to the introductory chapter, Chapter 1. Two new sections, on local iterated function systems and fractal transformations, have been added to Chapter 2. The new concept of a local fractal function is introduced in Chapter 5, and several recent results have been included. A new chapter, Chapter 6, on fractels and self-referential functions has been added. Both concepts have been quite recently introduced into the theory, and they deserve to be presented as they clarify, generalize, and extend some of the fundamental properties of fractal sets and fractal functions. The introduction to wavelets, Chapter 8, has been updated and some results that were not fully developed when the original monograph was published have been included. Finally, Chapter 10 introduces the concept of wavelet set and shows the connections between fractal surfaces, affine Weyl groups, and wavelet sets. The purpose of this second edition is to correct and extend some of the results in the first edition and to reduce the number of misprints therein. I would like to thank all my colleagues who pointed out these misprints and errors. Peter R. Massopust Munich, Germany February 2016

List of symbols (, F , μ) 2S ∗  S M h dim dimP dimB dimb dimM dimH B˙ sp,q  ˙ s F˙ p,q C E N Nn R Z Ps S(Rn ) D(Rn ) B(X) Bn Cs Cw F (X, Y) F k [0, 1] Hh KK (X) L(X, Y) Mh MK (X) Pd RF(X, Y) S  (Rn ) d h Twv Vk

measure space power set convolution semiconvolution product set complement forward difference operator Lyapunov dimension packing dimension Billingsley dimension box dimension Minkowski dimension Hausdorff dimension homogeneous Besov space disjoint union homogeneous Triebel-Lizorkin space complex numbers Euclidean space natural numbers {1, . . . , n} real numbers integers s-dimensional packing measure Schwartz space space of infinitely differentiable functions on Rn of compact support σ -algebra of all μ-measurable subsets of X ⊂ Rn Borel σ -algebra of Rn Zygmund space address structure class of fractal functions X → Y class of real-valued Ck fractal functions on [0, 1] h-Hausdorff measure linear space of functions X → K with compact support space of continuous linear operators h-net measure K-vector space of measures on X with values in K class or real-valued polynomials of order d class of recurrent fractal functions X → Y space of tempered distributions Hutchinson metric Hausdorff metric fractal transformation approximation space generated by fractal functions

xviii

Wk w R ∂S   σ (T)  n τγ τw XA cl S dom id im f int S osc(f ; I) osc(f ; x0 ) supp  W  lim − → lim ← − B(X) Bc (R) BC(X) C(X, Y) Ck C0 (R) CK (X) Cb (X, Y) dE f f,k fK ∞ (X , R) LN i

L∞ (X, Y) Lp (, F , K, μ) Mmn (R) T∗ V[φ 1 , . . . φ A ] W s,q Wφ †

· H(X) diag(ai )

List of symbols

wavelet space generated by fractal functions tops code space completed real line set boundary Read-Bajraktarevi´c operator injective mapping spectrum of an operator T code space finite code space translation operator tops function attractor of recurrent iterated function system set closure domain of a function identity function image of the function f set interior oscillation of f on I oscillation of f at x0 support of a function affine Weyl group surjective mapping direct or inductive limit inverse of projective limit bounded linear operators on X Banach space of R-valued functions that are bounded on compact subsets of R set of continuous functions X → R that are bounded on bounded subsets of X collection of all continuous functions X → Y class of all functions having continuous and bounded derivatives up to order k continuous R valued functions vanishing at infinity K-algebra of continuous functions X → K space of bounded continuous functions X → Y Euclidean metric on Rn fixed point of Read-Bajraktarevi´c operator hidden-variable fractal function Kiesswettter’s fractal function N

X L∞ (Xi , R)

i=1

complete metric space of bounded functions X → Y Lebesgue space on a measure space R-vector space of m × n matrices adjoint operator finitely generated shift-invariant space Sobolev space wavelet transform Hermitian conjugate floor function hyperspace of all nonempty compact subsets of X diagonal matrix with entries ai

List of symbols

Sim0 (Rn ) Sim(Rn ) #f i Bsp,q Ck,α s Fp,q

xix

set of contractive similitudes on Rn set of similitudes on Rn f ◦ u−1 i inhomogeneous Besov space Hölder space inhomogeneous Triebel-Lizorkin space

Mathematical preliminaries

1

Abstract This chapter provides most of the mathematical preliminaries necessary to understand the results in the following chapters. It is a mere collection of definitions and theorems given without proof (the only exceptions are the Banach fixed-point theorem, and the existence theorems for free semigroups and free groups). The bibliography contains a list of references such as [9, 22, 23, 28, 36, 43, 50, 52, 81, 100, 116, 174–179] in which the basis for all these results is given and the results are proved. In a sense, this chapter compiles the notation and terminology and serves as a reference guide for the remainder of the book. The relevant material is discussed in four sections: analysis and topology, probability theory, algebra, and function spaces. The first section covers basic topics such as vector spaces, normed and metric spaces, point-set topology, measures, and the different notions of convergence encountered in analysis. In the second section, probability measures, distribution functions, random variables, and their interconnections are considered. Then the (measure-theoretic) Lebesgue spaces are defined, the Riesz representation theorem is stated, and a brief overview of Markov processes and Markov chains is given. The third section is about algebra, and deals with diagrams, semigroups, groups, and semigroup and group endomorphisms and introduces free semigroups and free groups. A brief review of category theory and direct and inverse limits is also provided. The concepts of affine Weyl group and foldable figures are introduced and briefly discussed. Four types of function spaces (ie, Lebesgue, Hölder, Sobolev, and Triebel-Lizorkin) and their properties are presented in the last section.

1 Analysis and topology Throughout this monograph N := {1, 2, 3, . . .} denotes the set of natural numbers, N0 := N ∪ {0}, Z the ring of integers, and R the field of real numbers. Let K be a subfield of C, the field of complex numbers, and suppose that the mapping α: C → C, z = x + iy → z = x − iy maps K into itself. Such an α is called an involuntary automorphism of C. For an integer n ∈ N, we denote the set {1, . . . , n} by Nn and the set {0, 1, . . . , n} by N0n . An open connected subset of Kn will be called a domain. The interior of a set S (ie, the union of all open sets O contained in S) is written as int S. The closure of S (ie, the intersection of all closed sets C containing S) is denoted by cl S, and the boundary of S is denoted by ∂S := cl S \ int S. The complement of a set S in a set T is written as T \ S or, if T is a fixed universal set, as S. The power set of S (ie, the class of all subsets Fractal Functions, Fractal Surfaces, and Wavelets. http://dx.doi.org/10.1016/B978-0-12-804408-7.00001-1 Copyright © 2016 Elsevier Inc. All rights reserved.

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Fractal Functions, Fractal Surfaces, and Wavelets

of S) is denoted by 2S . The domain of a function f : X → Y is written as dom f , the image of f is written as im f , and its support is written as supp f . The symbol ∧ when used in the definition of a set represents the logical “and.” Notation. In the following N will always denote an integer greater than 1. Definition 1. Suppose X is a K-vector space (ie, a linear space over K). 1. A mapping ϕ: X → K is called semilinear or a semilinear form iff: (a) ∀x, x ∈ X: ϕ(x + x ) = ϕ(x) + ϕ(x ); (b) ∀x ∈ X ∀k ∈ K: ϕ(kx) = kϕ(x). If K ⊆ R, all semilinear forms are linear. 2. A mapping ϕ: X × X → K is called sesquilinear or a sesquilinear form iff: (a) ∀x, x , y ∈ X: ϕ(x + x , y) = ϕ(x, y) + ϕ(x , y); (b) ∀k ∈ K: ϕ(kx, y) = kϕ(x, y); (c) ∀x, y, y ∈ X: ϕ(x, y + y ) = ϕ(x, y) + ϕ(x, y ); (d) ∀k ∈ K: ϕ(x, ky) = kϕ(x, y). If K ⊆ R, all sesquilinear forms are bilinear. 3. A sesquilinear form ϕ is called Hermitian iff for all x, y ∈ X: ϕ(x, y) = ϕ(y, x). If K ⊆ R, a Hermitian form is called symmetric. 4. A sesquilinear form ϕ is called positive definite or positive semidefinite iff for all x ∈ X \ {0}, ϕ(x, x) > 0, or for all x ∈ X, ϕ(x, x) ≥ 0, respectively.

Definition 2 (Inner Product Space). An inner product on a K-vector space X is a positive definite Hermitian sesquilinear form ϕ: X × X → K. The pair (X, ϕ) is called an inner product space. If X is a linear space over R or C, (X, ϕ) is called a Euclidean space or a unitary space, respectively. Notation. Instead of writing ϕ(x, y), x, y ∈ X, we sometimes use the shorter notation x, y . An inner product ϕ on X can be used to define the norm of an element x ∈ X, and the distance between two elements x, y ∈ X. More precisely, the norm of x ∈ X is defined as  (1.1)

x ϕ := ϕ(x, x), and the distance between x, y ∈ X is defined by dϕ (x, y) := x − y ϕ =



ϕ(x − y, x − y).

(1.2)

Proposition 1 (Cauchy-Schwarz Inequality). Let (X, ϕ) be an inner product space over K. Then for all x, y ∈ X, |ϕ(x, y)| ≤



 ϕ(x, x) ϕ(y, y) = x ϕ y |ϕ ,

(1.3)

with equality iff there exists a k ∈ K such that x = ky. Definition 3. Suppose that X is a K-vector space. A functional · : X → R is called a norm on X iff the following conditions hold: 1. ∀x ∈ X: x ≥ 0, 0 = 0. 2. ∀x ∈ X ∀k ∈ K: kx = |k| x .

Mathematical preliminaries

5

3. ∀x, y ∈ X: x + y ≤ x + y . 4. x = 0 ⇒ x = 0.

If only properties 1–3 are satisfied, · is called a seminorm on X. The pair (X, · ) is called a normed linear space or a normed vector space over K. Proposition 2. Suppose that (X, ϕ) is an inner product space over K. Then · ϕ : X → K as defined in Eq. (1.1) is a norm on X. Definition 4. Suppose M is a set. A mapping d: M × M → R is called a metric on M iff the following conditions are satisfied: 1. 2. 3. 4.

∀x, y ∈ M: d(x, y) ≥ 0, d(x, x) = 0. ∀x, y ∈ M: d(x, y) = d(y, x). ∀x, y, z ∈ M: d(x, z) ≤ d(x, y) + d(y, z). d(x, y) = 0 ⇒ x = y.

If only properties 1–3 hold, then d is called a semimetric on M. The pair (X, d), where d is a (semi)metric on the set X, is called a (semi)metric space. Proposition 3. Suppose · is a norm or a seminorm on a K-vector space X. Then d(x, y) := x − y ,

x, y ∈ X,

(1.4)

defines a metric or a semimetric, respectively, on X. A norm · : X → R on a K-vector space X induces in a canonical way a topology on X, the so-called norm or strong topology. At this point the definition of topology on a set M is recalled. Definition 5. Let M be an arbitrary set and let M be a collection of subsets of M. Then M is called a topology on M provided the following conditions apply: 1. For all i in some index set I, Oi ∈ M ⇒ 2. O1 , . . . , On ∈ M ⇒ 3. ∅ ∈ M and M ∈ M.

n 



Oi ∈ M.

i∈I

Oi ∈ M.

i=1

The elements of M are called open sets and the pair (M, M) is called a topological space. The norm topology on a K-vector space X is then defined as follows: Let A ⊆ X be nonempty and let Br (a) := {x ∈ X : x − a < r} denote the ball of radius r > 0 centered at a ∈ X. The set A is called open iff for each a ∈ A there exists a ball Br (a), r > 0, contained entirely in A. It is easy to show that X · := {A ⊆ X : A is open} is a topology on X. The topological space (X, X · ) is also Hausdorff . Definition 6. A topology M on a set M is called Hausdorff iff two distinct points x, y ∈ M can be separated by two disjoint sets U  x and V  y in M; that is, iff for all x, y ∈ M, x = y, there exists U, V ∈ M such that x ∈ U, y ∈ V, and U ∩ V = ∅. Suppose that X is a K-vector space and · i : X → R, i = 1, 2, are arbitrary norms on X. · 1 and · 2 are called equivalent, written · 1 ≈ · 2 , iff there exist positive real numbers c1 and c2 such that for all x ∈ X,

x 1 ≤ c1 x 2

and

x 2 ≤ c2 x 1 .

(1.5)

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Fractal Functions, Fractal Surfaces, and Wavelets

Proposition 4 1. All norms on Rn are equivalent. 2. All norms on Rn generate the same topology.

Definition 7. Suppose (X, X) is a topological space. B ⊆ X is called a basis of X iff every open set is a union of elements of B:  O ∈ X ⇒ O = B, B∈B

with B ⊆ B. The following result gives the necessary conditions for a subset B of X to be a base. Proposition 5. Let (X, X) be a topological space and let B ⊆ X. If B is a base of X, then 1. X =



B;

B∈B 2. ∀B1 , B2 ∈ B ∀x ∈ B1 ∩ B2 ∃Bx ∈ B: x ∈ Bx ⊆ B1 ∩ B2 .

The concept of topology allows one to precisely define notions such as distance, convergence, and continuity. Definition 8. Let D be a nonempty set. A relation  on D is called directed iff it has the following properties: 1. Reflexivity: ∀α ∈ D: α  α. 2. Transitivity: ∀α, β, γ ∈ D: α  β, β  γ ⇒ α  γ . 3. ∀α, β ∈ D∃γ ∈ D: α  γ , β  γ .

A directed set is a set with a directed ordering. Remark 1. Some authors define a directed set as a nonempty partially ordered set satisfying condition 3 above. Let (X, X) be a topological space. A net in X consists of a directed set D and a mapping δ: D → X. It is common to write the image of α ∈ D under δ in X as xδ instead of δ(α). Nets are then denoted by {xα }α∈D , or simply by {xα } if it is understood which directed set is meant. Clearly, every sequence in X is a net in X: take D = N and  := ≤. Recall that a set N ⊆ X is called a neighborhood of x ∈ X iff N is a superset of an open set containing x. Definition 9. Let (X, X) be a topological space and let {xα } be a net in X. A point x ∈ X is called a limit point of {xα } iff for any neighborhood N of x there exists an α0 ∈ D such that all xα with α0  α are points in N. X

Notation. If x is a limit point of a net {xα }, then one writes for short: xα − → x or simply xα → x when the topology is understood. One obtains the classical characterization of convergence (in the strong topology) by choosing X to be a normed linear space, X := X · , and D := N: xn → x ⇐⇒ ∀ε > 0∃n0 ∀n ≥ n0 : xn − x0 < ε. Let (X, X) and (X  , X ) be two topological spaces and let F: X → X  be a mapping of sets. Then F is continuous iff xα → x in (X, X) implies F(xα ) → F(x) in (X  , X )

Mathematical preliminaries

7

for every net {xα } in X. A mapping F: X → X  is called a homeomorphism iff F is bijective and F and its inverse F −1 are continuous. Definition 10. Let (X, X) be a topological space and let R := R ∪ {±∞} be the completed real line. A function f : X → R is called upper semicontinuous or lower semicontinuous at x0 ∈ X iff for all α ∈ R with α > f (x0 ) or α < f (x0 ) there exists a neighborhood N of x0 in X such that for all x ∈ N one has α > f (x) or α < f (x), respectively. A function f is called upper semicontinuous or lower semicontinuous on X iff it is upper semicontinuous or lower semicontinuous at each x0 ∈ X. It is clear that if f is upper semicontinuous then −f is lower semicontinuous. Also, if a function f is both upper and lower semicontinuous at x0 ∈ X, then f is continuous at x0 . The next proposition characterizes lower semicontinuous functions. Proposition 6. A function f from a topological space (X, X) into the completed real line R is lower semicontinuous iff for all α ∈ R the set f −1 (α, ∞] is open in X, or equivalently the set f −1 [−∞, α] is closed in X. Definition 11. Let (X, X) be a topological space and let {xα } be a net in X. A point x ∈ X is called an accumulation point of {xα } iff for every neighborhood N of x and any α0 there exists an α  α0 such that xα ∈ N. Nets provide a simple characterization of compactness in a topological space (X, X). Theorem 1. Let (X, X) be a topological space. X is compact iff every net in X has an accumulation point. The next theorem is of great importance in other areas of mathematics as well. Theorem 2 (Tychonov’s Theorem). The (Cartesian) product of any collection of compact topological spaces is compact. Nets are also a useful tool for characterizing the closure, cl A, of a set A ⊆ X in a topological space (X, X). Theorem 3. The closure, cl A, of a set A consists precisely of the limit points of all the nets in A. A set A ⊆ X is closed iff it contains every limit point of every net in A. The characterization of completeness in terms of so-called Cauchy nets requires the concept of a uniform space. Such spaces are generalizations of metric spaces; they allow the comparison of neighborhoods located at different points in X. Every metric space is in a canonical way a uniform space and every uniform space is in a canonical way a topological space. To this end, let X be a set and let U be a subset of X × X. Define U −1 := {(x, x ) ∈ X × X : (x , x) ∈ U}, and let D := {(x, x) ∈ X × X : x ∈ X}. If U, U  ∈ X × X, define UU  := {(x, x ) ∈ X × X : ∃x ∈ X so that (x, x ) ∈ U ∧ (x , x ) ∈ U  }. Definition 12. Let X be a set and let U be a collection of subsets of X × X. The pair (X, U ) is called a uniform space (with uniformity U ) iff the following conditions hold:

 1. U∈U = D. 2. ∀U ∈ U : U −1 ∈ U. 3. ∀U ∈ U ∃U  ∈ U : U  U  ⊆ U.

8

Fractal Functions, Fractal Surfaces, and Wavelets

4. ∀U, U  ∈ U : U ∩ U  ∈ U. 5. ∀U ∈ U ∀U  ⊇ U: U  ∈ U.

The elements of U are called uniformities or entourages. From the above definition, one sees that every metric space and thus every normed linear space is a uniform space. (The uniformities are given by the neighborhoods of D in X × X.) Let (X, U ) be a uniform space. The definition of a net in a uniform space is just as above, with topological replaced by uniform. A Cauchy net {xα }α∈D in (X, U ) is a net with the following property: for all U ∈ U there exists an α ∈ D such that (xα  , xα  ) ∈ U whenever α  α  and α  α  . Theorem 4. A uniform space (X, U ) is complete iff every Cauchy net converges to a point of X. The next definition introduces two important classes of complete linear spaces. Definition 13. A complete inner product space (X,  , ) is called a Hilbert space, and a complete normed linear space (X, · ) is called a Banach space. A class of K-vector spaces that combine the linear algebraic structure and a topology are called topological vector spaces. Definition 14. A topological vector space X is a K-vector space endowed with a topology X such that scalar multiplication K × X → X, (λ, x) → λx, and vector addition X × X → X, (x, y) → x + y, are continuous with respect to X. It is not hard to verify that all normed spaces are topological vector spaces. Hence all Banach spaces and all Hilbert spaces are naturally topological vector spaces. Next, mappings between K-vector spaces are considered. For this purpose, let X and Y be K-vector spaces. A mapping F: D ⊆ X → Y, D = ∅, is called a function on X with domain D. If X = Y, then F is usually called an operator on X with domain D, and if Y = K, F is called a functional. The adjoint or dual space V ∗ of a K-vector space V is the space of all continuous linear functionals V → K. For f ∈ V and f ∗ ∈ V ∗ , f ∗ , f is called the canonical or adjoint pairing and denotes the composition f ∗ (f ). This pairing is sometimes also called an inner product. It follows immediately that V ∗ is a linear space and that the inner product ·, · is bilinear. Definition 15. For conjugate or dual spaces V and V ∗ and an operator F: V → V, the adjoint operator, F ∗ , is that operator for which f ∗ , Ff = F ∗ f ∗ , f ,

∀f ∈ V ∀f ∗ ∈ V ∗ .

Proposition 7. If F: V → V is a linear operator, then its adjoint F ∗ exists and is linear. Let (X, X) be a topological space and Y be a normed linear space over K. Denote by Cb (X, Y) the K-vector space of bounded continuous functions X → Y. Theorem 5. If Y is a Banach space, then Cb (X, Y) is also a Banach space when endowed with the norm

f Cb := sup f (x) Y . x∈X

Mathematical preliminaries

9

Remark 2. Theorem 5 also holds for Y being a complete metric space (Y, dY ). The K-vector space Cb (X, Y) is then a complete metric space with metric dCb (f , g) := sup dY (f (x), g(x)). x∈X

If (X, X) is a compact topological Hausdorff space and Y is either a Banach space or a complete metric space, then the K-vector space C(X, Y) of all continuous functions X → Y is a Banach space or a complete metric space, respectively. Now assume that X and Y are normed spaces. A function F: D ⊆ X → Y is called Lipschitz continuous iff there exists an L > 0, called the Lipschitz constant, such that

F(x) − F(x ) ≤ L x − x ,

∀x, x ∈ D.

If L < 1, F is called contractive or a contraction. Let T: D ⊆ X → X be an operator. A point x ∈ X is called a fixed point of T iff Tx = x. If TD ⊆ D, one defines the powers or iterates of T by T 0 := idD , T n := TT n−1 ,

∀n ∈ N.

Here idD denotes the identity function on D. The next theorem will be used extensively in the following chapters. Its proof uses a technique that will be encountered numerous times (see also Ref. [9]). Theorem 6 (Banach Fixed-Point Theorem). Let X be a normed linear space over K and let T be a Lipschitz continuous operator with complete domain D ⊆ X. Assume that TD ⊆ D. Furthermore, let Ln denote the Lipschitz constant of T n , n ∈ N0 , and assume that ∞ 

Ln < ∞.

n=0

Then T has exactly one fixed point x∗ ∈ D. Moreover, if x0 ∈ D is chosen arbitrarily and if xn+1 := Txn , n ∈ N0 , then the sequence {xn } converges to x∗ , independently of x0 . Proof. Existence. Let , m, n ∈ N0 be such that m < n ≤ . Let x0 ∈ D be arbitrary. Then m−1  m−1      k+1    xk+1 − xk  ≤

xm − xn ≤  T x0 − T k x0    k=n k=n  ∞ m− −1   k k

T x −1 − T x ≤ Lk x +1 − x . ≤ k=n−

k=n−

10

Fractal Functions, Fractal Surfaces, and Wavelets

∞

Now ∀ε > 0 ∃N ∈ N such that ( yields

xm − xn ≤

∞ 

k=n− Lk ) x1

− x0 ≤ ε. Setting = 0 therefore

Lk x1 − x0 < ε.

k=n

The Cauchy sequence {xn } has, by the completeness of D, a limit point x∗ ∈ D. Letting n → ∞ and using the fact that T is Lipschitz continuous, we obtain Tx = x. Uniqueness. Suppose x and x are fixed points of T. Then for all n ∈ N0 ,

x − x = T n x − T n x ≤ Ln x − x . Hence L ≥ 1 for all n ∈ N0 . This contradiction now yields uniqueness. The fact that xn → x∗ , independent of x0 , follows from the foregoing arguments. Now some of the properties of functions between normed spaces are presented. Let F: D ⊆ X → Y be a continuous linear function on D = ∅ (note that D is necessarily a linear subspace of X). It is well known that each such function has a smallest Lipschitz constant L ≥ 0; that is, there exists a least nonnegative number L with the property

Fx ≤ L x ,

∀x ∈ D, x = 0.

This smallest Lipschitz constant is equal to the supremum of the set {M ∈ R : M = Fx / x ∧ x ∈ D ∧ x = 0} in the case of dim D > 0; otherwise it is 0. (Here dim denotes the dimension of the linear subspace D.) It is easy to show that under the usual definition of addition and scalar multiplication of functions, and equipped with the norm of the smallest Lipschitz constant  0 if dim D = 0;

F := sup{ Fx / x : 0 = x ∈ D} if dim D > 0, the set of all continuous linear functions F: D ⊆ X → Y forms a normed linear space over K. This normed linear space will be denoted by L(D, Y). In the case where Y is a Banach space, L(D, Y) is also a Banach space. Uniform convergence in L(D, Y) implies norm convergence, which then implies pointwise convergence. Definition 16. Suppose X and Y are normed linear spaces over K. A continuous function F: D ⊆ X → Y is called equicontinuous iff for every bounded sequence {xn } ⊆ X the sequence {Fxn } contains a convergent subsequence. The following fixed-point theorem is due to Juliusz Schauder.

Mathematical preliminaries

11

Theorem 7 (Schauder Fixed-Point Theorem). Let X be a Banach space and let T: D → D be an equicontinuous operator with bounded, closed, and convex domain D ⊆ X. Then T has a fixed point in D. Recall that a subset C of a K-vector space X is called convex iff x, x ∈ C implies λx + (1 − λ)x ∈ C for all λ ∈ [0, 1]. The next result is of great importance in real analysis as well as functional analysis. Only its normed space version is stated. Theorem 8 (Hahn-Banach Extension Theorem). Suppose that X is a normed linear space over K and U ⊆ X is a linear subspace. Then every continuous linear functional ϕ: U → R can be linearly and continuously extended to all of X in such a way that it preserves the norm; that is, there exists a continuous linear functional ψ: X → R with the properties ∀u ∈ U: ψu = ϕu

(1.6)

ϕ = ψ .

(1.7)

and

Remark 3. The extension of a continuous linear functional—as guaranteed by the Hahn-Banach extension theorem—may not be unique. Now suppose X is a normed space over K. The set of all continuous linear functionals ϕ: X → K is called the dual (space) X ∗ of X. Clearly, if addition and scalar multiplication on X ∗ is defined in the usual way and X ∗ is equipped with the norm of the smallest Lipschitz constant, then X ∗ = L(X, K), and hence X ∗ is a Banach space. The elements of X ∗ are usually denoted by x∗ . Proposition 8. dim X = dim X ∗ . Let ε > 0 be given, and let x1∗ , . . . , xn∗ be a finite set of elements of X ∗ . One can define a topology X on X as follows. For each x0 ∈ X let N(x0 ; ε, x1∗ , . . . , xn∗ ) := {x ∈ X : xi∗ (x) − xi∗ (x0 ) < ε, i ∈ Nn } be a basis of neighborhoods. Clearly, this defines a locally convex topology on X, the weak topology. Let {xn }n∈N be a sequence in X. This sequence is called weakly convergent to an x ∈ X iff ∀x∗ ∈ X ∗ : lim x∗ (xn ) = x∗ (x). n→∞

(1.8)

The element x is called the weak limit of the sequence {xn }. Instead of Eq. (1.8), the w shorter notation xn → x or xn x is used.

12

Fractal Functions, Fractal Surfaces, and Wavelets

A subset U of a normed linear space X over K is called weakly compact if every sequence {xn } ⊆ U has a subsequence which converges weakly to an element of U. Definition 17. A subset U of a normed linear space X is called separable iff it satisfies the following equivalent conditions: 1. There exists a sequence {xn } ⊆ X such that each u ∈ U is the limit of a subsequence of {xn }. 2. There exists a sequence {xn } ⊆ X that is dense in X.

Definition 18. Let X be a normed linear space X over K. A sequence {en } ⊆ X is called a Schauder basis of X iff every element x ∈ X has a unique representation of the form  αn en (1.9) x= n∈N

for {αn } ⊆ K. Remark 4. The sum in Eq. (1.5) is understood as a limit in the norm topology. The following result will be used later. Proposition 9. Finite-dimensional normed linear spaces and infinite-dimensional normed linear spaces with Schauder basis are separable. Let X be a normed linear space over K and let X ∗ be its dual (considered as a Banach space). One can define the dual of X ∗ , X ∗∗ , to be the collection of all continuous linear functionals : X ∗ → K, ( ϕ)(x) := ϕ(x),

∀x ∈ X.

(1.10)

It is straightforward to show that

= x . One can give the dual of X a topology that is even weaker than the weak topology on X. This topology is called the weak* topology on X ∗ . The basis of neighborhoods of a point x0∗ ∈ X ∗ is defined by N(x0∗ ; ε, x1 , . . . , xn ) := {x∗ ∈ X ∗ : x∗ (xi ) − x0∗ (xi ) < ε, i ∈ Nn }, where ε > 0 and x1 , . . . , xn is a finite set of elements of X. This defines a locally convex topology on X ∗ . In general, X = X ∗∗ . Banach spaces for which X = X ∗∗ are called reflexive. A sequence {xn∗ } ⊆ X ∗ is called weak* convergent to x∗ ∈ X ∗ iff lim x∗ (x) n→∞ n

= x∗ (x), ∀x ∈ X.

(1.11) w∗

One calls x∗ the weak* limit of the sequence {xn∗ } and writes: xn∗ → x∗ . A subset U ∗ ⊆ X ∗ is called weak* compact iff every sequence {xn∗ } ⊆ U ∗ has a weak* convergent subsequence whose limit is in U ∗ . A topological space (X, X) is called metrizable if its topology can be defined by a metric on X, and locally compact if every point of X possesses a compact neighborhood.

Mathematical preliminaries

13

Let X be a compact and metrizable topological space. The K-algebra of all continuous functions f : X → K together with the norm

f := sup{|f (x)| : x ∈ X} becomes a Banach space. It will be denoted by CK (X). The above norm is usually referred to as the sup(remum) norm or Chebyshev norm. A measure μ on X is an element of the dual of CK (X); that is, a continuous linear functional f → μ(f ) on CK (X) satisfying |μ(f )| ≤ a f ,

(1.12)

for some a ∈ R+ and all f ∈ CK (X). Now suppose X is a separable, metrizable, and locally compact topological space. For every compact subset K ⊆ X, denote by KK (X|K) the linear subspace of CK (X) generated by those functions whose support is contained in K. Let KK (X) be the linear space of all functions f : X → K with compact support; that is, KK (X) :=



KK (X|K).

K⊆X

A measure on X is a linear functional μ on KK (X) with the property that for each compact subset K ⊆ X, there exists a nonnegative constant aK , depending on K, such that ∀f ∈ KK (X): |μ(f )| ≤ aK f .

(1.13)

The Lebesgue measure λ on R is, for instance, obtained as follows: Let f ∈ KR (X) λ and consider the linear functional f −→ R f (t) dt on KK (X). (Note that since f is compactly supported, the integral is well defined.) Letting K := [a, b], a, b ∈ R, a < b, yields



f (t) dt ≤ (b − a) f

R

for all f ∈ KK (X|R). Clearly, if μ and ν are measures on X, and if a ∈ K, then μ + ν and aμ are also measures on X. Hence the collection of all measures on X is a linear subspace of KKK (X) . This subspace is denoted by MK (X). Since MK (X) is a linear subspace of KKK (X) , it can be endowed with a weak topology. Therefore, a sequence {μn } of measures in MK (X) is called weak* convergent to w∗

a measure μ ∈ MK (X) if μn (f ) −→ μ(f ) in K for all f ∈ KK (X). Let J∗ (X) be the set of all lower semicontinuous functions on X with values in the completed real line R that are minorized by functions from KR (X); that is, if f ∈ J∗ (X), then there exists a g ∈ KR (X) such that g(x) ≤ f (x) for all x ∈ X. The set

14

Fractal Functions, Fractal Surfaces, and Wavelets

J∗ (X) is nonempty, for any lower semicontinuous nonnegative function is an element of J∗ (X). Similarly, one defines the set J ∗ (X) of all upper semicontinuous functions on X with values in R that are majorized by functions from KR (X). Note that J ∗ (X) = −J∗ (X). For each function f ∈ J∗ (X) define

μ∗ (f ) :=

sup

g≤f ,g∈KR (X)

μ(g).

(1.14)

The number μ∗ (f ) is an element of (−∞, ∞]. It follows that if f ∈ KR (X), then = μ(f ), and that μ∗ is a subadditive linear functional on J∗ (X). (The scalars are assumed to be positive reals here.) Analogously, one defines

μ∗ (f )

μ∗ (f ) :=

inf

g≥f ,g∈KR (X)

μ(g).

(1.15)

Then μ∗ (f ) ≤ μ∗ (f ) for all functions f : X → R. Definition 19. A function f : X → R is said to be μ-integrable on X iff μ∗ (f ) and μ∗ (f ) are finite and equal; their common value is called the integral of f with respect to μ and is denoted by X fdμ. Remark 5. The integral of f with respect to μ is sometimes also denoted by μ(f ), μ, f , or X f (x) dμ(x). We close this section with a few remarks about Hilbert spaces. If (X,  , ) is a Hilbert space over K and if x ∈ X, then ϕ := ·, x

(1.16)

defines a linear functional on X (ie, an element of X ∗ with ϕ = x ). Conversely, every ϕ ∈ X ∗ is of the form Eq. (1.16). The element x is uniquely determined by the conditions u, x = 0, u ∈ ker ϕ, and ϕ(x) = ϕ 2 . Here ker ϕ := {x ∈ X : ϕ(x) = 0} denotes the kernel of ϕ. Let T be a linear and continuous (ie, bounded) operator on (X,  , ). Denote the set of all such operators by B(X). Proposition 10. For each operator T ∈ B(X) there exists a unique bounded operator T ∗ , called the adjoint of T, such that x , T ∗ x = Tx, x for all x, x ∈ X. If X := Rn , then a bounded linear operator T on Rn can be represented by its associated n × n matrix A = A(T). An n × n diagonal matrix will be denoted by diag(ai ), where the ai ∈ R are the diagonal entries. Definition 20 1. A matrix A is called positive iff all its elements are positive. If A is positive, one writes A > 0. 2. A vector v ∈ Rn is called positive iff all its components are positive and is called nonnegative if all its components are nonnegative. In the former case, one writes v > 0, and in the latter case, one writes v ≥ 0.

Mathematical preliminaries

15

The next two theorems (see Ref. [10]) will be used in the following chapters. Theorem 9 (Perron-Frobenius Theorem). Suppose that A is a positive matrix. Then there exists a unique eigenvalue λ = λ(A) of A which has the greatest absolute value. This eigenvalue is positive and simple, and its associated eigenvector may be taken to be positive. Theorem 10. Let A be a positive matrix and let λ(A) be defined as above. Denote by S(λ) the set of all nonnegative λ ∈ R for which there exist nonnegative vectors x ∈ Rn such that Ax ≥ λx, and by T(λ) the set of positive λ ∈ R for which there exist vectors y ∈ Rn such that Ay ≤ λy. Then  max{λ: λ ∈ S(λ)}; λ(A) = min{λ: λ ∈ T(λ)}. Now let T ∈ B(X). The spectrum σ (T) of T is defined as σ (T) := {λ ∈ K : λI − T is singular}.

(1.17)

Proposition 11. Let T ∈ B(X). Then σ (T) is a nonempty compact subset of K and σ (T) ⊆ {λ ∈ K : |λ| ≤ T }. Definition 21. For T ∈ B(X), the spectral radius, r(T), of T is defined by r(T) := sup{|λ| : λ ∈ σ (T)}.

(1.18)

The following theorem gives Gelfand’s formula for the spectral radius. Theorem 11 (Spectral Radius Theorem). If T ∈ B(X), then r(T) = lim T ν 1/ν . ν→∞

2 Measures and probability theory In Section 1 measures were introduced as elements of the dual of a function space. However, there is another approach to measures—namely, via numerically valued set functions. This approach is usually undertaken when measures are first introduced in probability theory. In this section probability measures are defined as numerically valued set functions and some of their basic properties are presented. Once this has been done, relations between probability measures, random variables, and distribution functions are considered. This then leads to the Lebesgue spaces Lp (, R, μ) and the Riesz representation theorem. Markov processes are then defined via nonexpansive positive operators on L1 (, R, μ). Finally, the concept of a sequence space is used to introduce countable Markov chains.

16

Fractal Functions, Fractal Surfaces, and Wavelets

Definition 22. Let  be a nonempty set. A collection F of subsets of  is called a σ -algebra or a Borel field iff: 1. ∅ ∈ F and  ∈ F; 2. B ∈ F ⇒ B ∈ F ; and Bj ∈ F . 3. Bj ∈ F , j ∈ N ⇒ j∈N

Definition 23. Let  be a nonempty set and F be a Borel field of subsets of . A function μ: F → R is called a measure on F provided: 1. μ∅ = 0.  

2. ∀Bi , Bj ∈ F , Bi ∩ Bj = ∅ ⇒ μ j∈N = j∈N μBj .

A function π : F → R is called a probability measure on  iff the following additional conditions are satisfied: 1. ∀B ∈ F : π B ≥ 0. 2. π  = 1.

The triple (, F , μ) or (, F , π ) is called a measure space or a probability space, respectively. The sets in F are called μ-measurable or π -measurable, respectively. A measure space (, F , μ) is called σ -finite iff it is the countable union of μmeasurable sets of finite measure. A set A ⊆  is termed μ-measurable iff for each set B ⊆ , μB = μ(B ∪ A) + μ(B \ A).

(1.19)

The support of a measure μ is defined to be the closed set supp μ :=  \ ∪{O : O is open ∧ μO = 0}.

(1.20)

For any nonempty collection C of subsets of  there exists a smallest Borel field containing C . This minimal Borel field is said to be generated by C . A measure μ is called complete if whenever A ⊆ B, B ∈ F , and μB = 0, A is ,  μ) is said to be the completion of a μ-measurable and μA = 0. A measure space (, F measure μ iff  μ is complete, each μ-measurable set B is  μ-measurable with  μB = μB,  is given by  B = B ∪ Z, where B ∈ F and Z is a subset of a and each element  B∈F set of μ-measure zero. Every measure μ has a completion. Indeed, given the measure space (, F , μ), define  := {E ⊆  : E = B ∪ Z ∧ E ∈ F ∧ Z ⊆ Z  ∈ F ∧ μZ  = 0}. F ,  μ) is a measure space, where if B ⊆ E and E \ B is a subset Then it follows that (, F of a set of μ-measure zero,  μE = μB. An important example of a probability space is the triple (R, B1 , μ), where B1 is the (linear) Borel field generated by C := {(a, b] : −∞ < a < b < +∞}. Note that the Borel-Lebesgue measure λ is not a probability measure on R; however, the measure space (R, B1 , λ) is σ -finite. The next definition introduces certain classes of measures that admit good approximations of various types.

Mathematical preliminaries

17

Definition 24. 1. The smallest σ -algebra of Rn that contains all the open subsets of Rn is called the Borel σ -algebra B n of Rn . 2. A measure μ on  is called regular iff for each A ⊆  there exists a μ-measurable set B such that A ⊂ B and μA = μB. 3. A measure μ on Rn is called a Borel measure iff every Borel set is measurable. 4. A measure μ on Rn is Borel regular iff μ is Borel and for each A ⊆ Rn there exists a Borel set B with the property that A ⊆ B and μA = μB. 5. A measure μ on Rn is called a Radon measure iff μ is Borel regular and for every compact set K ⊆ Rn , μK < ∞.

Remark 6. The above definitions also hold with Rn replaced by any metrizable topological space X. Definition 25. A function f :  → R is said to be measurable (with respect to the measure μ) iff for every open set G ⊆ R, f −1 G ∈ F . A function F: R → R is called a distribution function iff it is increasing, right continuous, and satisfies limx→−∞ F(x) = 0 and limx→+∞ F(x) = 1. The classical theory of the Lebesgue-Stieltjes integral gives the following result. Proposition 12. Each probability measure μ on B1 determines a distribution function F through the correspondence F(x) := μ(−∞, x]

(1.21)

for all x ∈ R. Conversely, every distribution function F determines a probability measure μ on B1 through Eq. (1.21). Definition 26. Let (, F , π ) be a probability space. A random variable on (, F , π ) is any π -measurable function X :  → R; that is, X satisfies ∀B ∈ B1 : X −1 B ∈ F .

(1.22)

Any random variable on (, F , π ) induces canonically a probability measure μ on (R, B1 , μ) via the correspondence μB := π X −1 (B) = π X{ω ∈ B}

(1.23)

for all B ∈ B1 ; that is, the following diagram commutes (a precise definition of diagram and commute will be given in the next section): F ⏐ ⏐ π

R

X

−→ μ

B1

18

Fractal Functions, Fractal Surfaces, and Wavelets

or symbolically, μ = π ◦ X −1 . The distribution function F is given by F(x) = μ(−∞, x] = π{X(ω) ∈ (−∞, x]} =: π{X ≤ x}. Next we briefly restate what it means for a sequence of probability measures {μn } on B1 to weak* converge to a probability measure μ on B1 . This type of convergence is sometimes also called vague convergence. Proposition 13   w∗ f (x)dμn → f (x)dμ, (1.24) μn −→ μ ⇐⇒ ∀f ∈ K(K): R

R

where K(K) := KR (R|K). Remark 7. Proposition 13 also holds for all f ∈ C0 (R), the set of all continuous real-valued functions f with domain R and lim|x|→∞ f (x) = 0. A sequence {Xn } of random variables on (, F , π ) is said to converge in probability p to a random variable X on (, F , π ), written Xn → X, iff ∀ε > 0: lim π {|Xn − X| > ε} = 0,

(1.25)

n→∞

or equivalently lim π {|Xn − X| ≤ ε} = 1.

(1.26)

n→∞

This last statement can also, less rigorously of course, be rephrased as “Xn → X with probability 1.” A sequence of random variables {Xn } on (, F , π ) is said to converge in distribution d

to a distribution function F, written Fn → F, iff the associated sequence {Fn } of distribution functions converges weak* to F. Proposition 14. Let {Xn } be a sequence of random variables, X a random variable, p

d

and {Fn } and F the associated distribution functions. If Xn → X, then Fn → F. The next two theorems deal with the limit of partial sums of a sequence of random variables and are known as the strong law of large numbers and the weak law of large numbers, respectively. Theorem 12 (Strong Law of Large Numbers). Assume that {Xn } is a sequence of random variables. Suppose

that for all n ∈ N the expectation E(Xn ) :=  Xn (ω)dπ(ω) < ∞. Let Sn := j∈Nn Xj . Then

 

Sn − E(Sn )

≤ ε = 1. lim π

n→∞ n

Mathematical preliminaries

19

Theorem 13 (Weak Law of Large Numbers). Suppose {Xn } is a sequence of random variables with the property that for all n ∈ N the expectation E(Xn ) < ∞.

If Sn := j∈Nn Xj , then Sn − E(Sn ) →0 n for π -a.e. ω ∈ . Now the Lebesgue spaces are introduced and the Riesz representation theorem is presented. Definition 27. Suppose (, F , μ) is a measure space and μB ≥ 0 for all B ∈ F . Let 0 < p ≤ ∞. The Lebesgue space Lp (, F , K, μ) is the collection of all classes of μ-measurable functions f :  → K (one identifies functions that are equal μ-a.e.) with the property that  |f (ω)|p dμ < ∞. If the Borel field F is understood, Lp (, F , K, μ) is simply written as Lp (, K, μ). For p = ∞ the above integral is replaced by ess sup{|f (ω)| : ω ∈ } < ∞. Under the addition and scalar multiplication of equivalence classes of functions, the sets Lp (, K, μ) form linear spaces. If one defines 

f Lp :=

1/p |f (ω)| dμ p

(1.27)



for 1 ≤ p < ∞ and

f L∞ := ess sup{|f (ω)| : ω ∈ },

(1.28)

for p = ∞ then · p becomes a norm. Moreover, for this range of p-values the Lebesgue spaces Lp (, K, μ) are Banach spaces. Lebesgue spaces will be encountered again in Section 4 under a slightly different but equivalent viewpoint. Sometimes the case p = 0 is also of interest. The space L0 (, K, μ) is defined as the space of all classes of measurable, almost everywhere finite functions from  into K endowed with the topology of convergence in measure. (Convergence in measure is defined as in Eq. (1.25) with “probability” replaced by “measure.”) The next theorem is one of several versions of the Riesz representation theorem and certainly not the most general one. However, for our purposes this particular version will suffice. Theorem 14 (Riesz Representation Theorem). Let (, F , K, μ) be a σ -finite measure space. Let 1 ≤ p ≤ ∞ and suppose that q is such that 1/p + 1/q = 1, or q = ∞ if p = 1. Let ϕ ∈ (Lp (, K, μ))∗ . Then there exists a unique function g ∈ Lq (, K, μ) such that  ϕ(f ) =

f (ω)g(ω)dμ 

for all f ∈ Lp (, K, μ). Moreover, ϕ = g q .

(1.29)

20

Fractal Functions, Fractal Surfaces, and Wavelets

Definition 28. A Markov process is a quadruple (, F , μ, P), where P is a nonexpansive positive operator on L1 (, R, μ). Note that nonexpansive refers to the L1 -norm of P; that is

P = sup{|Pf | : f L1 ≤ 1} ≤ 1.

(1.30)

Positive means that 0 ≤ f ∈ L1 (, R, μ) ⇒ Pf ≥ 0.

(1.31)

Proposition 7 gives the unique adjoint P∗ of P acting on (L1 (, R, μ))∗ = L∞ (, R, μ), the set of all functions f :  → R with ess sup|f | < ∞. By the Riesz representation theorem, the correspondence 

f · f ∗ dμ,

L (, R, μ)  f → 1

(1.32)



for a unique f ∗ ∈ L∞ (, R, μ), defines a linear functional; that is, an element of L∞ (, R, μ). This duality will be denoted by f , f ∗ =



f · f ∗ dμ.

(1.33)



Hence Pf , f ∗ = f , P∗ f ∗ for all f ∈ L1 (, R, μ) and f ∗ ∈ L∞ (, R, μ). Since P∗ is also a nonexpansive and positive operator (on L∞ (, R, μ), however), P or P∗ is called a Markov operator. Now let χA : A → R denote the characteristic function on A ∈ F . Define P(ω, A) := (P∗ χA )(ω).

(1.34)

Then P(·, A) is a function from  into R. Furthermore 1. im P(·, A) = [0, 1]; 2. P(·, A) is F -measurable for all fixed A ∈ F ; 3. if {Ai }i∈N is a countable collection of disjoint sets in F , then ⎛ P ⎝·,

 i∈N

⎞ Ai ⎠ =



P(·, Ai );

i∈N

4. if μA = 0, then P(ω, A) = 0.

A function P(·, A) that satisfies conditions 1–3 everywhere is called a transition probability for the Markov process. The value P(ω, A) is the probability of transfer of ω ∈  into A.

Mathematical preliminaries

21

Conversely, one can show that, given a σ -finite measure space (, F , μ) and a function P(·, A), A ∈ F , fixed satisfying conditions 1–3, one can obtain a Markov process by defining P∗ on L∞ (, R, μ) by 

∗

f (ω )P(ω, dω ).

(P f )(ω) :=

(1.35)



It is not difficult to see that the product of two Markov operators is again a Markov operator. In particular, (P∗ )n is a Markov operator for any n ∈ N, and 

Pn (ω , A)P(ω, dω ).

Pn+1 (ω, A) = 

In later chapters mostly discrete Markov processes, called Markov chains, are considered. One way of obtaining these Markov chains is by setting  := N, F := 2N , the power set on N, and by letting μ be a counting measure; that is, μ{n} = 1 for n ∈ N. In this case the operator P is a positive matrix (Pnk )n,k∈N satisfying n∈N Pnk ≤ 1.

Also, P(ω, A) = n∈A Pnk , ⊆ N, and (Pf )(k) =



Pnk f (n),

(1.36)

n∈N

(P∗ f ∗ )(n) =



Pnk f ∗ (k).

(1.37)

k∈N

The characterization of a Markov process follows directly from its description in terms of random variables. A sequence {Xn }n∈N0 of random variables on (, F , μ) is called a Markov process iff π{Xn+1 ∈ B|X0 , . . . , Xn } = π {Xn+1 |Xn }

(1.38)

for all n ∈ N0 and for all B ∈ B1 . (Here π {X|Y)} denotes the conditional probability of X given Y.) In the following chapters a certain class of countable Markov processes is considered. To give an equivalent characterization of such Markov chains the concept of a sequence space has to be introduced. These sequence spaces will continue to play an important role in the theory of iterated function systems and fractal functions. Let S := Nn be a subspace of N, called the state space. Let  := SN and let n := SNn . The elements of  and n are infinite sequences and n-tuples, respectively, of the form i = (i0 i1 · · · in · · · ) and i(n) = (i0 i1 · · · in ), respectively, where ij ∈ Nn . The set n is called a finite tree of length n with N branches at each branch point ij . The set  is called a sequence space, its elements i are called codes, and in is called the nth outcome on i. The nth outcome is said to occur at time n. An element of n is called a finite code of length n.

22

Fractal Functions, Fractal Surfaces, and Wavelets

Let S0 , . . . , Sn be subsets of the state space S. Let Fn denote the collection of all unions of sets in  of the form {i = (i0 i1 · · · in · · · ): i0 ∈ S0 , . . . , in ∈ Sn }, ∞ 

and let F :=

(1.39)

Fn . Although each Fn is a Borel field, F is in general not. A measure

n=0

will be defined on the smallest Borel field G containing F , and then the completion of G is taken. To achieve this, a few more definitions and facts are needed. A set of F is called a cylinder set. If C is a cylinder set of Fn , then C can be written as C=



Bnk ,

(1.40)

k∈N0

where the basic cylinder sets Bnk are defined by Bnk := {i : i0 = c0 , . . . , in = cn }.

(1.41)

Now let νBnk :=

n 

pim ,

(1.42)

m=0

where pim denotes the probability that outcome im will occur. Then ν is a measure on Fn , and ν =

n  n∈N m=1

⎛ pim = ⎝

n 

⎞N pj ⎠ = 1.

j=1

It is possible to extend ν to a probability measure μ on G , and then to complete μ. The completion of μ is denoted by (, F , μ). (Here, to ease the notation, the symbol  is deleted from F and μ.) It is possible to define a metric, the so-called Fréchet metric, on the set . To this end, let dF :  ×  :→ R be given by dF (i, j) :=

 |in − jn | (N + 1)n

(1.43)

n∈N

for all i = (i1 · · · in · · · ), j = (j1 · · · jn · · · ) ∈ . The metric space (, dF ) is (by Tychonov’s theorem) compact, and one can verify that  is homeomorphic to the classical Cantor set on N symbols. (, F , μ) or (, dF ) is called a code space.

Mathematical preliminaries

23

Now suppose S is a countably finite or infinite state space and {Xn }n∈N0 is a sequence of random variables defined on a code space (, F , μ) with values in S; that is, Xn :  → S for all n ∈ N. The sequence {Xn }n∈N0 is called a countable Markov chain if π{Xn+1 = in+1 |Xn = in , . . . , X0 = i0 } = π {Xn+1 = in+1 |Xn = in }

(1.44)

for all i0 , . . . , in+1 ∈ S. The starting or initial probability distribution is a vector p whose components pi are given by pi = π{X0 = i}, and the transition matrix is defined by Pij = π{Xn+1 = j|Xn = i}, provided π {Xn = i} > 0. Assume that the matrix P = (Pij ) has a fixed point p∗ ; that is, Pp∗ = p∗ . Then p∗ is called a limiting probability or a stationary (probability) distribution (that p∗ = (p∗i )

∗ ∗ is a probability vector, ie, a vector satisfying pi ≥ 0 and i pi = 1, follows directly from Eq. 1.36). If the cardinality of S is finite, say, |S|c = N, and if the N × N transition matrix P is positive, then the following result holds. Theorem 15. Let pi (n) denote the probability that at time n, Xn = i, and let p(n) := (p1 (n), . . . , pN (n)) be the corresponding probability vector. Suppose that p(n + 1) = Pp(n). Then lim p(n) = p∗ ,

n→∞

(1.45)

where p∗ is a probability vector, independent of p(0). Furthermore, p∗ is an eigenvector of P with associated eigenvalue 1.

3 Algebra In this brief section on abstract algebra, the reader is reminded of such fundamental concepts as diagram, semigroup, group, endomorphisms, free semigroups, and free groups. These notions will be used in the next chapter when Dekking’s approach to fractal sets is introduced, and also later in connection with dimension calculations for fractals. A rather short review of category theory and direct and inverse limits is also presented. In addition, reflection groups and root systems and the concepts of a Coxeter group, an affine Weyl group, and a foldable figure are introduced. All these concepts play an important role in Chapter 10.

24

Fractal Functions, Fractal Surfaces, and Wavelets

3.1 Free groups, semigroups, and groups Definition 29. A diagram D of mappings is a quadruple (A, F, α, β) consisting of the following: 1. A system of sets A = (Ai )i∈I , where I is an index set. 2. A system of mappings F = (Fj )j∈J indexed by a set J. 3. Mappings α: J → I and β: J → I, satisfying: (a) dom Fj = Aα(j) , im Fj = Aβ(j) ; that is, Fj : Aα(j) → Aβ(j) for all j ∈ J; (b) α(J) ∪ β(J) = I.

The diagram D is called finite iff J is finite. The elements of F are called arrows, and an n-tuple τ := (Fj1 , . . . , Fjn ) of arrows is called a path in D iff for all ∈ Nn−1 , the arrow Fj ends where the arrow Fj +1 begins. The arrow Fj1 or Fjn is called the initial point or the terminal point, respectively, of the path τ . Along a path τ , one can form the composition Fτ of the mappings Fjn ◦· · ·◦Fj1 . Definition 30. 1. A diagram D is called commutative iff for two paths τ and τ  having the same initial and terminal point in D, Fτ = Fτ  . 2. A diagram D is called a sequence iff: (a) at each Ai , i ∈ I, there begins or ends at most one arrow; (b) for any two distinct sets Ai1 and Ai2 , there exists exactly one path whose initial point is Ai1 and whose terminal point is Ai2 for all i1 , i2 ∈ I.

Next semigroups and groups are defined. Let X be a set of objects which are called the elements of the set. A binary operation in X is a function θ : X × X → X, (a, b) → θ (a, b). Following the usual convention, one writes ab for θ (a, b), although ab may not mean ordinary multiplication. Moreover, ab is called the product of a with b. A binary operation θ on X is called associative if ∀a, b, c ∈ X: θ (θ (a, b), c) = θ (a, θ (b, c)),

(1.46)

or simply ∀a, b, c ∈ X: (ab)c = a(bc).

(1.47)

An element e in X is called a unit or a neutral element if ∀a ∈ X: ae = ea = a.

(1.48)

It is easy to establish that the unit in a set X (if it exists) is unique. Definition 31. A semigroup S is a pair (X, θ ), where X is a set and θ is an associative binary operation on X. A subset of T of a semigroup S is called stable (with respect to the binary operation θ in S) provided that for all a, b ∈ T, ab ∈ T. If T is a stable subset of S, the restriction ρ of θ to T × T defines a binary operation on T. Together with this binary operation ρ: T × T → T, T becomes a semigroup. Since T ⊆ S, T is called a subsemigroup of S.

Mathematical preliminaries

25

Now let X be an arbitrary subset of a semigroup S. If the smallest (with respect to set containment) subsemigroup of S that contains X is S itself, then X is said to be a set of generators for S and it is said that S is generated by X. Now suppose S and S are semigroups. A homomorphism of S into S is a function h: S → S such that ∀a, b ∈ S: h(ab) = h(a)h(b).

(1.49)

An homomorphism is called a monomorphism iff it is injective, and an epimorhism iff it is surjective. A bijective homomorphism is called an isomorphism. If S = S, then h is called an endomorphism of S. Isomorphic endomorphisms are called automorphisms. Notation. For an injective mapping f from a set A into a set B, one writes f : A  B f

or A  B, and for a surjective mapping g from a set A into a set B, one writes g: A  B g

or A  B. Let X be an arbitrary set. A free semigroup on the set X is a semigroup S together with a function f : X → S such that for every function g: X → S from the set X into a semigroup S there exists a unique homomorphism h: S → S such that the following diagram commutes: X ⏐ ⏐ g

f

−→

S

h

(1.50)

S This definition expresses what is called the universality of a free semigroup. The following facts about free semigroups are straightforward to prove. Proposition 15. Assume that the semigroup S together with the function f : X → S is a free semigroup on the set X. Then f is injective and its image f (X) generates S. Theorem 16 (Uniqueness Theorem). Suppose that (S, f ) and (S , f  ) are free semigroups on the same set X. Then there exists a unique isomorphism j: S → S such that the following diagram commutes: X ⏐ ⏐ f 

f

−→ j

S (1.51)

S The proof of the following existence theorem for free semigroups is given because it introduces notation and terminology that will be used later. Theorem 17 (Existence Theorem). For any set X there exists a free semigroup on X.

26

Fractal Functions, Fractal Surfaces, and Wavelets

Proof. Let S := {(x1 , . . . , xn ) : x1 , . . . , xn ∈ X ∧ n ∈ N}. Define a binary operation on S as follows: If ξ = (x1 , . . . , xn ) and η = (y1 , . . . , ym ) are in S, define ξ η to be the concatenated finite sequence ξ η = (x1 , . . . , xn , y1 , . . . , ym ). It is easy to show that this binary operation is associative. Hence S together with binary operation is a semigroup. Next define f : X → S by f (x) = (x), where (x) is the finite sequence consisting of the single element x. To prove that the pair (S, f ) is a free semigroup on X, let g: X → S be a given arbitrary function from the set X into a semigroup S . Define a function h: S → S as follows: h(ξ ) = g(x1 ) · · · g(xn ) for all ξ = (x1 , . . . , xn ) ∈ S. Since the binary operation on S is associative, the function h is a homomorphism. Furthermore, ∀x ∈ S: (h ◦ f )(x) = h(f (x)) = h((x)) = g(x); that is, h ◦ f = g. To show that h is unique, let h : S → S be an arbitrary homomorphism satisfying  h ◦ f = g. The following equalities hold for all ξ = (x1 , . . . , xn ) ∈ S: h (ξ ) = h ((x1 ) · · · (xn )) = h ((x1 )) · · · h ((xn )) = h (f (x1 )) · · · h (f (xn )) = g(x1 ) · · · g(xn ) = h(ξ ), and therefore h = h . Remark 8 1. Every set X determines (up to isomorphisms) a unique free semigroup (S, f ). By Proposition 15, X is identified with its image fX in S. Hence X becomes a subset of S that generates S, and every function g: X → S from X into an arbitrary semigroup S extends to a unique homomorphism h: S → S . (S, f ) is referred to as the semigroup generated by X. 2. The set X is also called the alphabet, and the elements of the free semigroup generated by X are called words.

The next result is an easy application of the existence theorem for free semigroups. Proposition 16. Let X be a set of generators of a semigroup (S, f ). Then every element of S can be written as the product of a finite sequence of elements in X. For our later purposes the concept of a group is needed, and especially that of a free group. A group G is a semigroup that has the following additional properties: 1. ∃e ∈ G ∀g ∈ G : ge = eg = g. 2. ∀g ∈ G ∃g−1 ∈ G : gg−1 = g−1 g = e.

Mathematical preliminaries

27

For any two subsets H, K of a group G, define the following new subsets: H −1 := {h−1 : h ∈ H}; HK := {hk : h ∈ H ∧ k ∈ K}; HK −1 := {hk−1 : h ∈ H ∧ k ∈ K}. Definition 32. A nonempty subset H of a group G is called a subgroup of G iff HH −1 ⊆ H.

(1.52)

There exist special subgroups in a group G that can be used to abelianize it. Definition 33. A group G is called abelian or commutative iff gh = hg for all g, h ∈ G. To justify the foregoing remark, a few more definitions and results from elementary group theory are needed. For this purpose, suppose H is an arbitrary subgroup of a group G. One can define a relation ∼ in the set G as follows: For any two elements g, h ∈ G, let g ∼ h iff g−1 h ∈ H. It is not at difficult to show that the relation ∼ : G × G → G is reflexive, symmetric, and transitive, and hence an equivalence relation on the set G. This equivalence relation divides the set G into disjoint subsets whose union is G. These subsets are called the equivalence classes modulo ∼. Denote by Q the totality of all equivalence classes modulo ∼. This set is termed the quotient set of the group G over its subgroup H; symbolically, Q = G/H. An interesting question is under what conditions on the subgroup H the quotient set Q = G/H becomes a quotient group. The following proposition gives the answer to this question. But first we require a definition. Definition 34. A subgroup H of a group G is called normal iff gH = Hg for all elements g ∈ G. (Here gH := {gh: h ∈ H}.) Proposition 17. Suppose H is a normal subgroup of a group G, and Q = G/H is the quotient set of G over H. The binary operation in Q defined by (gH)(g H) := (gg )H for all g, g ∈ G, makes Q into a group, the quotient group of G over H. Furthermore, the natural projection p: G  Q, p(g) := gH, is an epimorphism whose kernel equals H. Now suppose X is a subset of a group G. Then there exists a smallest subgroup H (with respect to set containment) of G that contains X. This subgroup is called the subgroup generated by X. If H = G, G is said to be generated by X and X is said to be a set of generators for G. The next result gives some information about the form of the elements of a group that is generated by a given set. Proposition 18. Let X be a set of generators for a group G. Then every element in G can be expressed as a product of a finite sequence of elements in X ∪ X −1 . Let g, g ∈ G be any two elements in a group G. The element gg g−1 g−1 is called the commutator of g and g . The subgroup (G) generated by all commutators in G is

28

Fractal Functions, Fractal Surfaces, and Wavelets

called the commutator subgroup of G. Clearly, (G) is normal, and thus G/ (G) is a group; moreover, it is an abelian group. Therefore, G/ (G) is called the abelianization of G. Since every group is also a semigroup, the preceding definitions of homomorphism, monomorphism, endomorphism, isomorphism, and automorphism defined for semigroups carry over to groups. Proposition 19. Suppose h: G → G is a homomorphism of a group G into a group G . Then 1. h maps the neutral element eG of G into the neutral element eG of G. 2. h(x−1 ) = (h(x))−1 .

Next the direct product of an arbitrary family of groups is introduced. For this purpose, let I be an arbitrary index set indexing  a given family of groups {Gi }i∈I . Denote by G the union of the sets Gi and by i∈I Gi the Cartesian productof the family of sets Gi . By the definition of the Cartesian product, each element of i∈I Gi is a function f :I → G such that f (i) ∈ Gi for every i ∈ I.  One defines a binary operation θ on i∈I Gi as follows: for any two elements f , g ∈ i∈I Gi , let θ (f , g) be the function given by (fg)(i) := f (i)g(i) ∈ Gi

for all i ∈ I.

 The neutral element of i∈I Gi is the function e: I → G given by e(i) = ei ∈ Gi , with f −1 : I → G ei being the neutral element in Gi , and the inverse element  is the function  G , θ is called the direct defined by f −1 (i) = [f (i)]−1 for all i ∈ I. The pair i∈I i product of the given family of groups. If the Gi are abelian groups, the direct product is also called the direct sum of the family {Gi }i∈I of abelian groups and is written as (⊕i∈I Gi , θ ). Now suppose X is an arbitrarily given set. A free group on the set X is a group F together with a function f : X → F such that for every function g: X → G from the set X into a group G there exists a unique homomorphism h: F → G with the property h ◦ f = g. The following two results are the analogues of Proposition 15 and Theorem 16. Proposition 20. Let (F, f ) be a free group on the set X. Then the function f : X → F is injective and its image f (X) generates F. Theorem 18 (Uniqueness Theorem for Free Groups). Assume that (F, f ) and (F  , f  ) are free groups on the same set X. Then there exists a unique isomorphism j: F → F  such that j ◦ f = f  . Now the existence theorem for free groups is established. Theorem 19 (Existence Theorem for Free Groups). Let X be any set. Then there exists a free group on X. Proof. Given the set X, define a new set Y by Y := X × {−1, 1}.

Mathematical preliminaries

29

Set x1 := (x, 1) and x−1 := (x, −1). The set Y generates a free semigroup S, whose words are finite formal products of elements of Y. A word w is called reduced iff for any x ∈ X, x1 never stands next to x−1 in w. The symbol e will stand for the empty word. Now define F as the collection of all reduced words in S together with e. To make F into a group, a binary relation has to be defined on it, and it must be shown that this binary relation satisfies the group axioms. So suppose that u, v ∈ F are arbitrary. If u = e, define uv = v, and if v = e, define uv = u. If neither u nor v equals e, u and v are reduced words in S, and so uv is in S. Two cases are possible: either uv = e or uv = w, where w is a reduced word one obtains by canceling from uv all pairs of the form x1 x−1 and x−1 x1 . Hence define a binary operation on F by  uv :=

e; w,

according to the two cases mentioned earlier. It is not difficult to verify that F with this binary operation is a group with unit e. Now define a function f : X → F by setting f (x) := x1 ∈ F for all x ∈ X. It remains to be shown that (F, f ) is a free group on X. To this end, suppose that G is an arbitrary group and that g: X → G is an arbitrary function from the set X into G. Let w ∈ F be arbitrary. Then w is either the empty word e or is of the form w = x1ε1 x2ε2 · · · xnεn , where εi := ±1, i ∈ Nn . Define a function h: F → G by  h(w) :=

if w = e; eG ε ε n 1 (g(x1 ) ) · · · (g(xn ) ) otherwise.

(eG is the unit in G.) Then h is a homomorphism satisfying h ◦ f = g. The uniqueness of h is established by use of an argument similar to the one given in Theorem 17. The uniqueness Theorem 18 implies that every set determines essentially a unique free group (F, f ). The injectivity of the function f allows us to identify the set X with its image f (X) is F. This identification is called the embedding of X into F; in symbols, X → F. Hence X is a set of generators for F. Furthermore, every function g: X → G from the set X into an arbitrary group G extends to a unique homomorphism h: F → G. Therefore, F is called the free group generated by the set X. It is sometimes useful to indicate the set X that generates a free semigroup or a free group. The notation S[X] or F[X] is used to express the fact that the free semigroup or the free group, respectively, is generated by the set X. Before we close this section, the relation between a free semigroup S[X] and a free group F[X] that are both generated by the same set X is investigated. By Theorems 17 and 19, and especially their respective proofs, it is not difficult to see that there is an obvious embedding S[X] → F[X] of a free semigroup S[X] into a free group F[X]: The free group F[X] is clearly a semigroup, and by the universality of S[X] there exists a homomorphism h from S[X] into the semigroup F[X]. The injectivity of h follows

30

Fractal Functions, Fractal Surfaces, and Wavelets

from the observation that the generators of S[X] are mapped onto generators of F[X], and that distinct words x1 · · · xn and y1 · · · ym in S[X] can be identified with the distinct words x11 · · · xn1 and y11 · · · y1m in F[X]. Assume now that the set X has finite cardinality, say, |X|c = n. Let F[X] be the free group generated by X. The abelianization of F[X] is the free abelian group ZX ; the generators {x1 , . . . , xn } of F[X] correspond to the n-tuples {(1, 0, . . . , 0), . . . , (0, . . . , 1)} in ZX . Thinking of ZX primarily as the set of all functions from X into Z, we can write the natural projection p: F[X]  ZX as p(x)(y) = δxy

(1.53)

for all x, y ∈ X. Here δxy denotes the Kronecker delta, defined by  δxy :=

if x = y; otherwise;

1 0

that is, the generator xi is mapped onto the n-tuple (0, . . . , 1, . . . 0) in ZX , where 1 is in the ith position. We briefly mention the notion of a category, which is one of the most important and far-reaching tools in modern mathematics: it is a unifying and clarifying device for apparently different and complicated concepts. It is therefore worthwhile to remind the reader of the definition of category and also a few related results from category theory. Some of these results will be used implicitly in later chapters, thus giving a more complete insight into the nature of the subject. Definition 35. A category K consists of: 1. a class obj(K) of objects; 2. disjoint sets K(A, B) for each ordered pair (A, B) of elements of obj(K); 3. compositions K(B, C) × K(A, B) → K(A, C) (for short, (f , g) → fg for g ∈ K(A, B) and f ∈ K(B, C)) for each triple (A, B, C) of elements of obj(K) such that: (a) ∀A ∈ obj(K)∃IA ∈ K(A, A): IA f = f and gIA = g for all f ∈ K(B, A) and g ∈ K(A, B); (b) ∀f ∈ K(A, B) ∀g ∈ K(B, C) ∀h ∈ K(C, D): h(gf ) = (hg)f .

The elements of K(A, B) are called K-morphisms from A to B and are denoted by f

f : A → B or A → B. The collection of all K-morphisms is denoted by morph(K). The elements IA , A ∈ obj(K), are called identities. If obj(K) is a set, the category is called small. The following are some examples of categories. Example 1. Let obj(S) be the class of all sets and let morph(A, B) be the set of all functions from A to B. This is the category S of sets. Example 2. Let obj(G) be the class of all groups and let morph(A, B) be the set of all group homomorphisms. Then G is the category of groups. Example 3. If obj(T) is the class of all topological spaces and morph(A, B) is the set of all continuous functions from A to B, then T is the category of topological spaces.

Mathematical preliminaries

31

Definition 36. Suppose K is a category. A K-morphism f : A → B is called a retraction iff there exists a K-morphism g: B → A such that f ◦ g = idB . B is called the retract of A under f . β Definition 37. A directed system {Gα , rα } of abelian groups over a directed set (D, ) is a function that assigns to each element α ∈ D an abelian group Gα and β to each pair (α, β) of elements of D with α  β a homomorphism rα : Gα → Gβ such that: 1. ∀α ∈ D: rαα = idGα ; γ β

γ

2. ∀α  β  γ : rβ rα = rα .

Remark 9. Analogously, one may define directed systems of groups, rings, modules, homomorphisms, etc. β Suppose that {Gα , rα } is a directed system of abelian groups. Regard the Gα as pairwise disjoint sets and define the disjoint union  ˙

Gα :=

α∈D



{α, Gα }.

α∈D

 An equivalence relation on ˙ Gα can be defined as follows: For gα ∈ Gα and gβ ∈ Gβ α∈D

let γ

gα ∼ gβ ⇐⇒ ∃γ ∈ D: α  γ ∧ β  γ ∧ rαγ (gα ) = rβ (gβ ). Denote the equivalence class of gα by [gα ] and the set of all equivalence classes by G. We can make the set G into an abelian group by defining +: G × G → G, γ γ ([gα ], [gβ ]) → [gα ] + [gβ ] as the equivalence class of rα (gα ) + rβ (gβ ), and −[gα ] as [−gα ]. The identity elements 0α in Gα are to represent the identity element 0 ∈ G. The abelian group thus obtained is called the direct or inductive limit of the directed system β {Gα , rα } and is written as G = lim Gα = − →

 ˙

Gα / ∼ .

α∈D

For a fixed α ∈ D, a homomorphism rα : Gα → G can be defined by Gα  gα → β [gα ] ∈ G. Clearly, if α  β, then rα = rβ rα . In a similar fashion, one can make G for a directed system of rings or modules into a ring or module, respectively. Example 4. Let G be a group and let Gα be the collection of all finitely generated subgroups of G. We can introduce an ordering on these subgroups by setting αβ

⇐⇒

Gα ⊆ Gβ .

32

Fractal Functions, Fractal Surfaces, and Wavelets β

β

The inclusion maps Gα → Gβ are taken as the rα . Then {Gα , rα } is a directed system and its direct limit is G. The following result is sometimes used to define the direct limit of a directed system. β Theorem 20. Assume that {Gα , rα } is a directed system of abelian groups and  sα : Gα → G is a system of homomorphisms from Gα into an abelian group G β satisfying sα = sβ rα for all α  β. Then there exists a unique homomorphism s: G = lim Gα → Gα − → such that the following diagram commutes for all α ∈ D: Gα ⏐ ⏐ rα 

f

−→

G

#s

(1.54)

G One may define a dual object to a directed system and a direct limit. This dual object is called an inverse system, and the corresponding limit is called the inverse or projective limit. Definition 38. Let (D, ) be a directed set and let {(Sα , Sα ): α ∈ D} be a collection β of topological spaces. Suppose that for all α  β there exist maps pα : Sβ → Sα such that: 1. pαα = idSα .

γ

β γ

2. ∀α  β  γ : pα = pα pβ .

Then {Sα , pα } is called an inverse system of topological spaces over the directed system (D, ). The inverse limit S of {Sα , pα }, written S = lim Sα , ← −  is the subset of α∈D Sα consisting of mappings sα : S → Sα with the property that β sα = pα sβ for all α  β. The topology of S is that induced by the Cartesian product  α∈D Sα , and the map sα is called the α-coordinate of s ∈ S. Example 5. Consider the collection of all subsets Sα of a topological space (S, S ). These subspaces are ordered by inverse inclusion; that is, α  β ⇒ Sα ⊃ Sβ . If these β inclusions are denotedby pα , then {Sα , pα } is an inverse system and its inverse limit S Sα . is homeomorphic to α∈D

The next two results will be used implicitly in later chapters. Theorem 21. The direct limit exists in the category of groups. Theorem 22. The inverse limit exists in the category of sets.

Mathematical preliminaries

33

3.2 Reflection groups and root systems In this section we introduce root systems, reflections about an affine hyperplane, and the associated affine Weyl groups. We present only those concepts that are of importance for later developments and ask the reader to consult references such as [11–17] for a more in-depth presentation of reflection groups and root systems. We denote by E a fixed Euclidean space (ie, a finite-dimensional R-vector space endowed with a positive symmetric bilinear form ·, · ). Let H ⊂ E be a linear hyperplane (ie, a codimension 1 linear subspace of E). Definition 39. A linear transformation r: E → E is called a reflection about H if: 1. r(H) = H; 2. r(x) = −x for all x ∈ H ⊥ .

A straightforward computation yields an explicit representation of r: rα (x) = x −

2x, α α α, α

for fixed 0 = α ∈ H ⊥ .

(1.55)

A discrete group generated by a set of reflections in E is called a Eulidean reflection group. An abstract group that has a representation in terms of reflections is termed a Coxeter group. More precisely, a Coxeter group is a discrete group C with a finite set of generators {αi : i ∈ Nk } satisfying ! C := α1 , . . . , αk : (αi αj )mij = 1 ∧ 1 ≤ i, j ≤ k , where mii = 1 for all i and mij ≥ 2 for all i = j. (mij = ∞ is used to indicate that no relation exists.) It can be shown that finite Coxeter groups are isomorphic to finite Euclidean reflection groups [13]. Example 6. Consider Klein’s four-group V or dihedral group D4 of order 4: " # V = D4 = α, β : α 2 = β 2 = (αβ)2 = 1 . Geometrically, V is the symmetry group of the unit square [− 12 , 12 ] × [− 12 , 12 ] centered at the origin. Definition 40. A root system R is a finite set of vectors α1 , . . . , αk ∈ E \ {0} with the properties that: 1. 2. 3. 4.

E = span{α1 , . . . , αk }; α, tα ∈ R iff t = ±1; ∀α ∈ R: rα (R) = R, where rα denotes the reflection through the hyperplane orthogonal to α; ∀α, β ∈ R: β, α ∨ ∈ Z, where α ∨ := 2α/α, α .

The dimension of E is called the rank of the root system. The elements of a root system are termed roots.

34

Fractal Functions, Fractal Surfaces, and Wavelets

We note that condition 4 restricts the possible angles between roots. Let α and β be two roots and let n(β, α) := β, α ∨ = 2

α, β ∈ Z. α, α

Denote by |α| := α, α 1/2 the length of α and by θ the angle between α and β. Then α, β = |α||β| cos θ, and thus n(β, α) = 2

|β| cos θ. |α|

(1.56)

It follows from Eq. (1.56) that n(β, α)n(α, β) = 4 cos2 θ . Hence n(β, α) ∈ Z ⇒ 4 cos2 θ ∈ {0, 1, 2, 3, 4}. The following table [17] shows the possible angles θ and thus the relation between the two roots α and β:

n(β, α) n(β, α) n(β, α) n(β, α) n(β, α) n(β, α) n(β, α)

= = = = = = =

0, 1, −1, 1, −1, 1, −1,

n(α, β) n(α, β) n(α, β) n(α, β) n(α, β) n(α, β) n(α, β)

= = = = = = =

0, 1, −1, 2, −2, 3, −3,

θ θ θ θ θ θ θ

= = = = = = =

π/2. π/3, 2π/3, π/4, 3π/4, π/6, 5π/6,

|β| |β| |β| |β| |β| |β|

= = = = = =

|α|. |α|. √ √2|α|. √2|α|. √3|α|. 3|α|.

Note that the value of θ determines both n(α, β) and {|α|/|β|, |β|/|α|}. Example 7. Three examples of root systems in E2 are given in Fig. 1.1. b

b A1× A1 a

Fig. 1.1 Three root systems of rank 2.

A2

a

b B2 a

Mathematical preliminaries

35

Roots may be divided into two classes. To this end, choose v ∈ E so that v, α = 0 for all roots α ∈ R. We define the set of positive roots R+ := {α ∈ R : v, α > 0} and ˙ ˙ − , where ∪ the set of negative roots R− := {α ∈ R : v, α < 0}. Clearly, R = R+ ∪R denotes disjoint union. A root in R+ is called simple if it cannot be written as the sum of two elements of R+ . The set  ⊂ R+ of simple roots forms a basis of E with the property that every α ∈ R can be written in the form  ki δi (ki ∈ Z ∧ δi ∈ ), α= i

where all ki > 0 or all ki < 0. We summarize some properties of roots and root systems below. Proposition 21. Let R be a root system and R+ be the set of positive roots: 1. R+ is that subset of R for which the following conditions hold: (a) Given α ∈ R, then either α ∈ R+ or −α ∈ R+ . (b) Let (α, β)∈ R+ with α = β. If α + β ∈ R then α + β ∈ R+ . 2. For α, β ∈  with α = β, α, β ≤ 0 (ie, the angle between two simple roots is always obtuse). 3. Assume that α, β ∈ R, α, and β are not multiples of each other, and α, β > 0. Then α − β ∈ R.

Now suppose that α, β ∈ R. Let H be the hyperplane orthogonal to α. We denote by α ∗ ∈ (E)∗ the unique element in the algebraic dual of E such that α ∗ (H) = 0,

and

α ∗ (α) = 2.

Then we may rewrite Eq. (1.55) in the form rα (β) = β − α ∗ (β)α. Since E and E∗ are isomorphic via the Euclidean inner product ·, · , there exists a unique element α ∨ ∈ E such that α ∗ = α ∨ , · . α ∨ is called the coroot of α. The lattice in E spanned by the roots R or coroots R∨ is called the root lattice or coroot lattice, respectively. The finite family of hyperplanes {Hα : α ∈ R}, where Hα is the hyperplane orthogonal  to α, partitions E into finitely many regions. The connected components Hα are called the (open) Weyl chambers of E. of E \ α∈R

The fundamental Weyl chamber C relative to  is defined by C :=

$

{v ∈ E : v, δ > 0}.

δ∈

36

Fractal Functions, Fractal Surfaces, and Wavelets

Then C is a simplicial cone, and hence convex and connected; see Fig. 1.2. b C a

Fig. 1.2 A root system, its Weyl chambers (regions between dashed lines) and the fundamental Weyl chamber C.

Definition 41. The subgroup of the isometry group of a root system that is generated by the reflections through the hyperplanes orthogonal to the roots is called the Weyl group W of the root system R. As the root system is finite, W is a finite reflection group. Moreover, W acts simply transitively on the Weyl chambers; that is, if C1 and C2 are two Weyl chambers and xj ∈ Cj , j = 1, 2, then there exists a unique r ∈ W such that r(x1 ) = x2 .

3.3 Affine Weyl groups and foldable figures An affine hyperplane with respect to a root system R is defined by Hα,k := {x ∈ E : x, α = k},

α ∈ R, k ∈ Z.

We can also consider reflections rα,k about affine hyperplanes. Employing conditions 1 and 2 in Definition 39 applied now to affine hyperplanes, we obtain the following expression for such reflections: rα,k (x) = x −

2(x, α − k) α =: rα (x) + kα ∨ . α, α

(1.57)

Definition 42. Let R be a root system and {Hα,k : α ∈ R, k ∈ Z} its system of affine % for R is the infinite group generated by the hyperplanes. The affine Weyl group W reflections rα,k about the affine hyperplanes Hα,k : ! % := rα,k : α ∈ R ∧ k ∈ Z . W The next result characterizes the affine Weyl group of a root system and relates it to the finite Weyl group and the lattice generated by the coroots. First, we need a definition.

Mathematical preliminaries

37

Definition 43. Let (G, ·) be a group with identity element e. Suppose H is a subgroup of G and N is a normal subgroup of G. Further suppose that G = H · N and H ∩ N = {e}. Then G is called the semidirect product of H and N, written H  N. % of a root system R is Theorem 23 (Bourbaki’s Theorem). The affine Weyl group W the semidirect product W  , where  is the abelian group generated by the coroots % and W is the isotropy group α ∨ . Moreover,  is the subgroup of translations of W (stabilizer) of the origin. The group W is finite and  infinite. In this context, also note the particular form of an affine reflection (1.57); it is the sum of a reflection across a linear hyperplane plus a translation along the lattice spanned by the coroots. % % has a fundamental domain C ⊂ E in the sense that no r ∈ W It can be shown that W % such maps a point of C to another point of C, and for all x ∈ E there exists an r ∈ W that r(x) ∈ C. Furthermore, C is a compact and convex simplex. All affine Weyl groups (and therefore their fundamental domains) can be classified. For a given dimension n ∈ N there exist only a finite number of possible groups and thus fundamental domains. The classification of so-called irreducible root systems (root systems that cannot be written as the union of two root systems R1 and R2 such that α1 , α2 = 0, for αi ∈ Ri , i = 1, 2) follows from the representation theory of simple Lie algebras. The classification yields four infinite families—An (n ≥ 1), Bn (n ≥ 2), Cn (n ≥ 3), and Dn (n ≥ 4)—and five exceptional cases (E6 , E7 , E8 , F4 , and G2 ); the subscript indicates the rank of the root system. For the cardinality of these families and the explicit construction of their root system as well as the geometric description of the fundamental domains, see Refs. [11, 16]. Fig. 1.3 shows the three irreducible root systems of rank 2, their fundamental domains C, and their coroot lattices. Note that the fundamental domains consist of an equiangular triangle (root system A2 ), a 90 degree-45 degree-45 degree triangle (root system B2 ), and a 90 degree-30 degree-60 degree triangle (root system G2 ). % Now suppose we are given a collection H of affine hyperplanes of E. Denote by W the group of affine isometries generated by reflections rH with H ∈ H. A collection %-invariant iff rH H = H, for all rH ∈ W %. The group W % is said to be H is called W %-invariant family of hyperplanes H that is an affine reflection group if there exists a W locally finite; that is, every point in E has a neighborhood which intersects only finitely many H ∈ H. A given collection H of hyperplanes partitions E into convex cells, these cells being defined by f = c, f > c, or f < c, where f = c defines a hyperplane H ∈ H, c ∈ R. The support of a cell is the linear subspace L defined by the inequalities f = 0 that occur in its description. (If there are no such equalities, then L := E.) The dimension of a cell is the dimension of its support. A chamber is a cell of maximal dimension dim E. The chambers are the connected components of the complement of H in E. The walls of a chamber are the supports of its codimension 1 faces. Now choose a chamber C and denote by R the set of all reflections with respect to the walls of C. Then the following facts hold.

38

Fractal Functions, Fractal Surfaces, and Wavelets

Ha + b,1

Ha

b

Hb

C

–a V

Ha

bV

Ha ,1

a+b

aV + b V

b

Hb

C

a Ha + b

3a + 2b Ha

a = aV

Hb

b a

Ha ,1

Fig. 1.3 Root systems A2 , B2 , and G2 .

Theorem 24 1. 2. 3. 4.

% R generates W. % W is simply transitive on the chambers. % H is the set of all hyperplanes H ∈ E with rH ∈ W. % on E; that is, no rH ∈ W % The closure C of C is a fundamental domain for the action of W % such that maps a point of C to another point of C, and for all x ∈ E there exists an rH ∈ W rH (x) ∈ C.

The next theorem addresses some finiteness questions. Theorem 25 1. C has only finitely many walls, and thus R is finite. 2. There are only finitely many linear hyperplanes H0 such that H contains a translate of H0 . % Then W is a finite reflection 3. Denote by W the set of linear parts of all elements in W. group.

The following theorem deals with the structure of C. Recall that a reflection group is called irreducible iff it cannot be expressed as a product of reflection groups. Theorem 26. Suppose that W is essential and irreducible. Then the chamber C has % is infinite. exactly 1 + dim E walls and is a simplex in E. Furthermore, W

Mathematical preliminaries

39

%. Since W % acts on E, the stabilizer of x ∈ E is Finally, consider the structure of W % % % the subgroup Wx of W given by {rH ∈ W : rH x = x}. It can be shown that there exist %x is isomorphic to W . points x ∈ E such that the stabilizer W The final concept we need to introduce to later set up the connection between affine fractal surfaces, affine Weyl groups, and wavelet sets is that of a foldable figure. Definition 44. A compact connected subset F of E is called a foldable figure iff there exists a finite set S of affine hyperplanes that cuts F into finitely many congruent subfigures F1 , . . . , Fm , each similar to F, such that reflection in any of the cutting hyperplanes in S bounding Fk takes it into some F . The next theorems summarize the connections between affine Weyl groups and their fundamental domains, and foldable figures. For proofs, see Refs. [11, 18]. Theorem 27. Let G be a reflection group and On be the group of linear isometries of E. Then there exists a homomorphism φ: G → On given by φ(g)(x) = g(x) − g(0), g ∈ G , x ∈ E. The group G is called essential if φ(G ) fixes only 0 ∈ E. The elements of ker φ are called translations. If G is essential and without fixed points, then G has a compact fundamental domain. Theorem 28. The reflection group generated by the reflections about the bounding % of some root system. hyperplanes of a foldable figure F is the affine Weyl group W % has F as a fundamental domain. Moreover, W Theorem 29. There exists a one-to-one correspondence between foldable figures and reflection groups that are essential and without fixed points. Remark 10. It follows from Theorem 23 that the finite reflection group W leaves the lattice  invariant. Such groups are also called crystallographic. There is a classification of these crystallographic Coxeter groups in terms of so-called Coxeter or Dynkin diagrams. For each n := dim E there is only a finite number of such groups (see the earlier comments about classification). Each such diagram not only represents the underlying group but also gives the explicit geometry of the chambers, which in % are simplices in E. These simplices the case of an essential and irreducible group W are used as domains for multivariate fractal functions to generate nested subspaces of L2 (Rn ) in Chapter 10. Example 8. Examples of foldable figures in E := R2 and E := R3 are shown in Figs. 1.4 and 1.5.

Fig. 1.4 Foldable figures corresponding to the reducible root system A1 × A1 (left) and the irreducible root system B2 (right).

40

Fractal Functions, Fractal Surfaces, and Wavelets

Fig. 1.5 The unit cube [0, 1]3 and the prism σ 2 × [0, 1].

4 Function spaces In this last section we introduce four types of function spaces that are relevant for further developments. These are the Lebesgue spaces, the Hölder spaces, the Sobolev spaces, and the Besov and Triebel-Lizorkin spaces. There is a tremendous literature on each of these function spaces; we mention only Refs. [19–27], which form the basis of the presentation here.

4.1 Lebesgue spaces In Definition 27 we introduced the Lebesgue spaces Lp (, F , K, μ). Here we consider the special and important case that  is a nonempty open subset of Rn . Then F is the Borel σ -algebra Bn () and μ is Lebesgue measure dx. In this case we write Lp () instead of Lp (, Bn (), Rn , dx). In this setting one can interpret the Lebesgue spaces Lp (), 1 ≤ p ≤ ∞, as obtained by completion of the space CR () := C(, R) of real-valued continuous functions  → R with respect to the Lp -norm: 

1/p

f Lp :=

|f (x)|p dx

.

(1.58)



For 1 ≤ p ≤ ∞, the spaces Lp () are known to become Banach spaces when endowed with the norms (1.58). For 0 < p < 1, the spaces Lp () are defined as above, but instead of a norm a metric is used to obtain completeness. More precisely, define p

dp (f , g) := f − g Lp ,

(1.59)

where · Lp is the norm introduced above. The completion, (Lp (), dp ), of Lp () with respect to the metric dp is an F-space (see later). Note that the inequality (a + b)p ≤ ap + bp holds for all a, b ≥ 0. For more details, see Ref. [28].

Mathematical preliminaries

41

Definition 45. A topological vector space X is called an F-space if its topology is induced by a complete translation-invariant metric d. Recall that a metric d: X × X → R is called complete if every Cauchy sequence in X converges with respect to d to a point of X, and translation-invariant if dX (x + a, y + a) = dX (x, y),

∀x, y, a ∈ X.

Instead of working with the F-spaces Lp , 0 < p < 1, we can consider the entire range of Lp spaces, p > 0, in another way. We do this by introducing the concepts of a quasi-norm and quasi-normed spaces. To this end, let E be a K-vector space. A mapping · : E → R+ 0 is called a quasinorm if it satisfies all the usual conditions imposed on a norm except for the triangle inequality, which is replaced by

x + y ≤ c( x + y ),

(1.60)

where c ≥ 1 is a constant. If c = 1, then · is a norm. A complete quasi-normed space is called a quasi-Banach space. The Lebesgue spaces Lp (Rn ) for 0 < p < 1 defined on Rn are quasi-Banach spaces (c = 2p ), and for 1 ≤ p ≤ ∞ are Banach spaces as seen above. For later purposes, we also require the following closed subspace of Lp (Rn ). Let X ⊂ Rn be a domain (ie, an open subset of Rn ). Define Lp (X) := {f ∈ Lp : suppf ⊂ X}, where X := clX. As Lp (X) inherits its quasi-norm from Lp (Rn ), it is also a quasiBanach space.

4.2 Hölder spaces Throughout this section,  denotes a nonempty open subset of Rn , n ≥ 1. Let k ∈ N0 and 0 < α ≤ 1. To fill in the gaps between the discrete ladder of smoothness spaces k () := Ck (, R), k ∈ N , we introduce the so-called (inhomogeneous) Hölder CR 0 spaces. Notation. For notational simplicity, we drop the subscript R from all function spaces whose elements map to R. The Hölder space Ck,α () is defined as the linear space of all k-fold continuously differentiable bounded functions f :  → R such that there exists a constant c > 0 with |(D f )(x) − (D f )(y)| ≤ c < ∞,

x − y α

∀x, y ∈ .

Here, we used multiindex notation, writing D :=

∂ 1 + 2 +···+ n ∂x1 1 ∂x2 2 · · · ∂xn n

42

Fractal Functions, Fractal Surfaces, and Wavelets

for the differential monomial D of order | | := 1 + 2 + · · · + n . As usual, D0 := I, where I denotes the identity operator on Rn . It can be shown that the linear spaces Ck,α () when endowed with the norms 

f k,α := sup {|(D f )(x)|} + sup | |≤k x∈Rn

| |=k x,y∈Rn x=y

|(D f )(x) − (D f )(y)|

x − y α



become Banach spaces. For completeness we also define C0,0 () := C(),

Ck,0 () := Ck ().

Definition 46. Suppose X and Y are normed spaces. An embedding of X into Y, written X → Y, is such that: 1. X is a subvector space of Y; j

2. the identity mapping X  x −→ j(x) := x ∈ Y is continuous; that is, ∃c > 0

∀x ∈ X: j(x) Y ≤ c x X .

The mapping j is called an embedding (of X into Y). Using the definition of the respective norms, we can easily show the validity of the following embeddings: Let 0 < α < β ≤ 1. Then 1. Ck+1 () → Ck (); 2. Ck,α () → Ck (); 3. Ck,β () → Ck,α (),

where the spaces Ck ( and Ck,α () are understood as subspaces of Ck () and Ck,α (), respectively, consisting of all functions in Ck () or Ck,α (), which together with all their derivatives are continuous on the boundary ∂ of .

4.3 Sobolev spaces Next properties of functions are related to properties of their partial derivatives. To obtain a relation that is widely applicable, a “weak” version of the partial derivative of k (Rn ) denote the collection of all a function is needed. For k ∈ N0 ∪ {∞}, let Kk = KR k n functions in C (R ) with compact support, and let D = D(Rn ) denote the class of all infinitely differentiable functions on Rn with compact support. Now suppose two locally integrable functions f and g are given. Then g = Dα f , α ∈ Ns a multiindex, in the weak sense or f is the weak derivative of g iff  Rn

f (x)(Dα ϕ)(x)dx = (−1)|α|

 Rn

g(x)ϕ(x)dx

for all ϕ ∈ D.

(1.61)

Note that if f had partial derivatives up to order |α| in the usual (strong) sense, the preceding equation would easily follow from integration by parts.

Mathematical preliminaries

43

Definition 47. Let 1 ≤ q ≤ ∞ and let s ∈ N0 . The Sobolev space W s,q = W s,q (Rn ) s (Rn ) with respect to the norm is defined as the completion of Ks = KR 

f Wqs :=

Dα f Lq ,

(1.62)

|α|≤s

where · Lq denotes the Lq -norm. Equivalently, W s,q is the collection of all R-valued functions f ∈ Lq for which all α D f exist and Dα f ∈ Lq whenever |α| ≤ s. It should also be clear that for s > 0, W s,q ⊂ Cs−1 , and that W 0,p can be identified with Lq . If q = 2, then the space W s,2 is also denoted by H s = H s (Rn ) and is a Hilbert space. The importance of Sobolev spaces is well summarized in the next theorem; see also Ref. [23]. It shows how restrictions on the partial derivatives of a function give rise to restrictions on the function itself. Theorem 30. Let s and q be as before, and let p be such that 1/p = 1/q + s/n. 1. If p < ∞, then W s,q ⊂ Lp and the inclusion mapping j: W s,q → Lp is continuous; that is, j is an embedding of W s,q into Lp . 2. If p = ∞, then the restriction of f ∈ W s,q to a compact subset X of Rn is an element of Lr for any r < ∞. 3. If q > n/s, then every f ∈ W s,q can be modified on a set of measure zero so that the resulting function is continuous.

A proof of this theorem can be found in Ref. [23]. The embedding W s,q → Lp is to be interpreted in the following manner: Let f be a representative of the equivalence class [f ] ∈ W s,q . Then f is equal a.e. (on Rn ) to a function g ∈ [g] ∈ Lp .

4.4 Besov and Triebel-Lizorkin spaces As seen earlier the Sobolev spaces are defined by our requiring that the partial derivatives are in some Lp space. It is therefore natural to ask whether or not there exist function spaces that are intermediate to Sobolev spaces. Two equivalent definitions of these intermediate Besov spaces are given. The first follows naturally from our development of function spaces; the second is needed for later purposes. We recall that the Mth order forward difference operator M h , M ∈ N, of step size h ∈ Rn acting on functions f : Rn → R is defined by (M h f )(x) :=

M  μ=0

(−1)M−μ

  M f (x + μh). μ

If f is defined on a bounded domain X ⊂ Rn , we set  M h f (x; X)

:=

M h f (x) if x + μh ∈ X for μ = 0, 1, . . . , M; 0 otherwise.

(1.63)

44

Fractal Functions, Fractal Surfaces, and Wavelets

In the following the canonical Euclidean norm in Rn by | · |. For 0 < p ≤ ∞, let 

σn,p

 1 := n − 1 ≥ 0, min{p, 1}

and for 0 < p < ∞, 0 < q ≤ ∞, let σn,p,q :=

n . min{p, q}

Definition 48. [Ref. [24, Section 2.5.12]] Let 0 < p, q ≤ ∞ and let s > σn,p . Assume that M ∈ N is such that M − 1 ≤ s < M. Then a function f ∈ Lp = Lp (Rn ) belongs to the •

homogeneous Besov space B˙ sp,q = B˙ sp,q (Rn ) iff

|f |B˙ s

p,q

•

⎧ ⎨

1/q −sq M f q dh < ∞, n |h| p |h|n R h L := ⎩sup M −s 0=h∈Rn |h| h f Lp < ∞,

0 < q < ∞; q = ∞;

(1.64)

inhomogeneous Besov space Bsp,q := Bsp,q (Rn ) iff

f Bsp,q := f Lp + |f |B˙ s < ∞.

(1.65)

p,q

It is known that for 1 ≤ p, q ≤ ∞, Bsp,q is a Banach space; otherwise it is a quasiBanach space. We remark that if P is a polynomial of order M, then P lies in the kernel of the Mth-order difference operator and, therefore, |P|B˙ sp,q = 0. Definition 49 (Ref. [24, Section 2.5.10]). Let 0 < p < ∞, 0 < q ≤ ∞, and suppose s > σn,p,q . If M ∈ N is such that M − 1 ≤ s < M, then a function f ∈ Lp is said to belong to the •

s (Rn ) iff homogeneous Triebel-Lizorkin space F˙ p,q

|f |F˙ s

p,q

⎧ 1/q    ⎪ −sq |M f (·)|q dh  < ∞, 0 < q < ∞; ⎨ |h| n n  R  p h |h| :=  L  ⎪  M −s ⎩ q = ∞; sup0=h∈Rn |h| |h f (·)| p < ∞,

(1.66)

L

•

s := F s (Rn ) iff inhomogeneous Triebel-Lizorkin space Fp,q p,q s := f Lp + |f | ˙ s < ∞.

f Fp,q F

(1.67)

p,q

s is a Banach space for 1 ≤ p, q ≤ ∞; otherwise it is a quasi-Banach space. Fp,q Polynomials of order M have again vanishing seminorm | · |F˙ s . p,q

Mathematical preliminaries

45

Remark 11. The homogeneous Besov and Triebel-Lizorkin spaces are more compactly described as the linear spaces of all f ∈ Lp such that        − nq −s  M (·) f (∗) p  | · | L

Lq

0 and s ∈ / N: Cs = Bs∞,∞ . k and W k,2 = Bk . Sobolev spaces: For 1 < p < ∞ and k ∈ N0 : W k,p = Fp,2 2,2 s . Slobodeckij spaces: For 1 ≤ p < ∞ and 0 < s ∈ / N0 : W s,p = Bsp,p = Fp,p s . Bessel potential spaces: For 1 < p < ∞ and s > 0: H s,p = Fp,2

For s ∈ R: H s,p := {f ∈ S  : F −1 [(1 + |ξ |2 )s/2 F f (ξ )](·) Lp < ∞}, where F denotes the Fourier transform on the Schwartz space of tempered distributions S  = S  (Rn ); see later for the definition of S  . Terminology. Here and in the following, equality of function spaces is always meant in the sense of equivalent quasi-norms. Now suppose that X is a domain in Rn . Let A denote either B or F. Definition 50. Let 0 < p, q ≤ ∞ (with p < ∞ for the F-spaces) and s ∈ R+ . Then s Ap,q (X) is defined as the closed subspace of Asp,q given by * + Asp,q (X) := f ∈ Asp,q : supp f ⊆ X . Asp,q (X) inherits its norm from Asp,q and is thus a quasi-Banach space. In addition to the Banach space C(X) of R-valued uniformly continuous functions on X and the classical smoothness spaces Ck (X), k ∈ N, we also require the class of Zygmund spaces C s , s ∈ R+ . These are defined in the following way. Definition 51 (Ref. [27, Definition 2]). Let s ∈ R+ be written as s = [s]− + {s}+ , where [s]− ∈ N0 and 0 < {s}+ ≤ 1. Then , C s := C s (Rn ) := f ∈ C(Rn ) : f C s < ∞ ,

46

Fractal Functions, Fractal Surfaces, and Wavelets

where

f C s := f C[s]− (Rn ) +



+

sup |h|−{s} 2h Dα f C(Rn ) .

|α|=[s]−

0=h∈Rn

Here, Dα denotes the ordinary differential operator with multiindex α ∈ Nn0 . Zygmund spaces on domains X ⊆ Rn are defined in the usual way: C s (X) := {f ∈ C s : supp f ⊂ X},

X := cl X.

It is worthwhile noting that the Zygmund spaces C s coincide with the classical Hölder spaces Cs if s ∈ Q+ \ N, but for s := k ∈ N, Ck  C k . The reason for the introduction of the Zygmund spaces lies in the fact that functions in C s are pointwise multipliers for the B-scale and F-scale of function spaces. A function g is called a pointwise multiplier for Asp,q provided the mapping f → g · f is a bounded linear operator on Asp,q [24, Section 2.8]. We quote a result from Ref. [26] which we will require later. The proof can be found in Ref. [26, Section 4.2.2]. Proposition 22. Let 0 < p ≤  ∞ (0 < p< ∞ for  the F-spaces), 0 < q ≤ ∞, and s ∈ R+ . Assume that  > max s, n 1p − 1 − s , where (·)+ := max{·, 0}. Then +

g · f Asp,q ≤ c g C  f Asp,q

(some c > 0)

for all g ∈ C  and all f ∈ Asp,q . Next a theorem about Besov spaces is stated and some embeddings are given. For more details and proofs, see Refs. [24, 26, 27]. Theorem 31 1. Let 0 < α < 1 and suppose f ∈ Lp . Then f ∈ Bαp,q iff 

∞



0

 q



∂   yα−1   ∂y u(x, y) p

dy y

1/q < ∞.

Furthermore, · Bα,q is equivalent to the norm p



· p +

0

∞



  q 1/q ∂  dy  yα−1  u(x, y) .  ∂y  y p

∂f 2. Let α > 1. Then f ∈ Bαp,q iff f ∈ Lp and ∂x ∈ Bα−1 p,q , j ∈ Nn . Moreover, the norms · Bαp,q j 

n   ∂(·)  and · ∞ + j=1  ∂x  α−1 are equivalent. j

Bp,q

Mathematical preliminaries

47

The following inclusion relations hold for Besov spaces: α

α

1 2 1. If either α1 > α2 > 0 or α1 = α2 > 0 and q1 ≤ q2 , then Bp,q 1 → Bp,q2 . 2. If α ∈ N and p ∈ (1, ∞), then (a) W α,p ∼ Bαp,q for p ≥ 2; (b) W α,p → Bαp,2 for p ≤ 2; (c) Bαp,p → W α,p for p ≤ 2.

Remark 12 1. The notation W α,p ∼ Bαp,p has to be interpreted as follows: for f ∈ [f ] ∈ W α,p , there exists a g ∈ [g] ∈ Bαp,p such that f = g a.e. (on Rn ). 2. One can also define Besov spaces using the concept of interpolation spaces. The basic idea is as follows: Suppose that (X0 , · 0 ) and (X1 , · 1 ) are Banach spaces contained in a linear Hausdorff space X such that the inclusion mappings jk : Xk → X , k = 0, 1, are continuous. The linear spaces X0 ∩ X1 and X0 + X1 := {f ∈ X : f = f0 + f1 ∧ fk ∈ Xk ∧ k = 0, 1} are Banach spaces under the norms

f X0 ∩X1 := max{ f k : k = 0, 1} and

f X0 +X1 := inf{ f0 0 + f1 1 : f = f0 + f1 ∧ fk ∈ Xk ∧ k = 0, 1}, respectively. The inclusions X0 ∩ X1 → Xk → X0 + X1 → X are then continuous. Any Banach space X embedded in X satisfying X0 ∩ X1 → X → X0 + X1 → X is called an intermediate space of X0 and X1 . Using the real interpolation method, one constructs the Besov spaces Bαp,p as real intermediate spaces of two appropriate Sobolev spaces. For more details, see Ref. [22].

To proceed, an important class of a function space and its dual needs to be defined. The Schwartz space S = S (Rn ) consists of all real-valued functions in C∞ all of whose derivatives remain bounded when multiplied by any polynomial. More precisely, a function ϕ ∈ C∞ is an element of S iff for arbitrary multiindices α and β, sup |xβ (Dα ϕ)(x)| =: cα,β (ϕ) < ∞.

x∈Rn

(1.70)

48

Fractal Functions, Fractal Surfaces, and Wavelets

This definition is equivalent to: ϕ ∈ C∞ is in S iff ∀α ∈ Nn ∀m ∈ N∃c ∈ R : |Dα ϕ| ≤

c . (1 + x )m

(1.71)

It follows from Eq. (1.70) that p(x)(Dα ϕ)(x) is bounded in Rn for all real-valued polynomials p. Convergence in S is defined as follows: If {ϕk }k∈N ⊂ S , then limk→∞ ϕk = 0 in S means: 1. ∀α, β ∈ Nn ∃cα,β ∈ R ∀k ∈ N ∀x ∈ R : |xβ Dα ϕ| ≤ cα,β . 2. ∀α ∈ Nn : limk→∞ (Dα ϕ)(x) = 0, uniformly in x ∈ Rn .

A linear continuous functional f defined on S is called a tempered distribution. The set of all tempered distributions forms a linear space denoted by S  . For f (ϕ) ∈ R one sometimes writes f , ϕ . If f : Rn → R is a locally integrable function satisfying f (x) ≤ c(1 + x )m for all m ∈ N and some constant c > 0, then the integral  f , ϕ =

Rn

f (x)ϕ(x)dx

exists and is a continuous linear functional, and hence a tempered distribution. In this case, f is called a regular tempered distribution. Convergence in S  is defined as follows: lim fk = f

k→∞

in S  ⇐⇒ ∀ϕ ∈ S : lim fk (ϕ) = f (ϕ). k→∞

(1.72)

For ϕ ∈ S and f ∈ S  , define the convolution of f with ϕ to be the (ordinary) function f ∗ ϕ given pointwise by (f ∗ ϕ)(ξ ) := f , ϕ(ξ − ·) .

(1.73)

If f is a regular tempered distribution, then Eq. (1.73) coincides with the ordinary definition of convolution:  f ∗ ϕ(ξ ) := f (x)ϕ(ξ − x)dx. Rn

For a function ϕ: Rn → R the notation ϕt (x) is means t−n ϕ(x/t) for any t > 0. The Fourier transform F : S → S is defined by  F ϕ(ξ ) :=  ϕ (ξ ) :=

Rn

ϕ(x)e−ix,ξ dx,

(1.74)

Mathematical preliminaries

49

where dx denotes the n-dimensional Lebesgue measure. The following fact, which is easy to establish, is well known (f ∗ ϕ)=  f ϕ. Now a definition of Besov spaces in terms of elements in S and S  is given. Using Calderón’s formula, one can show that this definition is equivalent to the one given earlier (for more details, see Ref. [20, 24]). ϕ ⊂ {ξ : 12 ≤ ξ ≤ 2} and  ϕ ≥ c > 0 To this end, choose a ϕ ∈ S so that supp  3 5 if 5 ≤ ξ ≤ 3 (the choice of these bounds is conventional, there are other choices that work as well). For α ∈ R, p, q ∈ (0, ∞], and f ∈ S  , we define 

f B˙ α,q := p



2 ϕ2−k ∗ f p kα

q

.1/q .

(1.75)

k∈Z

For ϕ ∈ S and f ∈ S  the convolution ϕ2−k ∗ f is a smooth function, and f B˙ α,q = 0 iff p k  ϕ2−k ∗ f is the zero function for all k ∈ Z. This, however, is equivalent to f ·  ϕ (2 ξ ) = 0 for all k ∈ Z. With use of the preceding conditions on  ϕ this is, in turn, equivalent to supp f = {0}, which implies that  f is a finite linear combination of the Dirac delta distribution δ and its derivatives. Thus f is a polynomial (recall that the Dirac delta distribution δ is defined on Schwartz functions by δ(ϕ) := ϕ(0)). Hence if f is regarded as an element of S /P , where P denotes the class of α,q is a Banach space norm, and B˙ p , the homogeneous Besov polynomials, then f B˙ α,q p is finite. We space, consists of all those functions in S /P whose norm · B˙ α,q p α,q obtain the corresponding inhomogeneous version Bp by defining a Banach space norm by  :=

∗ f p +

f Bα,q p



2 ϕ2−k ∗ f p kα

q

.1/q ,

(1.76)

k∈N

 ⊂ {ξ ∈ Rn : ξ ≤ 2} and

(ξ ) ≥ where ∈ S is chosen so that supp 5 (0) = 0, the need to consider distributions modulo c > 0 for all ξ ≤ 3 . Since polynomials disappears. The definition of the above Banach space norm is independent of the choices of ϕ and : different choices lead to equivalent norms. α,q α,q One can associate a sequence space b˙ p with the Besov space B˙ p in the following way. Let Q = Qk, := {x ∈ Rn : 2−k i ≤ xi ≤ 2−k ( i+1 + 1) ∧ i ∈ Nn } denote a dyadic cube in Rn , with k ∈ Z and = ( 1 , . . . , n ) ∈ Zn . The collection of all such dyadic cubes is denoted by Q := {Qk, : k ∈ Z ∧ ∈ Zn }. For a dyadic cube Q ∈ Q, let l(Q) be its side length and |Q| its diameter. Let s = {sQ } be a sequence in

50

Fractal Functions, Fractal Surfaces, and Wavelets

α,q Rn indexed by dyadic cubes in Q. For α ∈ R and p, q ∈ (0, ∞], let b˙ p be the space consisting of all sequences s = {sQ } for which

⎛

s b˙ α,q p

⎛  ⎞q ⎞1/q     ⎟ ⎟ ⎜ ⎜ −(α/n+1/2)  := ⎝ |Q|

sQ χQ  ⎝  ⎠ ⎠ < ∞.  k∈Z l(Q)=2−k

(1.77)

p

To introduce what will later be called the wavelet transform, the following lemma, whose proof can be found in Ref. [20], needs to be stated. ϕ ⊂ {ξ ∈ Rn : 12 ≤ ξ ≤ 2} and  ϕ (ξ ) ≥ c > Lemma 1. Let ϕ ∈ S satisfy supp  3 5 0 for all 5 ≤ ξ ≤ 3 . Then there exists a ψ ∈ S with the same properties as ϕ such that  (2−k ξ ) = 1 for ξ = 0.  ϕ (2−k ξ )ψ (1.78) k∈Z

Using this lemma and Calderón’s formula, one derives the following wavelet decomposition of f ∈ S  (see Ref. [20] for more details): f =

 f , ϕQ ψQ ,

(1.79)

Q

with ϕ and ψ as in Lemma 1. Here we set ϕQ := 2kn/2 ϕ(2k · − )

(1.80)

for Q = Qk, a dyadic cube, and similarly for ψ. The next theorem relates distributions α,q in S  /P to sequences in b˙ p . Theorem 32. Let α ∈ R and let p, q ∈ (0, ∞]. Suppose that ϕ and

ψ are as in Lemma 1. Assume that for f ∈ S  /P we have the decomposition f = Q f , ϕQ ψQ . α,q α,q Let sQ := f , ϕQ for each dyadic cube Q. Then f ∈ B˙ p iff s = {sQ } ∈ b˙ p , and ≈ s b˙ α,q .

f B˙ α,q p p Proof. See Ref. [20]. Next Theorem 32 is rephrased in terms of retractions and retracts and the ϕα,q transform is introduced. For this purpose, suppose that f ∈ B˙ p and ϕ is as in Lemma 1. α,q α,q For Q ∈ Q, let sQ := f , ϕQ , and define a mapping Wϕ : B˙ p → b˙ p by Wϕ f := s = {sQ }.

(1.81)

This mapping is called the ϕ-transform or the wavelet transform of f . The left ˙ α,q ˘ ψ : b˙ α,q inverse W p →B p of Wϕ is defined by ˘ ψ s := W

 Q∈Q

sQ ψQ .

(1.82)

Mathematical preliminaries

51

˘ ψ are bounded (for the latter this follows from It can be shown that Wϕ and W Theorem 32 and for the former from estimates involving the so-called atomic ˘ ψ ◦ Wϕ is seen to be the identity operator on B˙ α,q decomposition of f ). The mapping W p . ˘ ψ is a retraction, with B˙ α,q Moreover, with use of Definition 36, it follows that W p being α,q α,q α,q α,q the retract of b˙ p . Therefore, B˙ p is isomorphic to Wϕ B˙ p ⊂ b˙ p . This observation α,q can be used to characterize bounded linear operators on B˙ p in terms of the lift of Wϕ α,q to the space of linear operators on B˙ p . For more details, see Ref. [20]. Notice that, by our setting ϕ = ψ in Eq. (1.79), the resulting decomposition of f resembles a Fourier series. This resemblance can be further exploited by our considering the special case f ∈ L2 (R). An appropriate ψ ∈ S (R) whose dyadic dilates and integer translates ψQ form a complete orthonormal basis of L2 (R) needs to be found. (The dyadic cube Q in this case is, of course, a dyadic interval.) The proof of the following theorem can be found in Ref. [20]. Theorem 33. There exists a function ψ ∈ S (R) such that the family Bψ := {2k/2 ψ(2k · − ) : k, ∈ Z}

1 2 1 2  ⊂ − 8π , − 2π ∪ 2π , 8π , and is an orthonormal basis for L2 (R). Moreover, suppψ 3 3 3 3 thus R xm ψ(x)dx = 0 for all m ∈ N0 . Next the inhomogeneous case is briefly considered. As before it is assumed that ϕ satisfies the hypotheses of Lemma 1 and that satisfies the conditions leading to the . Furthermore, it is assumed that ϕ also definition of the Banach space norm · Bα,q p satisfies  (ξ ) +

 ϕ (2k ξ ) = 1. k∈N

Then Lemma 1 implies the existence of two functions ψ and  satisfying the same conditions as ϕ and , respectively, as well as (ξ )  (ξ ) +



(2−k ξ ) = 1 for ξ = 0.  ϕ (2−k ξ )ψ

(1.83)

k∈Z

For an f ∈ S  , the decomposition f =

 l(Q)=1

f , Q Q +



f , ϕQ ψQ

(1.84)

l(Q) 0. For ε > 0, define  Hεs (E)

:= inf

∞  i=0

 |Ui | : {Ui }i∈N0 countable ε-cover . s

Construction of fractal sets

55

The Hausdorff s-dimensional outer measure of E is then given by Hs (E) := lim Hεs = sup Hεs . ε→0

(2.3)

ε>0

A few remarks are in order now. Remark 13 1. Since Hεs increases as ε decreases, the limit in Eq. (2.3) exists, but may be infinite. 2. The restriction of Hs to the σ -algebra of Hs -measurable sets is referred to as the s-dimensional Hausdorff measure. 3. Suppose that X = Rn . If s = 1, then the one-dimensional Hausdorff measure is indeed the one-dimensional Lebesgue measure. For s = n, n ∈ N, n > 1, the n-dimensional Lebesgue measure is a constant multiple of the n-dimensional Hausdorff measure (the reason for this lies in the fact that the Lebesgue measure is defined in terms of the diameters of parallelepipeds). Therefore the s-dimensional Hausdorff measure generalizes the (integral) Lebesgue measure.

Now suppose that X = Rn . For any set E ⊆ Rn , Hs (E) is a nonincreasing function of s. If s < s, then 



Hs (E) ≤ ε s−s Hs (E), 

which implies that Hs (E) = ∞ if Hs (E) > 0. Hence there exists a unique number s∗ such that  ∞ if 0 ≤ s < s∗ ; s H (E) = 0 if s∗ < s < ∞. This unique value s∗ is called the Hausdorff (Hausdorff-Besicovitch) dimension of E. From now on, dimH E denotes the Hausdorff-Besicovitch dimension of a set E. It is important to note that Hs (E) at s = dimH E may be zero, finite, or infinite. A subset E of Rn is called an s-set if E is Hs -measurable and 0 < Hs (E) < ∞. As the n-dimensional Lebesgue measure of a set E scales as λn when E is magnified by a factor λ > 0, one might expect that the s-dimensional Hausdorff measure of a set scales as λs . The next proposition confirms this. Proposition 23. Let E ⊆ Rn and let λ > 0. Let λE denote the set {λe: e ∈ E}. Then Hs (λE) = λs Hs (E).

Proof. The proof is an exercise in applying the definition of the Hausdorff measure. A more general result relating the s-dimensional Hausdorff measure of a set to its image under a certain type of transformation can be proven. But first the following definition is needed.

56

Fractal Functions, Fractal Surfaces, and Wavelets

Definition 53. A function f : E ⊆ Rn → Rm satisfies a Hölder condition on E with exponent α > 0 iff there exists a positive constant c such that for all x, x ∈ E,

f (x) − f (x ) ≤ c x − x α .

(2.4)

When α = 1, f is called a Lipschitz function. A function f : E ⊆ Rn → Rm is said to belong to the class Lipα (E, Rm ), 0 < α ≤ 1, iff 

f (x) − f (x )

 : x, x ∈ E < ∞. sup

x − x α 

(2.5)

Proposition 24. Suppose that E ⊆ Rn and that f satisfies a Hölder condition on E with exponent α ∈ (0, 1]. Then for all s > 0, Hs/α (fE) ≤ cs/α Hs (E).

Proof. Let {Ui }i∈N0 be an ε-cover of E. Note that f (E ∩Ui ) ≤ c|Ui |, and therefore {f (E ∩ Ui )} is a c εα -cover of f (E). Hence 

f (E ∩ Ui ) s/α ≤ cs/α

i∈N0



|Ui |s .

i∈N0

Taking the infimum over all ε-covers and then the supremum over all ε > 0 yields the result. Now reconsider the classical Cantor set C. As seen earlier the Lebesgue measure fails to describe the Cantor set. However, with use of the Hausdorff measure the following well-known fact can be proven. Proposition 25. The Hausdorff-Besicovitch dimension of the Cantor set C is given 2 log 2 by log log 3 , and its log 3 -dimensional Hausdorff measure equals 1. Remark 14. The formula for the Hausdorff dimension of the Cantor set is derived later as a special case of a more general result. The Sierpi´nski gasket or Sierpi´nski triangle provides a two-dimensional example of a fractal set that was known long before the term fractal was introduced. This set had its origin in point-set topology, in particular in the theory of connectivity of sets. It was introduced by Sierpi´nski [30] in 1915. Its construction proceeds as follows: Let be the triangle with vertices (0, 0), (1, 0), and ( 12 , 1). Partition into four congruent subtriangles 1 , . . . , 4 and delete the center subtriangle 3 . Applying the above procedure to each of the remaining three subtriangles yields nine triangles 12 , 13 , 14 , . . ., 42 , 43 , 44 , together with the union (2) := 11 ∪ 21 ∪ 31 ∪ 41 of their deleted centers. The indefinite continuation of this process then yields a sequence { (k) }, where (k) denotes the union of the 3k−1 deleted centers after step k. The Sierpi´nski triangle is defined as ∞  S := \ (k) and is displayed in Fig. 2.1. k=1

Construction of fractal sets

57

1.0

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

Fig. 2.1 The Sierpi´nski triangle.

The one-dimensional Lebesgue measure of S is infinite, and the two-dimensional Lebesgue measure is zero. The set S is a connected subset of R2 containing no area. It is a consequence of the construction of S that any one of its subparts is similar to the entire figure itself. Also, note that S can be obtained as the union of successive applications of the three maps fi : → , i = 1, 2, 3, x y

, , 2 2 x+1 y f2 (x, y):= , , 2 2  x 1 y+1 + , , f3 (x, y):= 2 4 2

f1 (x, y):=

(2.6) (2.7) (2.8)

to each of their respective images. Later it is shown that the Hausdorff-Besicovitch 3 dimension of S is equal to dimH S = log log 2 .

1.2 Weierstraß-like fractal functions In connection with his work on infinite function series, Weierstraß [31] introduced in 1872 a function that is continuous but nowhere differentiable. Such functions are now

58

Fractal Functions, Fractal Surfaces, and Wavelets

called fractal functions since their graphs are in general fractal sets. Chapter 3 will focus entirely on these functions and their properties. Weierstraß’s (fractal) function W is given by the infinite series W(x) :=



λi(s−2) sin(λi x),

i∈N

with parameters 1 < s < 2 and λ > 1. In the 1930s, Besicovitch and Ursell [32] introduced a class of fractal functions f : R → R and calculated the HausdorffBesicovitch dimension of their graphs. Let us briefly summarize their results. Theorem 34. Suppose f ∈ Lipα (R) and let G := graph f . Then 1 ≤ dimH G ≤ 2 − α. Here Lipα (R, R) =: Lipα (R). Note that this result implies that if f has a finite derivative at all points x ∈ R, then dimH G = 1. Besicovitch and Ursell considered the following special class of functions in Lipα (R). Let φ: R → R be given by  φ(x) :=

2x for 0 ≤ x ≤ 12 ; φ(−x) = φ(x + 1) otherwise.

Define functions fφ : R → R by fφ (x) := φ(x) +



b−α i φ(bi x)

(2.9)

i∈N

for 0 < α < 1, bi > 0, i ∈ N. Besicovitch and Ursell [32] showed that if bi+1 ≤ Bbi for B > 1 and all i ∈ N, then fφ ∈ Lipα (R), but in no higher Lipschitz class. The following theorem gives the equality between 2 − α and dimH under certain conditions on α and the bi . Theorem 35 (Besicovitch-Ursell Theorem). Let fφ be defined as in Eq. (2.9) and let Gφ be its graph. Assume that the sequence {μi : i ∈ N} is such that μi ≥

1 − α 2 − dimH G · α dimH G − 1 μ

for all i ∈ N. Let b1 > 1 and bi+1 := bi i , i ∈ N. If limi→∞ bi+1 /bi = ∞ and μi → 1 as i → ∞, then dimH G = 2 − α. In Chapter 5 we will consider the functions fφ using the concept of an iterated function system (IFS).

Construction of fractal sets

59

2 Iterated function systems In this section the concept of an IFS is introduced. These IFSs are used to construct special fractal sets—namely, fractal functions (see Chapter 5) and fractal surfaces (see Chapter 9). IFSs are first defined via continuous operators on C(X), the Banach space of all continuous real-valued functions on a compact and metrizable topological space X endowed with the Chebyshev norm f := sup{|f (x)|: x ∈ X}.

2.1 Definition and properties of iterated function systems Let (X, d) be a complete metric space and let w := {wi : X → X}i∈NN be a collection of Borel measurable functions on X. Let p := {pi }i∈NN denote a set of probabilities. Define an operator T: C(X) → RX by (Tf )(x) :=

N 

pi (f ◦ wi )(x).

(2.10)

i=1

The following definition of an IFS is due to Barnsley and Demko [3]. Most of the results presented in this section can be found in Ref. [3]. Definition 54. The triple (X, w, p) is called an IFS (on X) with probabilities iff the associated set of probabilities p is such that the operator T defined in Eq. (2.10) maps C(X) into itself. An IFS with probabilities (X, w, p) is called contractive or hyperbolic with contractivity s iff there exists a constant 0 ≤ s < 1 such that  sup

 d(wi (x), wi (x ))  ∈ X ≤s : x, x d(x, x )

(2.11)

for i ∈ NN . Remark 15 1. If wi ∈ C(X) for all i ∈ NN , then (X, w, p) is an IFS with probabilities for any set of associated probabilities p. In this case the IFS is written as (X, w) and is referred to simply as an IFS. 2. The operator T clearly depends on w and p and should be more precisely denoted by T(w, p). However, to ease the notation, this less exact designation was chosen.

Let H(X) denote the collection of all nonempty compact subsets of the set X. The function h: H(X) × H(X) → R defined by   h(A, B) := max max min d(a, b), max min d(b, a) a∈A b∈B

b∈B a∈A

(2.12)

can be shown to be a metric on H(X). This metric is called the Hausdorff metric for X. If (X, d) is complete, then (H(X), h) also becomes a complete metric space, called

60

Fractal Functions, Fractal Surfaces, and Wavelets

the hyperspace of compact subsets (of the complete metric space X). The following elementary properties of h will be used later. For further details about this hyperspace and the Hausdorff metric, see Ref. [33]. Proposition 26 1. Suppose f : X → X is Lipschitz with Lipschitz constant s. Then for all A, B ∈ H(X), h(fA, fB) ≤ s h(A, B).

(2.13)

2. Let {Ai : i ∈ Nn } and {Bi : i ∈ Nn } be finite collections of sets in H(X). Then ⎛ h⎝

n 

n 

Ai ,

i=1

⎞ Bi ⎠ ≤ max{h(Ai , Bi ) : i ∈ Nn }.

(2.14)

i=1

One can associate with an IFS (X, w, p) a set-valued map w: H(X) → H(X) by setting w(A) :=

N 

wi (A).

(2.15)

i=1

If (X, w, p) is a contractive IFS with probabilities and contractivity s, one can show that w is a contraction on the complete metric space (H(X), h) with the same contractivity s (this follows from the properties of h stated in Proposition 26). Next the class of fractal sets that will be used most frequently are defined. Definition 55. Let (X, w, p) be a contractive IFS with probabilities, let w be the associated set-valued map (Eq. 2.15), and let E ∈ H(X). The attractor or fractal (set) generated by the contractive IFS (X, w, p) with probabilities is the unique compact set A defined by A := lim wn (E), n→∞

(2.16)

where the limit is taken in the Hausdorff metric. Remark 16. Eq. (2.16) has an algebraic interpretation as well: the attractor A is that set which is invariant under the semigroup S[w] generated by the maps wi , i ∈ NN . This interpretation will be taken up again in Chapter 6 when we consider fractels. That every contractive IFS (with probabilities) has an attractor and that this attractor is unique and independent of the set E follows from the Banach fixed-point theorem applied to the contraction w and the complete metric space (H(X), h). Furthermore, the uniqueness part of the theorem does not exclude the possibility that there exist sets A not in H(X) satisfying w(A) = A (eg, the empty set is an example). Note that one can also define an attractor for only an IFS with probabilities; however, this attractor does in general depend on the set E (the existence of an attractor in this case is guaranteed by the Schauder fixed-point theorem).

Construction of fractal sets

61

Also note that the attractor depends in a subtle way on the set of associated probabilities, even in the case where the wi are continuous. This dependence will be studied shortly. Two attractors of contractive IFSs have already been encountered. Example 9 (The Classical Cantor Set C). For this example, X = [0, 1] and d, the canonical metric in R, is restricted to [0, 1]. Choose as the wi the functions fi , i = 1, 2, given by Eqs. (2.1), (2.2), respectively. Since (X, w) is clearly a contractive IFS (with s = 13 ), the attractor, the classical Cantor set in this case, is uniquely determined by application of w to any initial interval [a, b], 0 ≤ a < b ≤ 1. Example 10 (The Sierpi´nski Triangle S). Here, for instance, X = [0, 1] × [0, 1] and wi = fi , i = 1, 2, 3, as defined in Eq. (2.6)–(2.8). It is easy to verify that (X, w) is a contractive IFS with s = 12 , and that the attractor, the Sierpi´nski triangle, is uniquely and independently determined by any nonempty compact set E ⊆ [0, 1] × [0, 1]. Given an IFS (X, w, p) with probabilities on X, it is sometimes useful to work with a compact subset K of X instead of using X itself. The following proposition (see Ref. [34]) shows that this can be easily achieved. Proposition 27. Suppose that K ⊆ X is compact. Then there exists a  K ∈ H(X) such that  K ⊇ K and w( K) ⊆  K.  i Proof. Set  K := clH(X) w (K). Then  K is compact, and by definition i∈N0

w( K) ⊆  K. Now let (X, w, p) be an IFS with probabilities. Consider the following discrete-time Markov process (X, B(X), μ, P), where B(X) denotes the algebra of Borel subsets of X, and where P(x, B) :=

N 

pi χwi (x) (B).

(2.17)

i=1

Recall that the dual of C(X) is the set of all (finite) regular Borel measures M(X) := MR (X). The operator T as defined in Eq. (2.10) maps the space C(X) continuously into itself, and thus the adjoint operator T ∗ maps the set P (X) of probability measures on X taking values in R weak*-continuously into itself. By the Schauder fixed-point theorem T ∗ has a fixed point μ ∈ P (X). Furthermore, ∗

(T ν)B =

N 

pi (w#i

◦ ν)B =

N 

i=1

i=1

 pi X

χwi (x) dν(x),

(2.18)

where (w#i ◦ ν)B := ν(w−1 i B). Therefore the fixed point μ satisfies μB =

N  i=1





pi X

χwi (x) dμ(x) =

P(x, B)dμ(x); X

(2.19)

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Fractal Functions, Fractal Surfaces, and Wavelets

that is, μ is the stationary probability measure for the Markov process defined above. This measure has a special name. Definition 56. The measure μ defined in Eq. (2.19) is called the p-balanced measure of the IFS (X, w, p) with probabilities. The p-balanced measure of a contractive IFS (X, w, p) with probabilities is also attractive for probability measures ν ∈ M(E), where E ⊆ X. More precisely, w∗

T ∗n ν −→ μ

as n → ∞

(2.20)

for all probability measures ν with support in E ⊆ X. To show this, the so-called Hutchinson metric d on the space P (X) ⊆ M(X) of probability measures on X must be introduced. Definition 57. Let μ, ν ∈ P (X) and denote by Lip(≤1) (X) the set of all Lipschitz functions φ: X → R with Lipschitz constant less than or equal to 1. Define d(μ, ν) := sup{μ(φ) − ν(φ): φ ∈ Lip(≤1) (X)}.

(2.21)

 Here the notation μ(φ) = X φ(x) dμ(x) was used. That d is indeed a metric is left as an exercise for the reader. Lipschitz functions map bounded sets to bounded sets, and thus φ ∈ Lip(≤1) (X) is also an element of BC(X), the collection of all continuous functions f : X → R that are bounded on bounded subsets of X. Moreover, the convergence in the topology induced by the d-metric implies convergence in the topology of M(X). It is left to the reader to establish the next proposition. Proposition 28. The linear space (P (X), d) is a complete metric space. The Hutchinson metric was designed to obtain the following result. Proposition 29. The operator T ∗ : P (X) → P (X), given by (T ∗ ν)B =

N 

pi (w#i ◦ ν)B,

(2.22)

i=1

is a contraction with contractivity s. Proof. Let μ, ν ∈ P (X), and let φ ∈ Lip(≤1) (X). Then d(T ∗ μ, T ∗ ν) = =

N  i=1 N 

pi (w#i ◦ μ)(φ) −

N 

pi (w#i ◦ ν)(φ)

i=1

pi (μ(φ ◦ wi ) − ν(φ ◦ wi )).

i=1

Now, recall that sup{d(wi (x), wi (x ))/d(x, x ): x, x ∈ X} ≤ s for i ∈ NN . Thus

Construction of fractal sets

d(T ∗ μ, T ∗ ν) = ≤

63

N  i=1 N 

pi s(μ(s−1 φ ◦ wi ) − ν(s−1 φ ◦ wi )) pi s d(μ, ν) = s d(μ, ν).

i=1

Here the fact that the Lipschitz constant of s−1 φ◦wi is less than or equal to s−1 ·1·s = 1 was used. The results obtained above are now summarized in a theorem. Theorem 36. Let (X, w, p) be a contractive IFS with probabilities, and let A be its attractor. Then there exists a unique probability measure μ on P (A) such that T ∗ μ = μ, where T ∗ is given in Eq. (2.22). Moreover, every probability measure ν ∈ P (X) is attracted to μ in the sense that T ∗n ν → μ, where the convergence is with respect to the Hutchinson metric d. Furthermore, μ is also the stationary probability measure for the discrete-time Markov process (Eq. 2.17). At this point it is natural to ask what the support of the measure μ is and how it is related to the attractor A. The setting in which these questions can be answered most easily involves the concept of a code space associated with an IFS with probabilities. Recall from Chapter 1, in particular Section 2, the definition of state space, code, a cylinder set, and the Fréchet metric; see Eqs. (1.39), (1.40), and (1.43). Here the same notation and terminology as in Section 2, is used; that is, the code space is defined by

:= NN N,

(2.23)

and the codes are given by i := (i1 i2 . . . in . . .),

(2.24)

where each in ∈ NN . Furthermore, recall that ( , dF ) is a compact metric space. If left-shift maps τi : →

τi (i1 i2 . . . in . . .) := (i i1 i2 . . . in . . .)

(2.25)

for all i ∈ NN are defined, then the pair ( , τ ) is a contractive IFS (for any choice of probabilities p). To see this, let us show that each map τi is Lipschitz with Lipschitz constant less than 1. Let i, i ∈ . Then dF (τi (i), τi (i )) = =

∞  |(τi (i))j − (τi (i ))j |

(N + 1)j

j=1 ∞ 

|ij − ij |

j=2

(N + 1)j


0 be given. Choose an integer n so large that sn |K| < ε. If i, i ∈ are such that they agree through n terms—that is, dF (i, i ) =

∞ |i − i |  j j j=n

(N

+ 1)j

then |γ (i) − γ (i )| < ε.


0 and E ∩ A = ∅. Then there would exist a function g ∈ CR (X) with range equal to [0, 1] and the property that  1 on E; g≡ 0 on an open set O ⊃ A.

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Fractal Functions, Fractal Surfaces, and Wavelets

Hence  κE ≤

 f dκ = =

f d(T ∗n κ) =



pi1 pi2 · · · pin

 T n f dκ  f (wi(n) (x)) dκ(x),

where the sum is taken over all (i1 i2 . . . in ). Theorem 37 implies that  wi(n) (x) → A uniformly in x ∈ X as n → ∞. Thus f (wi(n) (x)) → 0, and therefore f dκ = 0. This contradiction now shows that if κE > 0, then E ⊆ A. Our next goal is to describe the relationship between the p-balanced measure μ of a contractive IFS with probabilities (X, w, p) and the p-balanced measure ρ of the contractive IFS ( , τ . This will be done by “lifting” the surjection γ to the function space of continuous mappings. More precisely, let γ :  A be the surjection defined in Theorem 37. The contravariant linear operator : C(A) → C( ) defined by (f )(i) := f (γ (i))

(2.30)

is injective and has norm 1. Also, 1 = 1. Hence the mapping f → f

(2.31)

is well defined and a continuous linear surjective mapping of (C(A)) onto C(A). Now suppose ν ∈ M(A). Define κ0 : C( ) → R by κ0 (f ) := ν(f ); that is, κ0 is the composition of the mapping defined in Eq. (2.31) with ν. Thus κ0 is a continuous linear functional on (C(A)), and by the Hahn-Banach extension theorem there exists an extension κ ∈ M( ). Therefore ν = κ0 ◦  = κ ◦  =  ∗ κ, which shows that the adjoint  ∗ of  is a surjection from M( ) onto M(A). With use of the order-theoretic formulation of the Hahn-Banach extension theorem [36, Section 32], more can be shown: namely, that  ∗ maps P ( ) onto P (A). Now let  T be the restriction of T to C(A). Theorem 37 implies the commutativity of the following diagram: γ

−−−−→ ⏐ ⏐τ i γ

A ⏐ ⏐w  i

−−−−→ A

(2.32)

Construction of fractal sets

67

Recalling Definitions 2.10, 2.26, and 2.30, one thus obtains (T)(f )(i) = (Tf )γ (i) =

N 

pi (f ◦ wi )γ (i) =

i=1

N 

pi (f ◦ γ )τi (i)

i=1

= f (γ (i)) = (f )(i) = ( )(f )(i). Hence for any integer 1 < n ∈ N, this yields T n = n , and also T ∗n  ∗ =  ∗ ∗n . Given any κ ∈ P (A), there exists a ν ∈ P ( ) such that κ =  ∗ ν, and T ∗n κ = T ∗n  ∗ ν =  ∗ ∗n ν →  ∗ ρ, the p-balanced measure of the contractive IFS ( , τ ). Thus  ∗ ρ is the unique fixed point if T ∗ is in P (A) and by our earlier results also in P (X). The objective to characterize the relation between the p-balanced measure μ of a contractive IFS with probabilities (X, w, p) and the p-balanced measure ρ of the contractive IFS ( , τ ) has now been achieved. The above arguments yield the following theorem. Theorem 38. Suppose (X, w, p) is a contractive IFS with probabilities. Then the p-balanced measure μ is given by μE = (γ # ◦ ρ)E for all E ∈ F . Furthermore, the support of μ is the attractor A, independent of p.

2.2 Moment theory and iterated function systems Theorem 36 may be used to find a recursion relation for the moments of a certain class n of IFSs. For illustrative purposes it is assumed  that X is a compact subset of (K , | · |), where | · | is the metric given by |z| := j∈Nn zi zi . Define affine maps wi : X → X by wi (z) := Ai z + ζi for all i ∈ NN , where Ai is the diagonal matrix ⎛

ai1 ⎜0 ⎜ Ai := ⎜ ⎝0 0

0 ai2 0 0

... ... .. . ...

⎞ 0 0⎟ ⎟ ⎟, 0⎠ ain

(2.33)

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Fractal Functions, Fractal Surfaces, and Wavelets

where aij ∈ K with |aij | < 1 for all i ∈ NN and all j ∈ Nn , and ζi ∈ Kn . It follows from Eq. (2.33) that wi is a contraction with constant of contractivity c ≤ max max{|aij |: i ∈ NN ∧ j ∈ Nn } < 1. Hence (X, w) is a contractive IFS for any choice of probabilities p. Its p-balanced measure is denoted by μ. By Theorem 36,  f dμ =

N 

X

 f ◦ wi dμ

pi

i=1

(2.34)

X

for all f ∈ CK (X); that functions in CK (X) as well as functions in C(X) = CR (X) can be used follows from previous results. To proceed the following definition is needed. Definition 58. Let z ∈ Kn and let α ∈ Nn0 := (N0 )n . For any probability measure ν ∈ P (X), the αth moment of ν is defined by  zα dν(z).

M(ν, α) :=

(2.35)

X

Remark 18. In Definition 58 multiindex notation has been used. For z = (z1 , . . . , zn )t ∈ Kn and α = (α1 , . . . , αn )t ∈ Nn0 , set zα :=

n 

α

zj j .

j=1

Here superscript t denotes the transpose. If α, β ∈ Nn0 , define α ≤ β ⇐⇒ ∀j ∈ N: αj ≤ βj . Furthermore, let α − β := (α1 − β1 , . . . , αn − βn )t . Since zα ∈ CK (X), Eq. (2.34) can be used with f (z) := zα . Choosing α ∈ Nn , the αth moment of the p-balanced measure μ of the above-defined contractive IFS is then given by  zα dμ(z) =

M(μ, α) :=

  n

X

X j=1

α

zj j dμ(z) =

Each factor in the preceding equation reduces to [aij zi + ζij ]αj =

αj   αj βj =1

βj

β

α −βj βj zj ,

aijj ζij j

N  i=1

pi

  n X j=1

[aij zi + ζij ]αj dμ(z).

Construction of fractal sets

69

and thus using the definition of moments, one obtains M(μ, α) =

N  i=1

pi aαi

 α β α−β · M(μ, α) + · M(μ, β), a ζ β i i 0≤β dimb E. Since for the calculation of the box dimension of a set E covers Wε (E) consisting of disjoint balls of radius ε > 0 whose centers are in E can be used, one might try to define an analog of the Hausdorff measure and the Hausdorff dimension for these dense packings of E by disjoint balls. n Therefore let ε > 0 be given, let s ∈ R+ 0 , and let E be a set in R . Define    s s Pε (E) := sup |Wi | : Wi ∈ Wε (E) (3.21) i

(see Eq. 3.11). As Pεs (E) decreases as ε decreases, the limit s (E) := lim Pεs (E) P ε→0+

(3.22)

Dimension theory

111

s (E) is not a measure on Rn but one can be obtained by our setting exists. However, P  P (E) := inf s



s (Ei ) : E ⊆ P

i

∞ 

 Ei .

(3.23)

i=1

This measure is called the s-dimensional packing measure on Rn . Now one proceeds in the usual way to define the associated dimension. Let + s s dimP E := sup{s ∈ R+ 0 : P (E) = ∞} = inf{s ∈ R0 : P (E) = 0}.

(3.24)

This dimension is called the packing dimension of E. In the next section it will be related to the box dimension.

3 Probabilistic dimensions The previous sections dealt primarily with dimension from a geometric point of view. However, as already seen, a fractal has a much richer structure; namely, it is the support of an invariant measure. It is therefore natural to look at the “size” of the support of this measure when one is defining a dimension for the fractal. Suppose then that A is the attractor for an iterated function system (IFS) (with probabilities). As before, the p-balanced measure of the IFS is denoted by μ. Recall that μA = 1. Definition 86. The Hausdorff dimension of the measure μ is defined by dimH μ := inf{dimH E: E ⊆ A ∧ μE = 1}.

(3.25)

Remark 30. Since A = supp(μ) it is clear that dimH μ ≤ dimH A. In general one may have strict inequality. However, equality of both dimensions is assured if there exist numbers s > 0 and c > 0 such that for all sets C μ(E ∩ C) ≤ c|C|s . To see this let {Cν } be any cover of A. Then  0 < μA = μ

 ν

 E ∩ Cν

≤

 ν

μCν ≤ c



|Cν |s ,

ν

and thus choosing ε > 0 small enough and taking the infimum, we have Hεs (A) ≥ μA/c. Hence Hs (A) ≥ μA/c, and so dimH A ≥ s, implying dimH μ ≥ s. The Hausdorff dimension of a measure will be considered again in Chapter 4 when dynamical systems are introduced. Another type of a probabilistic dimension was introduced by Billingsley [55, 56]. One starts with a probability space (, B, μ) on which there is defined a stochastic

112

Fractal Functions, Fractal Surfaces, and Wavelets

process {Xn }n∈N0 with countable state space S and one defines an outer measure analogous to the Hausdorff outer measure but one allows only cylinders defined by {Xn }n∈N0 as covering sets. We establish that for  = [0, 1] and under a certain condition the Billingsley dimension agrees with the Hausdorff dimension. A set of the form Z := {ω ∈  : Xn (ω) = in ∧ in ∈ S ∧ n ∈ NN } is called an N-cylinder of . The unique cylinder Zn that contains ω ∈  is denoted by s Zn (ω). Let s be any nonnegative number. Define an s-dimensional outer measure HB of a subset 0 ⊆  by  s HB (0 )

:= lim inf ε→0+

∞ 

μZis :

0 ⊆



 Zi ∧ Zi N-cylinder ∧ μ(Zi ) < ε .

n=1

Note that if 0 does not possess a covering of the required form, then inf ∅ := ∞. Definition 87. The Billingsley dimension of a subset 0 of  is defined by s dimB 0 := sup{s: HB (0 ) = ∞}.

(3.26)

Furthermore, if it is assumed that limn→∞ μZn (ω) = 0 holds for all ω ∈ , then 0 ≤ dimB 0 ≤ 1 for all 0 ⊆  and dimB 0 = 1

if μ∗ (0 ) > 0,

where μ∗ denotes the outer measure corresponding to μ. Next it is shown that, by our defining an appropriate semimetric on , the Hausdorff dimension induced by this semimetric agrees with the Billingsley dimension. Consider the collection of cylinders Zn Z := {Zn (ω): μZn (ω) > 0,

∀n ∈ N∀ ω ∈ }

and denote by X the topology on  generated by these cylinders. Furthermore, for all ω1 , ω2 ∈  define a mapping d:  ×  → R+ 0 by d(ω1 , ω2 ) := inf{μZ: Z cylinder ∧ ω1 , ω2 ∈ Z}.

(3.27)

Then the following result holds [57]. Theorem 56. Suppose that (, B, μ) is a probability space and {Xn }n∈N0 is a stochastic process satisfying the condition that limn→∞ μZn (ω) = 0 holds for all

Dimension theory

113

ω ∈ . Then the function d defined by Eq. (3.27) is a semimetric on  and the topology X is semimetrizable with respect to d. Furthermore, for all 0 ⊆ , dimB 0 = dimH 0 , where dimH denotes the Hausdorff dimension induced by the semimetric d. Proof. Conditions (a) and (b) for a semimetric (Definition 4) are readily verified. To show the triangle inequality, assume that d(ω1 , ω2 ) > 0. Then there is a cylinder / Zn+1 (ω1 ). Hence d(ω1 , ω2 ) = μZn (ω1 ). If Zn (ω1 ) such that ω2 ∈ Zn (ω1 ) but ω2 ∈ / Zn+1 (ω1 ), then d(ω1 , ω3 ) ≥ ω3 ∈ Zn+1 (ω1 ), then d(ω1 , ω3 ) = μZn+1 (ω1 ), and if ω3 ∈ μZn (ω1 ). Therefore d(ω1 , ω2 ) ≤ max{d(ω1 , ω3 ), d(ω2 , ω3 )}. To show that X is semimetrizable by d, it suffices to note that if μZn (ω) > 0, then {ω2 ∈ : d(ω1 , ω2 ) < μZn (ω1 )} ⊆ Zn (ω1 ) ⊆ {ω2 ∈ : d(ω1 , ω2 ) ≤ μZn (ω1 )}, and if μ(Zn (ω1 )) = 0, then Zn (ω1 ) = {ω2 ∈ : d(ω1 , ω2 ) < μZn−1 (ω1 )}. To prove the last statement observe that any cover of 0 by cylinders of diameter μZ < ε is clearly a cover by sets of diameter less than ε. Hence dimH 0 ≤ dimB 0 . The reverse inequality follows from the fact that every covering set of diameter less than ε in any cover of 0 is contained in a cylinder of equal diameter. Thus dimH 0 ≥ dimB 0 . Example 21. Let  := [0, 1], let B be the σ -algebra of Borel sets, and let μ be the Lebesgue measure. Define a stochastic process {Xn } by ω=

∞ 

Xn (ω)2−n ,

n=1

where Xn (ω) = 0 or 1 and Xn (ω) = 1 infinitely often. Define X to be the topology that is generated by all half-open intervals with dyadic endpoints. Then the hypotheses of Theorem 56 are satisfied, and thus the Billingsley dimension agrees with the Hausdorff dimension. Now suppose that  is a nonempty set, B is a σ -algebra of subsets of , and μ and ν are two probability measures on B. Also, let {Xn }n∈N0 and {Yn }n∈N0 be two stochastic processes and let d1 and d2 be the induced semimetrics. Then if Ki (ω, r) := {ω ∈ : di (ω, ω ) < r}, K1 (ω, r) is a cylinder that contains ω. If in addition Xn = Yn for all n ∈ N0 , then d1 (K1 (ω, r)) is the μ-measure and d2 (K1 (ω, r)) is the ν-measure of this cylinder. Proposition 40 gives the next result (see also Ref. [56]).

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Fractal Functions, Fractal Surfaces, and Wavelets

Theorem 57. Let 0 ⊆ . Assume that

log νZn (ω) 0 = ω: lim inf ≥δ . n→∞ log μ(Zn (ω)) Then (2) dim(1) H 0 ≥ δ dimH 0 .

Remark 31. In the preceding theorem the following conventions about ratios of logarithms are used. If 0 < ξ , η < 1, then 1. log ξ/ log 0 = log 1/ log η = log 1/ log 0 := 0; 2. log 0/ log η = log ξ/ log 1 = log 0/ log 1 := ∞; 3. log 0/ log 0 = log 1/ log 1 := 1.

4 Dimension results for self-affine fractal sets This section is concerned with the calculation of the Hausdorff dimension and the box dimension of self-affine fractal sets generated by ordinary and recurrent IFSs. A fractal set F in a complete (semi)metric space (X, d) is called self-affine iff it is generated by a finite collection of maps wi of the form wi (x) := Ai x + vi ,

i ∈ NN ,

(3.28)

for some bounded, linear, and contractive operators Ai : X → X and some vi ∈ X. The collection of all bounded linear operators X → X is denoted by L(X) and the subset of all contractive operators is denoted by L0 (X). The fractal F is called self-similar iff it is constructed with contractive similitudes. Remark 32. If X = Rn it can be shown [2] that S: Rn → Rn is a similitude iff S = Hs ◦ τv ◦ O, for some homothety Hs : Rn → Rn , x → sx, s ∈ R, some translation operator τv : Rn → Rn , x → x + v, and some orthogonal transformation O: Rn → Rn ; that is, an element of the special orthogonal group SO(n). The space of all similitudes S: Rn → Rn will be denoted by Sim(Rn ) and the subspace consisting of contractive similitudes will be denoted by Sim0 (Rn ).

4.1 Dimension of self-similar fractals The first results concerning the Hausdorff dimension of self-similar fractal sets are due to Moran [58]. Moran’s theorem and proof are presented in the language of IFSs in Refs. [2, 54]. Falconer [53, 59, 60] has perhaps the most general theorem for the Hausdorff dimension and the box dimension of self-affine fractal sets generated by

Dimension theory

115

IFSs. It basically states that the Hausdorff dimension and the box dimension of a selfaffine fractal are “equal almost surely,” a notion that will be made more precise later. A number of other dimension results, such as the box dimension of an attractor of a recurrent IFS, are also given [37, 61]. The results are presented in Rn , although most of them hold in (semi)metric spaces. To obtain the formula for the Hausdorff dimension and the box dimension of selfsimilar fractal sets, one has to impose what Hutchinson called the open set condition (OSC). This condition ensures that the components Si (F) of a self-similar fractal F do not overlap “too much.” Definition 88 (OSC). Assume that F is the attractor of a contractive IFS (Rn , S, p) (with probabilities), where S := {Si ∈ Sim0 (Rn ): i ∈ NN }. The family S is said to satisfy the OSC iff there exists a nonempty bounded open set G ⊆ Rn such that N

G⊇

 ˙

Si (G),

(3.29)

i=1

˙ denotes the disjoint union of sets. where ∪ Remark 33. A simple volume argument shows that if S satisfies the OSC, then N 

sni < 1.

i=1

In what follows, we require the next lemma. Lemma 4. Suppose that {Gi } ⊆ Rn is a countable collection of disjoint open sets with the property that each Gi contains a ball of radius ρ1 r and is contained in a ball of radius ρ2 r. Then any ball B of radius r > 0 intersects at most (1 + 2ρ2 )n ρ1−n of the closures cl Gi . Proof. Suppose that cl Gi ∩ B = ∅. Then cl Gi is contained in a concentric ball of radius (1 + 2ρ2 )r. Denote by m the number of sets cl Gi that intersect B. The sum over the volumes of the interior balls of radii ρ1 r yields m(ρ1 r)n ≤ (1 + 2ρ2 )n rn . This gives the required result. Theorem 58. Assume F is the attractor of the contractive IFS (Rn , S, p) and that S satisfies the OSC. Then dimH F = dimb F = d,

(3.30)

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Fractal Functions, Fractal Surfaces, and Wavelets

where d is the unique positive solution of N 

sdi = 1.

(3.31)

i=1

Moreover, F is a d-set. Proof. We notice that the fixed-point property of F under the set-valued map N  S: H(Rn ) → H(Rn ), S(E) := Si (E), implies that i=1



F=

Fi(n) ,

i(n)∈n

where n = {1, . . . , N}Nn (see Section 2 and Eq. 2.29). Since the composition of the similitudes Si(n) is a similitude with contractivity si(n) , Eq. (3.31) implies that 



|Fi(n) |d =

i(n)∈n

⎛



(si(n) )d |F|d = ⎝

i(n)∈n

⎞

⎛



sdi1 ⎠ · · · ⎝

i1

⎞ sdin ⎠ |F|d = |F|d ,

in

where i(n) := (i1 i2 . . . in ). Given any ε > 0, one can always find an n ∈ N large enough such that |Fi(n) |d ≤ (max{si : i ∈ NN })d ≤ ε. Thus Hεd (F) ≤ |F|d , and consequently Hd (F) ≤ |F|d . To obtain a lower bound, a measure ν on n-cylinders Zi(n) := {i ∈ : i = i(n)  j} is introduced, where :  →  denotes the concatenation of codes: (i1 i2 . . . in )  (j1 j2 . . . jm . . .) := (i1 i2 . . . in j1 j2 . . . jm . . .). Define νZi(n) := (si(n) )d .  It follows from Eq. (3.31) that νZi(n) = i νZi(n)∧i and therefore ν = 1. Using Eq. (2.29), we extend ν to a measure  ν on F by setting  ν := γ # ◦ ν. Then  νF = 1.

Dimension theory

117

Let G be the nonempty bounded open set whose existence is guaranteed by the OSC. The fact that every compact set converges (in the Hausdorff metric) to the attractor F implies cl G ⊇ S(cl G) ⊇ · · · ⊇ Sn (cl G) → F. Note that cl G ⊇ F, and therefore cl Gi(n) ⊇ Fi(n) for all i(n), n ∈ N. Now let B be a ball of radius r < 1 intersecting F. Let i ∈  and let n be the first integer for which r min{si : i ∈ NN } ≤ si(n) ≤ r. Denote by  ∗ the set of all such codes. Observe that for any code i ∈  there is exactly one integer n such that i(n) ∈  ∗ . As {G1 , . . . , GN } is disjoint, so is {Gi(n)∧1 , . . . , Gi(n)∧N } for all i(n) ∈ n . Hence the collection {Gi(n) : i(n) ∈  ∗ } is disjoint, and therefore   Fi(n) ⊆ cl Gi(n) . F⊆ i(n)∈ ∗

i(n)∈ ∗

Now choose two positive real numbers ρ1 and ρ2 such that G contains a ball of radius ρ1 and is contained in a ball of radius ρ2 . If i(n) ∈  ∗ , the set Gi(n) contains a ball of radius si(n) ρ1 and thus one of radius (mini∈NN si )ρ1 r, and is contained in a ball of radius si(n) ρ2 and hence in one of radius ρ2 r. Now denote by  ∗∗ the set of all codes in  ∗ for which Gi(n) ∩ B = ∅. By the preceding lemma, there are at most m=

(1 + 2ρ2 )n (ρ1 min{si : i ∈ NN })n

codes in  ∗∗ . Then ⎛  νB =  ν(B ∩ F) ≤ ν{i(n) ∈ : γ (i) ∈ B ∩ F} ≤ ν ⎝



⎞ Zi(n) ⎠ .

i(n)∈ ∗∗

Thus  νB ≤

 i(n)∈ ∗∗

νZi(n) =

 i(n)∈ ∗∗

sdi(n) ≤



rd ≤ mrd .

i(n)∈ ∗∗

As every set U is contained in a ball of radius |U|,  ν(U) ≤ m|U|d . Remark 30 d −1 implies that H (F) ≥ m > 0, and thus dimH F = d. | ∗ |c It remains to be shown that dimb F = d. To this end, notice that the cardinality d ∗ −d −d of  is at most (mini∈NN si ) r . (This follows directly from i(n)∈ ∗ si(n) = 1 and the definition of  ∗ .) Hence if i(n) ∈  ∗ , then |cl Gi(n) | = si(n) |cl G| ≤ r|cl G|.

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Fractal Functions, Fractal Surfaces, and Wavelets

Therefore F can be covered by | ∗ |c sets of diameter r|cl G| for any r < 1. Eq. (3.11) together with Eq. (3.13) then implies that dimb F ≤ d. Hence dimb F = d. Now the packing dimension can be related to the box dimension. To do this, the strong OSC has to be introduced [62]. Definition 89. A collection of similitudes S satisfies the strong OSC iff there exists an open set G ⊆ Rn such that Eq. (3.29) is satisfied and G can be chosen such that G ∩ F = ∅. The following theorem is proven in Ref. [62]. Theorem 59. Suppose that F is the attractor of a contractive IFS whose collection of similitudes satisfies the strong OSC. Then dimP F = dimb F.

(3.32)

4.2 Dimension of self-affine fractals The calculation of the dimension for self-affine fractal sets in Rn is far more difficult. One of the most general results concerning the equality of the Hausdorff dimension and the box dimension for such fractals is due to Falconer [53]. His theorem is stated without our presenting the full proof since it requires potential-theoretic arguments and methods that will not be developed in this monograph. Recall that an affine mapping w: Rn → Rn is of the form w(x) = Ax + v, where A ∈ L(Rn ) ∼ = Mnn (R) and v ∈ Rn . Let L0 (Rn ) be the subspace of L(Rn ) consisting of all nonsingular and contractive linear mappings. Denote the eigenvalues of A∗ A by λ1 , . . . , λn . The singular values of A are defined as αi :=



λi ,

i ∈ Nn .

After they have possibly been reindexed, they are ordered according to 1 > α1 ≥ α2 · · · ≥ αn > 0. For a real number 0 ≤ s ≤ n, define the singular value function ϕ s of A by ϕ s (A) := α1 α2 . . . αr−1 αrs−r+1 ,

(3.33)

where r is that integer for which r − 1 < s ≤ r. It follows that ϕ s (A) is a continuous and strictly decreasing function of s. Proposition 43. For a fixed s ∈ [0, n], ϕ s : L0 (Rn ) → R+ is submultiplicative; that is, for A, B ∈ L0 (Rn ),

Dimension theory

119

ϕ s (AB) ≤ ϕ s (A)ϕ s (B). Proof. The proof is left to the reader. Let k ∈ N, let s ∈ [0, n] be fixed, and let wi , i ∈ NN , be a collection of affine mappings whose linear parts Ai ∈ L0 (Rn ). For a finite code i(k) ∈ k , define 

Ssk :=

ϕ s (Ai(k) ).

(3.34)

i(k)∈k

Then for any m ∈ N, Ssk+m =

 i(k+m)∈k+m

=





ϕ s (Ai(k+m) ) ≤ 

s

ϕ (Ai(k) )

i(k)∈k

ϕ s (Ai(k) )ϕ s (Ai(k+m) )

i(k+m)∈k+m

ϕ (Ai(m) ) = Ssk Ssm . s

i(m)∈m

Hence the sequence {Ssk }k∈N is submultiplicative and therefore {(Ssk )1/k }k∈N converges to a limit Ss∞ . Note that since the function ϕ s is strictly decreasing in s, so is Ss∞ . This then implies that if Sn∞ ≤ 1, there exists a unique s∗ ∈ [0, n] such that ⎛ 1 = Ss∞ = lim ⎝ k→∞



⎞1/k ϕ s (Ai(k) )⎠

.

(3.35)

i(k)∈k

Using the submultiplicativity of Ssk , one can express this last equation also in the form ⎧ ⎫ ∞  ⎨ ⎬  ϕ s (Ai(k) ) < ∞ . (3.36) s∗ = inf s ∈ [0, n]: ⎩ ⎭ k=1 i(k)∈k

Theorem 60. Let F be the attractor of a contractive IFS (Rn , w, p), where w is a collection of affine maps wi (x) = Ai x + vi , with Ai ∈ L0 (Rn ) and vi ∈ Rn , i ∈ NN . Then for almost all (v1 , . . . , vN ) ∈ RnN dimH F = dimb F = s∗ .

(3.37)

Remark 34. “For almost all . . . ” refers to the nN-dimensional Lebesgue measure. Proof. An argument quite similar to the one given in the proof of Theorem 58 yields dimH F ≤ s∗ . The lower bound requires potential-theoretic methods and is therefore omitted. The interested reader may consult [53].

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Fractal Functions, Fractal Surfaces, and Wavelets

4.3 Recurrent iterated function systems and dimension The next objective is the calculation of the box dimension of a fractal set F generated by a recurrent IFS. For this purpose, we follow Ref. [37]. But first it has to be clarified what is meant by the box dimension of a set in XH(X), where X is a compact (semi)metrizable topological space (X, d). Definition 90. Suppose that XE := (E1 , E2 , . . . , EN ) ⊆ XH(X). The box dimension of XE is defined as dimb XE := max{dimb Ek : k ∈ NN }.

(3.38)

Now suppose a contractive recurrent IFS (X, w, P), whose collection of maps w consists of similitudes with contractivities si ∈ [0, 1), i ∈ NN , is given. Recall that the connection matrix C = (cij ) is defined by  1 cij := 0

if pji > 0; if pji = 0.

Definition 91. The attractor A =



Ai of an IFS (X, w, P) is called nonoverlap-

i∈NN

ping iff Aj ∩ Ak = ∅ for all j, k ∈ I(i) such that j = k and i ∈ NN . Theorem 61. Let (X, w, P) be a recurrent IFS and F its attractor. Furthermore, assume that the maps wi in the collection w belong to Sim0 (Rn ). Let C be an irreducible N × N connection matrix. For all t ∈ R+ define a diagonal matrix D(t) by ⎛t s1 ⎜0 ⎜ D(t) := ⎜ ⎝0 0

0 st2 0 0

... ... .. . ...

⎞ 0 0⎟ ⎟ ⎟, 0⎠ stN

(3.39)

where si is the contractivity constant of the map wi , i ∈ NN . Assume also that F is nonoverlapping. Let t∗ be the unique positive number such that 1 is an eigenvalue of D(t∗ )C of greatest absolute value. Then dimb F = t∗ .

(3.40)

Proof. Let s := min{si : i ∈ NN } and s := max{si : i ∈ NN }. The compactness of each Fj ⊆ F implies the existence of an ε0 > 0 such that ∀i ∈ NN

∀(j, k ∈ I(i) ∧ j = k): d(wi (Fj ), wi (Fk )) > ε0 .

(3.41)

For i ∈ NN , denote by Ni,ε (F cover of Fi by balls of i ) the cardinality of a minimal wi (Fj ) and wi ∈ Sim0 (Rn ), together with Eq. (3.41) radius ε > 0. Since Fi = j∈I(i)

they give a system of functional equations of the form

Dimension theory

Ni,ε (Fi ) =

121



Nj,ε/si (Fj )

(3.42)

j∈I(i)

for all 0 < ε < sε0 . By the Perron-Frobenius theorem there exists a strictly positive eigenvector of D(t∗ )C corresponding to the eigenvalue 1. This eigenvector is denoted by e = (e1 , . . . , eN ). There exist positive constants c1 and c2 such that ∗

∗

∀(sε0 < ε ≤ ε0 ) ∀i ∈ NN : c1 ei ε−t ≤ Ni,ε (Fi ) ≤ c2 ei ε−t .

(3.43)

For n ∈ N suppose that Eq. (3.43) holds for all sn sε0 ≤ ε ≤ ε0 . Now assume that ε is chosen in the interval [sn sε0 , sε0 ]. Then sn sε0 ≤ ε/si ≤ ε0 , and thus Ni,ε (Fi ) =



∗

Nj,ε/si (Fi ) ≤ c2 ε−t sti

j∈I(i)

∗



∗

ej = c2 e2 ε−t ,

(3.44)

j∈I(i)

and analogously ∗

c1 ei εt ≤ Ni,ε (Fi ).

(3.45)

By induction on n the following functional inequalities, which prove the theorem, are obtained: ∀(0 < ε ≤ ε0 )

∗

∗

∀i ∈ NN : c1 ei ε−t ≤ Ni,ε (Fi ) ≤ c2 ei ε−t .

4.4 Recurrent sets and Mauldin-Williams fractals Next a dimension result for a class of recurrent sets is stated. This result was first conjectured by Dekking [45] and then fully proved by Bedford [44]. The full proof, which involves pressure-theoretic arguments, is omitted. First a few definitions are needed. Definition 92 1. Let w ∈ S[X]. Define | · |x : S[X] → N0 as the number of occurrences of the letter x in the word w, and | · |E : S[X] → N0 as the number of essential letters in w. 2. A recurrent set Kϑ (w) is called resolvable iff there exists an η > 0 such that λKη (ϑ ν (w)) > 0, ν→∞ |ϑ ν (w)|E

lim inf

(3.46)

where λ denotes the n-dimensional Lebesgue measure and Kη (ϑ ν (w)) denotes the η-parallel body of K(ϑ ν (w)).

Let ϑ be a semigroup endomorphism and let x ∈ X. Define a matrix ME := (mxy )x,y∈E

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Fractal Functions, Fractal Surfaces, and Wavelets

on the set of essential letters E whose entries are given by mxy := |ϑ(x)|y . Let λE be the largest nonnegative eigenvalue of ME . Definition 93. A semigroup endomorphism ϑ is called essentially mixing iff the matrix ME := (mxy )x,y∈E is mixing. In other words, iff there exists a ν ∈ N such that for all x, y ∈ E, x occurs in ϑ ν (y). Theorem 62. Suppose that ϑ: S[X] → S[X] is essentially mixing and that Lϑ is an expansive similitude with eigenvalue λ. Then dimH Kϑ (w) =

log λE log |λ|

(3.47)

iff Kϑ (w) is resolvable. For completeness, a formula for the Hausdorff dimension of Mauldin-Williams fractals is given. Recall that by Theorem 45 there exists a unique element (Xv )v∈V ∈  H(X v ) such that v∈V Xu =



Se Xv .

v∈V e∈Euv

Let X :=



Xu . Given a positive number d, define a matrix

u∈V

M(d) := (Muv (d))1≤u,v≤N , where Muv (d) :=



s(e)d

e∈Euv

and N := |V|c . The spectral radius of M(d) is denoted by r(d). By the Perron-Frobenius theorem, r(d) is the largest nonnegative eigenvalue of M(d). The following theorem is proven in Ref. [46]. Theorem 63. Assume (G, s) is a strongly connected and contracting MauldinWilliams graph. The Hausdorff dimension of X is that number d∗ for which r(d) = 1. Furthermore, X is a d∗ -set.

5 The box dimension of projections In this section the box dimension of a self-affine set in R2 is related to the box dimension of the orthogonal projection onto a coordinate axis. For this purpose, let (x, y) be a Cartesian coordinate system of R2 ; let XA = (A1 , . . . , AN ) be the attractor

Dimension theory

123

of the recurrent IFS (X, w, P), where X is—without loss of generality—the unit square in R2 , and the maps wi : X → X are affine and of the form  wi (x, y) :=

ai 0

0 bi

    x c + i , di y

(3.48)

with 0 < |ai |, |bi | < 1, i ∈ NN . Furthermore, it is assumed that the connection matrix C = (cij ) associated with the recurrent IFS (X, w, P) is irreducible (ie, cij = 0 or 1). For a set E ⊂ R2 , denote the orthogonal projection of E onto the y-axis by Ey . Note that (XA)y is the attractor of the recurrent IFS (Xy , wy , P) with (wi )y : Xy → Xy given by (wi )y (y) = bi y + di ,

i ∈ NN .

It follows from a result in Ref. [59] that dimb (XA)y exists and equals dimH (XA)y . The next theorem relates the box dimension of (XA)y to that of A. Theorem 64. Let (X, w, P) be a recurrent IFS as defined earlier. Assume w satisfies the OSC ∀i, j ∈ NN ,

i = j: wi (int X) ∩ wj (int X) = ∅. ∗

Let d∗ be the unique positive number such that r(diag(|ai |d )C) = 1 and let d be determined by the formula ! r diag(|bi |dy |ai |d−dy )C = 1, where dy := dimb (XA)y . If r diag(|bi |dy |ai |d

∗ −d

y

! )C > 1,

(3.49)

then dimb XA = d.   Proof. Since XA = Ai and Ai = wi Aj , it follows that i∈NN

j∈I(i)

dimb XA = max{dimb Ai : i ∈ NN } = dimb Aj . Furthermore, the irreducibility of the connection matrix C implies that each component Ai contains a nonsingular affine image of Aj , and thus dimb Ai = dimb Aj = dimb XA, i ∈ NN . Now let 0 < ε < 1 be given. Denote by ε the set of all finite codes i(n) = i(n)(ε) such that |ai−1 . . . ai−n | ≤ ε, |ai−1 . . . ai−n+1 | > ε,

(3.50) (3.51)

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Fractal Functions, Fractal Surfaces, and Wavelets

and i = i−1 ∈ ε if |ai−1 | ≤ ε. It is easy to see that ε generates a partition of the code space  into cylinder sets i(n). To simplify the notation, let ai(n) := |ai−1 . . . ai−n | and

bi(n) := |bi−1 . . . bi−n |,

and wi(n) := wi−1 ◦ · · · ◦ wi−n ,

Ai(n) := wi−1 ◦ · · · ◦ wi−n+1 Ai−n .

Note that the codes are “time reversed.” For i ∈ NN , let Ui :=



Aj and note that

cij >0

Ai(n) = wi(n) Ui−n . Let E be a bounded set in R2 , and let Cε be a class of covers such that each Cε ∈ Cε consists of (ε × ε)-squares with sides parallel to the coordinate axis. Denote by Nε (E) the cardinality of a minimal cover from Cε , and by Nε (Ey ) the minimum number of compact intervals of length ε needed to cover Ey . The proof is based on the following geometrical observations: Let ε > 0 and let i(n) ∈ ε . Then 1. Ai(n) ⊂ wi(n) X and wi(n) X is a rectangle of width ai(n) and height bi(n) ; 2. since ai(n) ≤ ε, any cover of Ai(n) that is in Cε may be arranged in such a way that each (ε × ε)-square meets both vertical sides of wi(n) X. (This is possible since each (ε × ε)-square is wider than wi(n) X ⊃ Ai(n) .) If Q is such a square, then w−1 i(n) Q is a rectangle of height ε/bi(n) that meets the lines x = 0 and x = 1. Hence Q ∩ Ai(n) = ∅ iff w−1 i(n) Q ∩ (Ui−n )y = ∅. This now defines a one-to-one correspondence between covers of Ai(n) in Cε and covers of (Ui−n )y consisting of compact intervals of length ε/bi(n) . Consequently, Nε (Ai(n) ) = Nε/bi(n) ((Ui−n )y ).

To establish the dimension result, the cardinality of ε is needed. For this purpose the following probability measure on ε is introduced. Let v = (v1 , . . . , vN )t be a right ∗ positive eigenvector of diag(|ai |d C) and define a row-stochastic irreducible matrix M = (mij ) by vj ∗ mij := |ai |d cij . vi Let δ = (δ1 , . . . , δN )t be the unique  stationary distribution associated with the matrix M. In other words, Mδ = δ and i∈NN δi = 1. Let μM denote the probability measure on  generated by M with initial distribution δ. Then for any i(n) ∈ ε , one has max vi min vi ∗ ∗ |ai(n) |d ≤ μM i(n) ≤ |ai(n) |d . max vi min vi Eqs. (3.50), (3.51) now imply that ∗

c1 εd ≤ μM i(n) ≤ c2 εd

∗

Dimension theory

125

for some positive constants c1 and c2 . Consequently, ∗

μM ε = 1 ≥ c1 εd |ε |c . The remainder of the proof involves estimates on certain sums. To obtain these estimates a second row-stochastic matrix is now introduced. For 0 ≤ β ≤ 1, let α be the unique positive number such that β α−β

r diag(bi ai

C) = 1,

(3.52) β α−β

and let u be a positive left eigenvector of diag(bi ai  := ( M mij ) be defined by β α−β

 mij := bi ai

cji

C) with eigenvalue 1. Let

uj . ui

μM Given the initial distribution  δ = ( N1 , . . . , N1 )t , the induced probability measure   satisfies  μM  =

1 1 ui−n β α−β mi−n+1 i−n = b a .  mi i · · · · ·  N −1 −2 N ui−1 i(n) i(n)

Hence 1 max ui 1 min ui β α−β  μM  μ  i(n).  i(n) ≤ bi(n) ai(n) ≤ N max ui N min ui M

(3.53)

Now all the tools are available to prove the dimension result. First it is shown that d is an upper bound for dimb XA. Note that, since ε generates a partition of , XA =



Ai(n) .

i(n)∈ε

Thus Nε (XA) ≤



Nε (Ai(n) ) =

i(n)∈ε



Nε/bi(n) ((Ui−n )y ).

i(n)∈ε

Let β > dy and α be as in Eq. (3.52). Then it follows from the Perron-Frobenius theorem that limβ→dy α = d. As β > dy , there exists a constant c > 0 with the property that Nε ((Uj )y ) ≤ cε−β

for all j ∈ NN . Let ε∗ := {i(n)(ε) ∈ ε : ε/b1 > 1}. Then

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Fractal Functions, Fractal Surfaces, and Wavelets

Nε/bi(n) ((Uj )y ) = 1

for all j ∈ NN , whereas if i(n) ∈ ε∗∗ := ε \  ∗ , then ε/bi(n) ≤ 1 and thus for some c > 0, Nε/bi(n) ((Uj )y ) ≤ cε−β bi(n) , β

j ∈ NN .

Therefore 

Nε (XA) ≤

Nε (Ai(n) ) =

i(n)∈ε



≤c

i(n)∈ε∗

⎛

Nε/bi(n) ((Ui−n )y )

i(n)∈ε β ε−β bi(n)



≤ c⎝



+ |ε∗∗ |c ≤ c



ε−β bi(n) + |ε∗∗ |c β

i(n)∈ε

⎞

ai(n) bi(n) ⎠ ε−α + |ε∗∗ |c . α−β β

i(n)∈ε

The last inequality follows from the fact that ai(n) ≤ ε for i(n) ∈ ε . Since |ε∗∗ |c ≤ |ε |c and since by Eq. (3.53)  α−β β 1 max ui 1 max ui =  μM ai(n) bi(n) ,  ε > N min ui N min ui i(n)∈ε

the preceding inequality becomes  Nε (XA) ≤ εα

εα−d c max ui + N min ui c1

∗

 .

Letting β → dy , yields dimb XA ≤ d. Finally it is established that d is also a lower bound for β → dy . Note that the OSC implies that wi (int X) ∩ wj (int X) = ∅ for all i(n), j(n) ∈ ε with i(n) = j(n). Thus any (ε × ε)-square in a cover belonging to the class Cε meets at most L = 2a−1 + 4 of the rectangles {wi(n) X: i(n) ∈ ε∗ }. Hence Nε (XA) ≥ L−1

 i(n)∈ε∗

⎛ Nε (Ai(n) ) = L−1 ⎝



⎞ Nε (Ai(n) ) − |ε∗∗ |c ⎠ .

i(n)∈ε

The last equality holds since Nε (Ai(n) ) = 1 for i(n) ∈ ε∗∗ . Choosing β < dy and proceeding as above yields the result.

Dimension theory

127

If (X, w, P) is an IFS—that is, C = (1)—Eq. (3.49) reduces to N 

|ai |d

∗ −d

y

|bi |dy > 1,

i=1

 ∗ with i∈NN |ai |d = 1. Hence the following corollary holds. Corollary 4. Let (X, w) be an IFS with attractor A and maps wi : X → X of the form Eq. (3.48). Suppose that the OSC i = j: wi (int X) ∩ wj (int X) = ∅

∀i, j ∈ NN , and

 i∈NN



|ai |d

∗ −d

y

|bi |dy > 1 hold. If d is the unique positive solution of

|ai |d−dy |bi |dy = 1,

i∈NN

then dimb A = d.

Dynamical systems and dimension

4

Abstract This chapter introduces the concept of a dynamical system. It will be seen how the geometric theory of dynamical systems can be used to describe attractors of iterated function systems. In particular, the Lyapunov dimension of an attractor of a dynamical system is defined, and it is shown how it relates to the Hausdorff dimension and the box dimension. Again, the limited scope of this monograph allows only the presentation of the most basic aspects of the theory, thus giving the reader a general overview of this fascinating subject. Also, results will be presented in Rn rather than on finite-dimensional Riemannian manifolds. Some of the references in the bibliography give a more precise and general introduction to dynamical systems [63, 64, 161, 198–204].

1 Ergodic theorems and entropy Let X be a subset of Rn and let μ: 2X → R be a measure on X. The σ -algebra of all μ-measurable subsets of X is denoted by B(X). Definition 94. Let f : X → X be a μ-measurable map. Assume that f is μ-invariant; that is, f # μB = (μ ◦ f −1 )B = μB for all B ∈ B(X). Then the quadruple (X, B(X), f , μ) is called an (n-dimensional) dynamical system. Remark 35. There are two types of dynamical systems: discrete and continuous. Here only discrete dynamical systems are considered (ie, only the behavior of the iterates f m of f as m → ∞ is studied). Examples of continuous dynamical systems are provided by the solutions of initial value problems for autonomous ordinary and partial differential equations. Definition 95. Suppose that (X, B(X), f , μ) is a dynamical system and that μX = 1 (ie, μ is a probability measure on B). Such a dynamical system is called ergodic iff ∀E ⊆ X: f −1 E = E ⇒ μE = 0 or 1.

(4.1)

Remark 36. Definition 95 expresses the simple fact that an ergodic dynamical system cannot be decomposed into nontrivial subsystems which do not interact with each other. The sequence of iterates orb f := {f m (x): m ∈ N} of f for a fixed x ∈ X is called the orbit of f . Definition 96 (Attractor of a Dynamical System). Suppose that (X, B(X), f , μ) is a dynamical system. A closed subset A of X is said to be the attractor of the dynamical system (X, B(X), f , μ) iff: Fractal Functions, Fractal Surfaces, and Wavelets. http://dx.doi.org/10.1016/B978-0-12-804408-7.00004-7 Copyright © 2016 Elsevier Inc. All rights reserved.

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Fractal Functions, Fractal Surfaces, and Wavelets

1. A is invariant under f (ie, fA = A) and no proper subset of A satisfies this requirement; 2. there exists an open set G, called the basin of attraction such that ∀ x ∈ G: dE (f m (x), A) → 0,

as k → ∞,

where dE denotes the Euclidean metric on Rn .

Some examples of attractors of dynamical systems are fixed points, limit cycles, and orbits of p-periodic points {y, f (y), . . . , f p−1 (y)}, where p is the least positive integer satisfying f p (y) = y and dE (f m (x), f i (y)) → 0 as m → ∞. Besides these classical attractors, A may be a fractal set. There are numerous examples of such fractal attractors in the literature: the Lorentz attractor, the Hénon attractor, the Rössler attractor, etc. If a dynamical system (X, B(X), f , μ) exhibits such a fractal attractor A, the map f is very often chaotic in the following sense: 1. There exists an x ∈ X such that cl {f m (x)}m∈N = A. 2. cl Per f = A, where Per f denotes the set of periodic points of f ; that is, points for which there exists a positive integer p such that f p (x) = x. 3. The map f has sensitive dependence on the initial conditions: ∀ε > 0 ∃ δ > 0 ∀x ∈ A ∃ y ∈ A ∃ m ∈ N: dE (x, y) < ε ⇒ dE (f m (x), f m (y)) ≥ δ.

An example of a chaotic map is the logistic map, which has its origin in population dynamics. It is the archetype of a one-parameter family of maps associated with a one-dimensional dynamical system. Example 22 (The Logistic Map). Let X := R, let μ be the Lebesgue measure, and let c ∈ R+ . Define a map fc : R → R by fc (x) := cx(1 − x).

(4.2)

As the parameter c increases, the p-periodic orbits of fc exhibit different behavior, and as c reaches the value c∞ ≈ 3.5699340, a Cantor-like attractor appears. For a more detailed description of the behavior of fc , see, for instance, Refs. [63, 64]. Next the geometric structure of dynamical systems is briefly investigated. For this purpose, the tangent map of f and also Oseledec’s multiplicative ergodic theorem [65] have to be introduced. Definition 97. Let f : X ⊆ Rn → Rk be a function and suppose that p ∈ X is a point of accumulation of X. Then f is called Fréchet differentiable at p iff there exists a map Tp f ∈ L(Rn , Rk ) such that

f (x) − f (p) − Tp f (x − p) = 0. x→p

x − p lim

(4.3)

The function f is called Fréchet differentiable on X if it is Fréchet differentiable at each point p ∈ X. Note that the linear map Tp f is a k × n matrix whose entries are the univariate ∂fi (p), i ∈ Nk and j ∈ Nn . R-valued functions ∂x j

Dynamical systems and dimension

131

The partial derivatives ∂j (p) := ∂x∂ j (p), j ∈ Nn , form an R-vector space basis for the tangent space at p; every tangent vector v(p) at p is of the form v(p) =

n 

cj ∂j (p)

j=1

for scalars cj ∈ R. The tangent space at p ∈ X is denoted by Tp X. The tangent space of X is then defined as TX :=

 ˙

Tp X.

p∈X

Now suppose that f : X ⊆ Rn → Y ⊆ Rm is a differentiable function. One can “lift” f to a mapping Tf : TX → TY between tangent spaces via Tp f : Tp X → Tf (p) Y, where Tp f maps tangent vectors at p ∈ X to tangent vectors at f (p) ∈ Y. More precisely, Tp f  ∂fi ∂ ∂ (p) −→ (p) (f (p)), ∂xj ∂xj ∂yi n

i=1



 where ∂y∂ i (f (p))



i∈Nn



∂ ∂xj (p) j∈N . m L(Rn , L(Rn , Rm )) which

is the tangent space basis induced by

is isomorRemark 37. The mapping Tf is an element of phic to L(Rn , Rm+n ). Definition 98. Let f : X ⊆ Rn → X be a bijective differentiable function. Then f is said to be a diffeomorphism on X iff its inverse f −1 is also differentiable on X. A Ck -diffeomorphism is a bijective k times continuously differentiable function whose inverse is also k times continuously differentiable. The next result is one of the most important theorems in ergodic theory [66]. Theorem 65 (Birkhoff’s Ergodic Theorem). Suppose (X,  B(X), f , μ) is a dynamical system and φ: X → R is an integrable function; that is, X φ dμ < ∞. Then the time average k−1 1 k→∞ φ(f m (x)) −→ φ k

(4.4)

m=0

converges for a.e. x ∈ X. Furthermore, if (X, B(X), f , μ) is ergodic, then φ equals the space average:  φ=

φ(x) dμ(x). X

(4.5)

132

Fractal Functions, Fractal Surfaces, and Wavelets

The matrix version of Birkhoff’s ergodic theorem is due to Oseledec [65]. Theorem 66 (Oseledec’s Multiplicative Ergodic Theorem). Let f be a diffeomorphism of X ⊆ Rn whose derivative is uniformly bounded and let μ be an invariant Borel probability measure on X. Then at a.e. x ∈ X, the tangent space Tx X can be decomposed into a direct sum of subspaces Tx X =

r(x) 

Wj (x)

j=1

such that: 1. (Tx f )Wj (x) = Wj (f (x)), ∀j = 1, . . . , r(x): 2. ∀x ∈ X ∀j = 1, . . . , r(x) ∃λj (x) ∈ R ∀w ∈ Wj (x): lim

1

k→∞ k

log Tx f ±1 w = ±λj (x);

(4.6)

3. if θx (i, j) denotes the angle between Wi (x) and Wj (x), then lim

1

k→∞ k

log | sin(θf ±k (x) (i, j))| = 0.

(4.7)

The numbers λj counted with multiplicity dim Wj (x) are called the Lyapunov exponents at x ∈ X. If the dynamical system (X, B(X), f , μ) is ergodic, then the Lyapunov exponents are independent of x. Remark 38 1. The Lyapunov exponents measure the exponential growth of an infinitesimal volume element dVx in the tangent space Tx X. 2. If f is an affine map, Tx f is constant, and therefore the Lyapunov exponents are independent of x. The eigenvalues i (k) of (Tx f )k =: Txk f are then related to the Lyapunov exponents via λi = lim

1

k→∞ k

log | i (k)|.

To predict the long-term behavior of a dynamical system, information has to be gathered through a series of experiments. A measure of the gain or loss of information is provided by the entropy. This invariant was first introduced in information theory by Shannon and in its present form by Kolmogorov. A simple example from probability theory will provide the basis for Kolmogorov’s definition. Suppose an experiment E that has k possible outcomes E1 , . . . , Ek occurring with probabilities p1 , . . . , pk is performed. To predict in advance the outcome of E , one has to associate a function H(E ) with E that measures the amount of uncertainty in the prediction. It has been proven that the only such function that agrees with intuition is—up to a multiplicative constant—given by H(p1 , . . . , pk ) = −

k  i=1

pi log pi .

(4.8)

Dynamical systems and dimension

133

To define the entropy for a dynamical system, the concept of a partition of a set in B needs to be introduced. Definition 99. Suppose that (X, B(X), f , μ) is a dynamical system. A finite collection {E1 , . . . , Ek } ⊆ B is called a partition of X iff k

 ˙

Ei = X,

(4.9)

i=1

where ˙ denotes the disjoint union of sets. For two partitions P and Q of X, the join P ∨ Q is defined by P ∨ Q := {P ∩ Q: P ∈ P ∧ Q ∈ Q},

(4.10)

and the preimage of P under f is defined by f −1 P := {f −1 P: P ∈ P }.

(4.11)

It is easy to see that P ∨ Q and f −1 P are again partitions of X. The k-fold join of k

partitions P1 ∨ · · · ∨ Pk is written as Pi . i=1

The preceding example can be related to the present setup by our asking for a measure of uncertainty in the prediction of the location of a point x ∈ X. Giving a partition P of X is equivalent to predicting the location of x in a specific element of P . This gives rise to the definition of entropy of the partition P of X: H(P ) := −



μP log μP.

(4.12)

P∈P

Now suppose Q is another partition of X. The conditional entropy of Q given P is defined by H(Q | P ) := H(P ∨ Q) − H(P ).

(4.13)

Next the entropy for a dynamical system is defined. The basic idea is as follows: For a given ε > 0, choose a partition P of X whose elements have diameter less than or equal to ε. To predict the location of the iterate f k (x) of a point x ∈ X, one has to know to which element of f −k (P ) x initially belonged. More generally, to predict the approximate location of the orbit {x, f (x), . . . , f k (x)}, one has to know the element of k

f −i (x) initially containing x. Therefore it is natural to define the entropy of f with i=0

respect to the partition P by  hμ (f , P ) := lim H f k→∞

−(k+1)

k  −i P f P . i=0

(4.14)

134

Fractal Functions, Fractal Surfaces, and Wavelets

A rather lengthy calculation shows that  k 

1 −i f P . hμ (f , P ) = lim H k→∞ k

(4.15)

i=0

The advantage of Eq. (4.15) lies in its interpretation: the entropy hμ (f , P ) can be thought of as the average uncertainty in predicting the first k iterates of x ∈ X. Finally, the entropy of a map is defined. Definition 100. Let (X, B(X), f , μ) be a dynamical system and let P(X) denote the collection of all partitions P of X. Then the entropy hμ (f ) is defined by hμ (f ) :=

sup hμ (f , P ).

P ∈P(X)

(4.16)

Suppose that μ and ν are two measures defined on the same σ -algebra B(X) of a space X. Recall that μ is said to be absolutely continuous with respect to ν, in symbols μ  ν, iff there exists a positive ν-measurable function g: X → X such that  ∀B ∈ B(X): μB = g(x) dν(x). B

The next result is known as Pesin’s formula [67]. Theorem 67. Let (X, B(X), f , μ) be a dynamical system, where X is compact and μ an invariant Borel probability measure. Assume that f is a C2 -diffeomorphism on X and that μ is absolutely continuous with respect to the Lebesgue measure. Then  hμ (f ) =

⎛ ⎝

X



⎞ λi (x)⎠ dμ(x).

(4.17)

λi (x)>0

Remark 39. If (X, B(X), f , μ) is ergodic, then Pesin’s formula simplifies to hμ (f ) =



λi .

(4.18)

λi >0

Since partitions are a special class of covers, one might expect that there exists a relationship between the Hausdorff dimension of the invariant measure μ and the Lyapunov exponents and the entropy of f . The existence of such a relationship was established in Ref. [68]. Theorem 68. Let (X, B(X), f , μ) be an ergodic two-dimensional dynamical system and let f be a C2 -diffeomorphism such that f # μ = μ, where μ is a Borel probability measure. Also, assume that the Lyapunov exponents of f satisfy λ1 ≥ λ2 . Then

Dynamical systems and dimension

 dimH (μ) = hμ (f )

1 1 − λ1 λ2

135

 ,

(4.19)

whenever the right-hand side of Eq. (4.19) is not equal to 0/0. In the proof of this theorem one uses a result that is worthwhile mentioning. Proposition 44. Let X ⊆ Rn be μ-measurable with μX > 0. Let B(x, r) denote the closed ball of radius r > 0 centered at x ∈ X. Assume that for every x ∈ X d ≤ lim inf r→0+

log μB(x, r) log μB(x, r) ≤ lim sup ≤ d. log r log r r→0+

(4.20)

Then d ≤ dimH X ≤ d. Remark 40 1. There is no loss of generality if in the preceding limits the continuous variable r is replaced by a sequence {rν }ν∈N with lim rν = 0 and lim log rν+1 / log rν = 1.

ν→∞

2. It can be shown [68] that if μ is a Borel probability measure, X is compact, and lim

r→0+

log B(x, r) = d, log r

(4.21)

for μ-a.e. x ∈ X, then dimH μ = dimb μ = d,

(4.22)

where dimb μ denotes the box dimension of the invariant measure μ. This dimension is defined as follows: dimb μ := sup inf{dimb Y: Y ⊆ X ∧ μY ≥ 1 − δ}.

(4.23)

δ→0+

2 Lyapunov dimension Theorem 68 gives a relationship between the Hausdorff dimension of the invariant measure μ and the Lyapunov exponents and entropy of f . However, there exists another notion of dimension associated with the invariant measure μ that relates it to the box dimension of the attractor A of a dynamical system. This dimension, which was

136

Fractal Functions, Fractal Surfaces, and Wavelets

introduced by Kaplan and Yorke [69], is called the Lyapunov dimension of the invariant measure μ on A. The definition of the Lyapunov dimension can be obtained by our looking at a simple example. Suppose that (X, B(X), f , μ) is an ergodic n-dimensional dynamical system and that f is an affine map leaving the Borel probability measure μ invariant. Assume that the Lyapunov exponents of f are such that λ 1 ≥ λ2 ≥ · · · ≥ λ n , and that λ1 > 0. Let A be the attractor of the dynamical system. To estimate the box dimension of A, one has to find a minimal cover of A consisting of n cubes of side ε > 0. Denote by Nε (A) the cardinality of this cover. Applying f k times to a cube in this cover yields a parallelepiped with sides (exp λ1 k)ε, . . . , (exp λn k)ε. Hence the volume of this ε-cube has changed by a factor of exp(λ1 + · · · + λn )k. Let k0 be the largest integer such that λ1 + · · · + λk0 > 0. Now consider a cover of A by cubes of side exp(λk0 +1 )ε. Then every (exp λ1 k)ε × · · · × (exp λn k)ε parallelepiped can be covered by approximately exp(λ1 + · · · + λn )k = exp(λ1 + · · · + λn − nλk0 +1 )k (exp λk0 +1 k)n such cubes. Since λk0 +1 ≥ λk0 +2 ≥ · · · ≥ λn , it can also be covered by approximately exp(λ1 + · · · + λk0 − k0 λk0 )k cubes of side (exp λk0 +1 k)ε. Denoting the cardinality of the associated minimal cover of A by N(exp λk0 +1 k)ε (A), one thus obtains N(exp λk0 +1 k)ε (A) ≈ (exp(λ1 + · · · + λk0 − k0 λk0 )k)Nε (A). If it is assumed that Nε (A) ∝ ε−db , where db := dimb A, then ((exp λk0 +1 k)ε)−db ≈ (exp(λ1 + · · · + λk0 − k0 λk0 )k)ε−db .

Dynamical systems and dimension

137

Taking logarithms and solving for db yields db ≈ k0 −

λ1 + · · · + λk0 λ1 + · · · + λk0 = k0 + λk0 +1 |λk0 +1 |

(4.24)

since λk0 +1 < 0. The right-hand side of Eq. (4.24) is called the Lyapunov dimension of μ. Definition 101. Let (X, B(X), f , μ) be an ergodic n-dimensional dynamical system. Assume that μ is a Borel probability measure and that f # μ = μ. Let λ1 ≥ λ2 ≥ · · · ≥ λn be the Lyapunov exponents of f . Let k0 := max{λ1 + · · · + λk > 0: k ∈ Nn }. For 1 ≤ k0 < n, the Lyapunov dimension of μ is defined by dim μ := k0 +

λ1 + · · · + λk0 . |λk0 +1 |

(4.25)

If no such k0 exists, dim μ := 0, and if k0 = n, then dim μ := n. Now Theorem 68 can be brought into the picture. Suppose that a two-dimensional ergodic dynamical system defined on a compact X ⊆ R2 is given and that λ1 > 0. Then, under the hypotheses of Theorem 68, dim μ = 1 + dimH μ = 1 + dimb μ.

(4.26)

Our next goal is to relate the attractor of an iterated function system (IFS) to the attractor of a dynamical system [70]. For X ∈ H(Rn ), let (X, w, p) be a contractive IFS with probabilities. Its attractor is denoted by F and the p-balanced measure is denoted by μ. There is a natural way of representing (X, w, p) as a dynamical system. Namely, let X ∗ := X × [0, 1]

and

μ∗ := μ × m,

where m denotes the uniform Lebesgue measure on [0, 1]. Furthermore, let B∗ (X ∗ ) be the smallest σ -algebra containing B(X) and the Borel sets on [0, 1]. Finally, define a map f ∗ : X ∗ → X ∗ by ⎧  ⎨ wi (x), p−1 (y − Si−1 ) , (x, y) ∈ X × [Si−1 , Si ], i ∈ NN−1 ; i  f ∗ (x, y) :=  −1 ⎩ wN (x), p (y − SN−1 ) , (x, y) ∈ X × [SN−1 , SN ], N (4.27)  where Si := ij=1 pj and S0 := 0. For similar constructions, see Refs. [71, 72]. The following result is almost immediate.

138

Fractal Functions, Fractal Surfaces, and Wavelets

Proposition 45. The quadruple (X ∗ , B∗ (X ∗ ), μ∗ , f ∗ ) is a discrete dynamical system. Note that the attractor F ∗ of (X ∗ , B∗ (X ∗ ), μ∗ , f ∗ ) is F × [0, 1]. Assumption. For the remainder of this section it is assumed that w := {Si := si Ai : i ∈ NN } ⊆ Sim0 (Rn ) and that w satisfies the open set condition. Under the preceding assumption, the tangent map Tx∗ f ∗ exists for μ∗ -a.e. x∗ ∈ X ∗ and equals the constant matrix Tx∗ f ∗ =



si Ai 0

0



p−1 i

for x∗ ∈ X × [Si−1 , Si ], i ∈ NN−1 , or x∗ ∈ X × [SN−1 , SN ]. Now let k ∈ N. The kth iterate of the tangent map, Txk∗ f ∗ , is given by ⎛

k 

k kj  k Aj j ⎜ sj ⎜j=1 j=1

Txk∗ f ∗ = ⎜ ⎝

0

⎞ 0

⎟ ⎟ , k −k ⎟  j⎠ pj

j=1

where kj denotes the number of times map wi is chosen. The moduli of the eigenvalues of Txk∗ f ∗ are | 1 | =

N  ki  1 i=1

pi

and

| 2 | = · · · = | n+1 | =

N 

ski i .

i=1

Using the law of large numbers (Theorems 12 and 13) to find the Lyapunov exponents yields λ1 = −

N 

pi log pi > 0 and

λ2 = · · · = λn+1 =

i=1

N 

pi log si > 0.

i=1

The Lyapunov dimension of the invariant measure μ∗ of the attractor F ∗ is now given by N dim(p) μ∗ = k +

sk−1 i i=1 pi log pi . N − i=1 pi log si

(4.28)

Dynamical systems and dimension

139

 Here k is the largest integer such that i∈NN ski ≤ 1, which implies that λ1 + · · · + λk+1 ≤ 0 and λ1 + · · · + λk > 0. To proceed the following lemma is needed. Lemma 5. Let g: [0, 1]N → R, N ∈ N, be defined by N g(x) :=

i=1

−

N

xi log axii

i=1

xi log bi

,

where 0 < ai , bi ≤ 1, i ∈ NN , and ◦

there exists an x in the interior of

 i∈NN [0, 1]N

ai > 1. Assume that



i∈NN xi

= 1. Then ◦

at which g attains its maximum value g. ◦  g Furthermore, this maximum value satisfies i∈NN ai bi = 1. Proof. The proof is by calculus. The next result is an immediate consequence of Lemma 5 and the preceding arguments (see also Ref. [70]). Theorem 69. Let (X ∗ , B∗ (X ∗ ), μ∗ , f ∗ ) be the dynamical system associated with the ◦ IFS (X, w, p). Then there exists a set p of probabilities that maximizes the Lyapunov ◦

dimension dim(p) μ∗ . This maximized Lyapunov dimension  satisfies ◦

 = 1 + dimH F = 1 + dimb F.

(4.29)

Proof. Use Lemma 5 with ai := sk−1 and bi := si . i Remark 41 1. Clearly, since F ∗ = F × [0, 1], ◦

dimH F ∗ = dimb F ∗ = . ◦

◦

2. The set p of probabilities that maximizes the Lyapunov dimension is given by pi = sdi , where d stands for either dimH μ∗ or dimb μ∗ .

Theorem 70. Let (X, w, p) be a contractive IFS with probabilities. Suppose that ◦ (X ∗ , B∗ (X ∗ ), μ∗ , f ∗ ) is its associated dynamical system. Let p be the unique set of ∗ probabilities that maximizes the Lyapunov dimension of μ = μ∗ (p). Then the box ◦ dimension and the Hausdorff dimension of the p-balanced measure supported on the ◦ attractor F of (X, w, p) equal (μ∗ (p)) − 1. Proof. By Remark 40 it suffices to show that lim

r→0+

log B(x, r) ◦ = (μ∗ (p)) − 1 log r

for μ-a.e. x ∈ X.

140

Fractal Functions, Fractal Surfaces, and Wavelets ◦

Let m ∈ N and let i(m) ∈ m . Remark 40 implies pi = sdi , and thus μwi(m) F ≥

m m m    ◦ ◦ ◦ pik μ(w−1 w F) = p μF = pik . ik i(m) i(m) k=1

k=1

k=1

Now suppose x ∈ F. By Theorem 37 there exists a code i ∈ such that x = γ (i). Let 0 < r < 1 be given and let q = q(r) be the least integer such that wi(q) (F) ⊆ B(x, r). Order the si in the following way: s 1 ≤ s2 ≤ · · · ≤ s N . Then



q 

 sik |F| ≥ rs1 ,

k=1

for if the reverse inequality held then  q   sik |F| < rs1 ≤ rsiq , k=1

contradicting the choice of q. Hence μB(x, r) ≥ μwi(q) F ≥

q  ◦ pik . k=1

Thus

q ◦ q ◦  log k=1 pik log k=1 pik log qk=1 sik log μB(x, r) ≤ = . q log r log r log r log k=1 sik

But since q 

s ik ≥

k=1

s1 r , |F|

one obtains q log k=1 sik log(s1 r)/|F| ≤1+ . log r log r Therefore,  q ◦    log k=1 pik log μB(x, r) log(s1 r)/|F| ≤ lim . lim 1+ q r→0+ r→0+ log log r log r k=1 sik

Dynamical systems and dimension

141

As q → ∞ as r → 0+, Theorems 12 and 13 again imply N

q

◦ p ik lim k=1 q r→0+ log k=1 sik

log

=

qi i=1 q lim N q i q→∞ i=1 q ∗ ◦

N ◦ ◦ p log pi = i=1 ◦i N log si i=1 pi log si ◦

log pi

= (μ (p)) − 1. Here qi denotes the number of times maps wi have been applied. Finally, lim

r→0+

log μB(x, r) ◦ ≤ (μ∗ (p)) − 1. log(r)

To show the opposite inequality, one has to use Lemma 4. Denote by G the open set whose existence is guaranteed by the open set condition. Suppose that G contains a ball of radius ρ1 and is contained in a ball of radius ρ2 . Let r > 0 be given. For each code j = (j1 j2 . . .) ∈ choose the least integer q such that s1 r ≤ sj1 r ≤ · · · ≤ sjq ≤ r. Denote the set of all such codes by J := {j(q) = (j1 j2 . . . jq ) ∈ : s1 r ≤ sj1 r ≤ · · · ≤ sjq ≤ r}.

Note that if i = (i1 i2 . . .) ∈ , there exists exactly one j(q) ∈ J such that i1 = j1 , i2 = j2 , . . . , iq = jq . This then implies that {wj(q) (G): j ∈ J } is a finite disjoint collection of open sets. Furthermore, each wj(q) (G) contains a ball of radius sj(q) ρ1 and hence of radius s1 rρ1 , and is contained in a ball of radius sj(q) ρ2 and thus in one of radius rρ2 . It follows from Lemma 4 that at most (1 + 2ρ2 )n (s1 ρ1 )−n of the closures of the wj(q) (G) can meet B(x, r), and therefore at most (1 + 2ρ2 )n (s1 ρ1 )−n of the wj(q) (F) can possibly intersect B(x, r). For all j(q) ∈ J , q q   ◦ μwj(q) (F) = p jk = sdjk ≤ rd , k=1

k=1

◦

where d := (μ∗ (p)) − 1. Thus log B(x, r) log(1 + 2ρ2 )n (s1 ρ1 )−n rd ≥ = d . r→0+ log r log r lim

Construction of fractal functions

5

Abstract In this chapter fractal functions are considered (ie, functions whose graphs are fractal sets and which are generated by certain classes of iterated function systems). The term fractal refers to the fact that the graph of such a function has, in general, a nonintegral dimension. It is shown that these fractal functions may be used for interpolation and approximation purposes, and are in this way analogous to (parameterized) splines. For a more in-depth relation between splines and fractal functions, see Massopust [78]. In Section 1.2 we encountered examples of classical fractal functions—namely, the Weierstraß function and its generalization by Besicovitch and Ursell. Here a general construction based on a Read-Bajraktarevi´c operator [73, 74] acting on the function space L∞ (X, Y) is presented. The Read-Bajraktarevi´c operator provides a natural framework for the description of fractal functions in terms of several different classes of contractive iterated function systems. Related constructions of fractal functions based on recurrent sets and iterative interpolation are also studied and set in perspective. Properties such as moments, integral transforms, Lipschitz continuity, and extreme values of fractal functions are presented and discussed. Our focus is primarily on univariate R-valued fractal functions defined on subsets of R, although one section is devoted to Peano curves and their relation to so-called hidden-variable fractal functions. We also introduce fractal functions that are Ck -smooth but for which the graph of the kth derivative has, in general, a nonintegral dimension. Biaffine fractal functions are briefly discussed, and the relatively new concept of a local fractal function is introduced. For the latter, we obtain conditions so that these functions belong to certain classes of function spaces, such as Lebesgue, Sobolev, Hölder, Besov, and Triebel-Lizorkin spaces. The references for this chapter are [8, 70, 75, 76, 78–80, 83, 87–95, 97, 99, 105–107, 122, 138, 160, 186, 205–219].

1 The Read-Bajraktarevic´ operator The fractal functions considered in this section are fixed points of a certain contractive operator. This operator has its origin in the theory of functional equations and was first investigated by Read [73] and then later by Bajraktarevi´c [74]. These operators were used by Dubuc [75, 76] and Bedford [77] to construct fractal functions and have become an important tool in the theory of fractal interpolation. Denote by L∞ (X, Y) the complete metric space consisting of all bounded functions X → Y endowed with the metric Fractal Functions, Fractal Surfaces, and Wavelets. http://dx.doi.org/10.1016/B978-0-12-804408-7.00005-9 Copyright © 2016 Elsevier Inc. All rights reserved.

146

Fractal Functions, Fractal Surfaces, and Wavelets

d(f , g) := sup dY (f (x), g(x)).

(5.1)

x∈X

Definition 102. Let X be a set and (Y, dY ) a complete metric space. Suppose u: X → X is an arbitrary function on X. Furthermore, assume that a mapping v: X × Y → Y is given. The Read-Bajraktarevi´c operator, for short RB operator, associated with u and v is defined by : L∞ (X, Y) → Y X ,

f → v(u, f ◦ u).

(5.2)

To construct fractal functions for fractal interpolation, some additional conditions have to be put on the mapping v. Two natural conditions are as follows: (A) For all x ∈ X, the mapping v(x, ·) ∈ Lip( 1. If the hypotheses of Theorem 71 are satisfied in this special case, then the unique fixed point f : [0, 1] → Rn of the RB operator  is called a vector-valued fractal function (see also Ref. [79]). A special class of such vector-valued fractal functions is considered next. Let := {(xj , yj ): 0 = x0 < x1 < · · · < xN = 1 ∧ j ∈ N0N } ⊆ [0, 1] × Rn and let F ([0, 1], Rn ) := {f ∈ C([0, 1], Rn ): f (xj ) = yj , ∀j ∈ N0N }. Define mappings ui : [0, 1] → [0, 1] by

154

Fractal Functions, Fractal Surfaces, and Wavelets

1.0

0.5

0.2

0.4

0.6

0.8

−0.5

Fig. 5.2 The graph of a fractal function with interpolation property.

ui (x) = (xi − xi−1 )x + xi−1 , and vi : Rn → Rn by ⎛

δyi,1 ⎜ .. vi (y) := ⎝ . δyi,n

si,1 0

... .. . ...

0

⎞ ⎟ ⎠ y + ti ,

si,n

where δyi,k := yi,k − yi−1,k − si,k (yN,k − y0,k ) and ti,k := yi−1,k − si,k y0,k ,

1.0

Construction of fractal functions

155

i ∈ NN and k ∈ Nn . Here yj := (yj,1 , . . . , yj,n ) ∈ Rn , j ∈ N0N , and ti := (ti,1 , . . . , ti,N ) ∈ Rn , i ∈ NN . The si,k ∈ (−1, 1) are free parameters. The RB operator  on F ([0, 1], Rn ) is then given by −1 f (x) := vi (u−1 i , f ◦ ui (x)),

x ∈ ui [0, 1],

(5.23)

and the unique fixed point of  is a continuous fractal function f : [0, 1] → Rn interpolating . This special class of vector-valued fractal functions will be considered again in Chapter 7, where dimension formulae for the graphs of fractal functions are presented. Theorem 74 can be used to give a geometric interpretation of the construction of a fractal function possessing the interpolation property. The contractive IFS associated with such a fractal function is given by

a wi (x, y) = i ci

0 si



x d + i , ei y

i ∈ NN ,

(5.24)

where the ai , ci , di , and ei are defined as in Eqs. (5.19)–(5.22) and the si ∈ (−1, 1) are free parameters. Recall that by Proposition 27 one can find a  K ∈ H(X) such that w( K) ⊆  K. This is used in Fig. 5.3 to depict the convergence of  K ⊆ R2 to the graph of a fractal function with interpolation property.

1.0

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

Fig. 5.3 The convergence of  K to the graph of a fractal function with interpolation property.

156

Fractal Functions, Fractal Surfaces, and Wavelets

In Section 1.2, Weierstraß-like fractal functions were introduced, in particular, the functions fφ (see Eq. 2.9) defined by Besicovitch and Ursell. Next we show that they can also be defined via IFSs. Recall that the function φ|[0,1] interpolates the points (0, 0), ( 12 , 1), and (1, 0). In the notation of Section 1.2, let bj := 2j and α := − log |s|/ log 2 for some 0 < |s| < 1. If we define u1 (x) := 12 x, u2 (x) := 12 (x + 1), λ1 (x) := x, and λ2 (x) := 1 − x, it follows that, with v1 and v2 defined as in Eq. (5.9), the operator −1  := λi ◦ u−1 i + s(·) ◦ ui

is contractive on the Banach space C∗ ([0, 1], R) := {f ∈ CR [0, 1]: f (0) = f (1) = 0}. Its unique fixed point is a continuous fractal function f satisfying f (0) = f (1) = 0. Claim. f ≡ fφ on [0, 1]. Proof. It suffices to show that fφ satisfies the fixed-point equation −1 f = λi ◦ u−1 i + s f ◦ ui

for all x ∈ ui [0, 1]. This, however, follows easily from the fact that φ|ui [0,1] = λi ◦ u−1 i for all x ∈ ui [0, 1]. The preceding arguments indicate that there might exist an infinite sum representation of a continuous fractal function. Such a representation will be derived for fractal functions generated by operators of the form Eq. (5.11). For the unique fixed point f of , one has −1 f = λi ◦ u−1 i1 + s f ◦ ui1

for all x ∈ ui1 [0, 1]. Hence for x ∈ ui1 ◦ ui2 [0, 1], −1 −1 f ◦ u−1 i1 (x) = λi2 ◦ ui2 (x) + sf ◦ (ui1 ◦ ui2 ) (x).

Since (X, u) is a contractive IFS with attractor X, for each x ∈ X there is a code ix ∈  such that x = γ (ix ). Let x ∈ X be arbitrary and let ix := (i1 i2 . . . in . . .) be its code. By the preceding argument, f (x) =

n 

−1 n sν−1 λiν ◦ u−1 iν (x) + s f ◦ uix (n) (x)

ν=1

for all x ∈ uix (n) [0, 1].Whence

Construction of fractal functions

f (x) =

∞ 

157

sν−1 λiν ◦ u−1 iν (x)

(5.25)

ν=1

for x = γ (ix ).

2 Local fractal functions In this section we present a generalization of fractal functions that is based on local IFSs. These local fractal functions provide more flexibility for fractal interpolation. Here we briefly discuss some of their properties (see Refs. [42, 80]). To this end, let X be a nonempty set and {Xi : i ∈ NN } be a family of nonempty subsets of X. Suppose {ui : Xi → X}i∈NN is a family of bijective mappings with the following property: (P) {ui (Xi )}i∈NN forms a partition X; that is , X = for all i = j ∈ NN .



ui (Xi ) and int ui (Xi ) ∩ int uj (Xj ) = ∅

i∈NN

Now suppose that (Y, dY ) is a complete metric space with metric dY . Recall that L∞ (X, Y) denotes the set of all bounded functions X → Y endowed with the metric d(f , g) := sup{dY (f (x), g(x)): x ∈ X}. (L∞ (X, Y), d) is a complete metric space. Similarly, we define L∞ (Xi , Y), i ∈ NN . Remark 44. Under the usual addition and scalar multiplication of functions, the spaces L∞ (X, Y) and L∞ (Xi , Y) become metric linear spaces. A metric linear space is a vector space endowed with a metric under which the operations of vector addition and scalar multiplication are continuous [81]. For i ∈ NN , let vi : Xi × Y → Y be a mapping with the property that there exists an  ∈ [0, 1) such that for all y1 , y2 ∈ Y dY (vi (x, y1 ), vi (x, y2 )) ≤  dY (y1 , y2 ),

∀x ∈ X.

(5.26)

In other words, vi is uniformly contractive in the second variable. Define an RB operator : L∞ (X, Y) → Y X by f (x) :=

N 

−1 vi (u−1 i (x), fi ◦ ui (x))χui (Xi ) (x),

(5.27)

i=1

where fi := f |Xi . We note that  is well defined and as f is bounded and each vi is contractive in the second variable, f ∈ L∞ (X, Y). Then Eq. (5.26) implies the following inequality: d(f , g) = sup dY (f (x), g(x)) x∈X −1 −1 −1 = sup dY (v(u−1 i (x), fi (ui (x))), v(ui (x), gi (ui (x)))) x∈X

−1 ≤  sup dY (fi ◦ u−1 i (x), gi ◦ ui (x)) ≤  dY (f , g). x∈X

(5.28)

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Fractal Functions, Fractal Surfaces, and Wavelets

N Here we set v(x, y) := i=1 vi (x, y)χXi (x). As  is a contraction on the complete metric space L∞ (X, Y), the Banach fixed-point theorem yields a unique fixed point f ∈ L∞ (X, Y). We refer to this unique fixed point as a local fractal function of class L∞ (X, Y). Now suppose that (Y, dY ) is an F-space. Suppose that the mappings vi : Xi × Y → Y has the special form vi (x, y) := λi (x) + Si (x)y,

i ∈ NN ,

(5.29)

where λi ∈ L∞ (Xi , Y) and Si : Xi → R is a function. We further require the metric dY to be homogeneous; that is, dY (αy1 , αy2 ) = |α| dY (y1 , y2 ),

∀α ∈ R ∀y1 , y2 ∈ Y.

If the function Si is bounded on Xi with bound in [0, 1), then vi given by Eq. (5.29) satisfies condition (5.26): dY (λi (x) + Si (x)y1 , λi (x) + Si (x)y2 ) = dY (Si (x)y1 , Si (x)y2 ) = |Si (x)|dY (y1 , y2 ) ≤ Si ∞,Xi dY (y1 , y2 ) ≤ s dY (y1 , y2 ). Here s := max{Si ∞,Xi : i ∈ NN } and  · ∞,Xi denotes the supremum norm with respect to Xi . Fixing the set of functions {λi }i∈NN and {Si }i∈NN , we find the RB operator (5.27) now has the form f =

N 

λi ◦ u−1 i χui (Xi ) +

i=1

N  −1 (Si ◦ u−1 i ) · (fi ◦ ui )χui (Xi ) , i=1

or equivalently fi ◦ ui = λi + Si · fi ,

on Xi , ∀ i ∈ NN ,

where fi = f |Xi . In the following we set N

∞ (Xi , Y) := X L∞ (Xi , Y) λ := (λ1 , . . . , λN ) ∈ LN i=1

and N

∞ (Xi , R) := X L∞ (Xi , R). S := (S1 , . . . , SN ) ∈ LN i=1

Construction of fractal functions

159

Theorem 75. Let Y be an F-space with homogeneous metric dY . Let X be a nonempty set and {Xi }i∈NN a family of nonempty subsets of X. Suppose {ui : Xi → ∞ (X , Y), X}i∈NN is a family of bijective mappings satisfying property (P). Let λ ∈ LN i ∞ (X , R), and set s := max{S  : i ∈ N }. S ∈ LN i i ∞,Xi N ∞ (X , Y) × L∞ (X , R) × L∞ (X, Y) → L∞ (X, Y) by Define a mapping : LN i i N (λ)(S)f =

N 

λi ◦ u−1 i χui (Xi ) +

i=1

N 

−1 (Si ◦ u−1 i ) · (fi ◦ ui )χui (Xi ) .

(5.30)

i=1

If s < 1, then the operator (λ)(S) is contractive on the complete metric space L∞ (X, Y) and its unique fixed point f satisfies the self-referential equation f =

N 

λi ◦ u−1 i χui (Xi ) +

i=1

N 

−1 (Si ◦ u−1 i ) · (f,i ◦ ui )χui (Xi ) ,

(5.31)

i=1

or equivalently f ◦ ui = λi + Si · f,i , on Xi , ∀i ∈ NN ,

(5.32)

where f,i = f |Xi . Proof. The statements follow directly from the arguments preceding the theorem. The fixed point f is called a bounded local fractal function. Remark 45. The local fractal function f generated by the operator defined by Eq. (5.30) depends not only on the family of subsets {Xi }i∈NN but also on ∞ (X , Y) and S ∈ L∞ (X , R). the two N-tuples of bounded functions λ ∈ LN i i N The fixed point f should therefore be written more precisely as f (λ)(S). However, for notational simplicity we usually suppress this dependence for both f and . Example 28. Let X := [0, 1] and Y := R. Let X1 := [ 13 , 23 ] and X2 := X3 := [ 12 , 1] and let u1 : X1 → X, x → x − 12 , u2 : X2 → X, x → 12 (3x + 1), and u3 : X3 → X, x → 1 3 1 2 2 (x + 1). Let λ1 := λ2 := λ3 := 1 and let S1 := 4 , S2 := 2 , and S3 := − 3 . In Fig. 5.4 the graph of the bounded local fractal function corresponding to these data is shown. The following result found in Ref. [82] and in more general form in Ref. [83] is the extension to the setting of local fractal functions. ∞ (X , Y) to Theorem 76. The mapping λ → f (λ) is a linear isomorphism from LN i ∞ L (X, Y). ∞ (X , Y). Injectivity follows immediately Proof. Let α, β ∈ R and let λ, μ ∈ LN i from the fixed-point Eq. (5.31) and the uniqueness of the fixed point: λ = μ ⇐⇒ f (λ) = f (μ).

160

Fractal Functions, Fractal Surfaces, and Wavelets

2.5 2.0 1.5

0.2

0.4

0.6

0.8

1.0

0.5 0.0

Fig. 5.4 A bounded local fractal function.

Linearity follows from Eq. (5.31), the uniqueness of the fixed point, and injectivity: N  (αλi + βμi ) ◦ u−1 f (αλ + βμ) = i χui (Xi ) i=1

+

N  −1 (Si ◦ u−1 i ) · (f,i (αλ + βμ) ◦ ui )χui (Xi ) i=1

and αf (λ) + βf (μ) =

N  (αλi + βμi ) ◦ u−1 i χui (Xi ) i=1

+

N  −1 (Si ◦ u−1 i ) · (αf  , i(λ) + βf,i (μ) ◦ ui )χui (Xi ) . i=1

Hence f (αλ + βμ) = αf (λ) + βf (μ). For surjectivity, define λi := f ◦ ui − Si · f , i ∈ NN . Since f ∈ L∞ (X, Y), we have ∞ (X , Y). Thus f (λ) = f . λ ∈ LN i   Next we investigate the set of discontinuities of the fixed point f of the RB operator (5.31); see also Ref. [42]. First we need a definition. Definition 105. Let f : dom f ⊂ R → R and let I ⊂ dom f be a nonempty open interval. The oscillation of f on I is defined as osc(f ; I) := sup f (x) − inf f (x) = sup |f (x1 ) − f (x2 )|, x∈I

x∈I

x1 ,x2 ∈I

and the oscillation of a function f at a point x0 inside an open interval contained in its domain is defined by

Construction of fractal functions

161

osc(f ; x0 ) := lim ω(f ; (x0 − δ, x0 + δ)), δ→0

δ > 0.

Theorem 77. Let X := [0, 1], Y := R, and let  be given as in Eq. (5.31). Assume that each ui is contractive and that the mapping λi is continuous on cl Xi , i ∈ NN . Then the set of discontinuities of f is at most countably infinite. Proof. By the Banach fixed-point theorem we can choose any bounded function, say, f0 = χ[0,1] , to construct a sequence of iterates fn := fn−1 , n ∈ N, that, under the given hypotheses, converge in the L∞ -norm to f . Each iterate fn may have finite jump discontinuities at the interior knots {xj }j∈NN−1 , xj := cl uj (Xj ) ∩ cl uj+1 (Xj+1 ), and also at the images ui1 ◦ ui2 ◦ · · · ◦ ui−1 (xj ) of the interior knots. As the sets Xi may contain only a subset of the interior knots, the number of possible discontinuities at level n is bounded above by N n−1 (N −  1). Let En be the En . E is at most finite set of all finite jump discontinuities at level n and let E := n∈N

countably infinite. Let x ∈ [0, 1] \ E and let ε > 0. From the fixed-point equation for f , f (ui (x)) = λi (x) + Si (x)f,i (x),

x ∈ Xi ,

we obtain that for all intervals I ⊂ Xi , osc(f ; ui (I)) ≤ s osc(f ∗ ; I) + |I|, where s := max{Si ∞,Xi : i ∈ NN } < 1 and   = maxi∈NN supx∈Xi |λi (x)|. Thus for Nm any finite code σ |K := σ1 σ2 · · · σK ∈  := N of length K ∈ N, one has that m∈N0



osc(f ; uσ |K (I)) ≤ sK osc(f ; I) +  |I| aσ2 ···σK + saσ3 ···σK + · · · + sK−2 aK + sK−1   ≤ sK osc(f ∗ ; I) +  |I| aK−1 + saK−2 + · · · + sK−2 a + sK−1 ≤ sK osc(f ∗ ; I) +  |I|

aK |a − s|



(5.33)

for all intervals I ⊂ Xi . Here we set a := max{ai : i ∈ NN } < 1. Notice that (X, (Xi , ui )i∈NN ) is a contractive local IFS with attractor [0, 1]. As the local IFS (X, (Xi , ui )i∈NN ) is point fibered, there exists a code σ ∈  = N∞ N such that  γ (σ ) = {x} = uσ |k (X). k∈N

Given any K ∈ N, we can find a nonempty compact interval IK such that x ∈ IK ⊂

K  k=1

uσ |K (X).

162

Fractal Functions, Fractal Surfaces, and Wavelets

−1 −1 The length |IK | of IK is bounded above by aK . Let J := u−1 σ |K (IK ), where uσ |K := uσK ◦ · · · ◦ u−1 σ1 . Employing Eq. (5.33) yields

osc(f ; IK ) = osc(f ∗ ; uσ |K (J)) ≤ sK osc(f ; J) +  |J|

aK . |a − s|

As f is bounded on [0, 1], |J| ≤ 1, and aK → 0 as K → ∞, one can choose K large enough so that sK osc(f ; J) < 2ε and  |J| aK /|a − s| < 2ε . Hence osc(f ; IK ) < ε, which proves the continuity of f at all points in [0, 1] \ E and completes the proof. Corollary 5. Under the assumptions of Theorem 77, the fixed point f is Riemann integrable over [0, 1]. Proof. This follows from Theorem 77 and, for example, Theorem 7.5 in Ref. [84]. Next we present the relation between the graph G of the fixed point f of  given by Eq. (5.27) and the local attractor of an associated contractive local IFS. For this purpose, we require that X is a closed subset of a complete metric space. Consider then the complete metric space X × Y and define mappings wi : Xi × Y → X × Y by wi (x, y) := (ui (x), vi (x, y)),

i ∈ NN .

We need to assume, in addition, that the mappings vi : Xi × Y → Y are uniformly Lipschitz continuous in the first variable; that is, that there exists a constant L > 0 such that for all y ∈ Y, dY (vi (x1 , y), vi (x2 , y)) ≤ L dX (x1 , x2 ),

∀x1 , x2 ∈ Xi , ∀i ∈ NN .

Denote by a := max{ai : i ∈ NN } the largest of the Lipschitz constants of the mappings {ui : Xi → X}i∈NN and let  := (1 − a)/(2L). It is readily verified that the mapping d : (X × Y) × (X × Y) → R,

d := dX +  dY ,

is a metric for X × Y, which is compatible with the product topology on X × Y. Theorem 78. The family wloc := (X × Y, (Xi × Y, wi )i∈NN ) is a contractive local IFS with respect to the metric d and graph f is a local attractor for wloc . Moreover, graph (f ) = wloc (graph f ),

(5.34)

where on the right-hand side wloc denotes the set-valued operator (Eq. 2.49) associated with the local IFS w loc . Proof. First we establish that (X × Y, (Xi × Y, wi )i∈NN ) is a contractive local IFS. To this end, let (x1 , y1 ), (x2 , y2 ) ∈ Xi × Y, i ∈ NN , and note that

Construction of fractal functions

163

d (wi (x1 , y1 ), wi (x2 , y2 )) = dX (ui (x1 ), ui (x2 )) + dY (vi (x1 , y1 ), vi (x2 , y2 )) ≤ a dX (x1 , x2 ) + dY (vi (x1 , y1 ), vi (x2 , y1 )) + dY (vi (x2 , y1 ), vi (x2 , y2 )) ≤ (a + L)dX (x1 , x2 ) +  s dY (y1 , y2 ) ≤ r d ((x1 , y1 ), (x2 , y2 )). Here we used Eq. (5.26) and set r := max{a + L, s} < 1. As graph f satisfies wloc (graph f ) = = = =

N 

wi (graph f ) ∩ Xi =

i=1 N 

N 

wi ({(x, f (x): x ∈ Xi )})

i=1

{(ui (x), vi (x, f (x))): x ∈ Xi }

i=1 N 

{(ui (x), f (ui (x))): x ∈ Xi }

i=1 N 

{(x, f (x)): x ∈ ui (Xi )} = graph f ,

i=1

it is a local attractor for wloc . Eq. (5.34) follows from the above computation and the fixed-point equation for f written in the form f ◦ ui (x) = vi (x, f (x)),

x ∈ Xi , i ∈ NN .

3 Fractal bases for fractal functions In this section we show that for a special choice of functions λi the associated space of fractal functions F | [0, 1] := F | ([0, 1], R) that have the interpolation property possesses a fractal basis of Lagrange type. For this purpose, denote by Pd the class of real-valued polynomials of order d ∈ N or equivalently of degree at most d − 1. Suppose that := {(xj , yj ) ∈ [0, 1] × R: 0 =: x0 < x1 < · · · < xN−1 < xN := 1} is a given interpolation set. Assume further that we are given contractive homeomorphisms ui : [0, 1] → [xi−1 , xi ] satisfying ui (x0 ) := xi−1 and ui (XN ) := xi ,

i ∈ NN ,

164

Fractal Functions, Fractal Surfaces, and Wavelets

and mappings vi : [0, 1] × R → R defined by vi (x, y) := λi (x) + si y, where λi ∈ Pd , si ∈ R, i ∈ NN , and required to satisfy conditions (A) and Eq. (5.16). Recall that (see Section 1) if we choose C∗ [0, 1] := {f ∈ CR [0, 1]: f (xj ) = yj ∧ j ∈ N0N }, the mappings vi also satisfy condition (B0 ) provided s := max{|si |: i ∈ NN } < 1. The associated RB operator  given in Eq. (5.11) is then contractive on C∗ [0, 1] and its unique fixed point f ∈ F | [0, 1] ∩ C∗ [0, 1]. Theorem 76 adapted to our current setting then states that the mapping N

X Pd  λ → f (λ) ∈ F | [0, 1] ∩ C∗ [0, 1]

i=1

is a linear isomorphism. Proposition 46. Let Sd [0, 1] := F | [0, 1] ∩ C∗ [0, 1]. Then dim Sd [0, 1] = N(d − 1) + 1. N

Proof. This follows from Theorem 76, the fact that dim X Pd = Nd, and that there i=1

are (N − 1) join-up conditions to guarantee continuity of f across [0, 1]. Remark 46. The space Sd [0, 1] is the fractal analog of a spline space for polynomials of order d. In terms of nonstandard analysis, the elements of Sd [0, 1] can be thought of as hyperfinite splines. The interested reader is referred to Refs. [34, 85, 86] for more details on this fascinating connection. Let us consider the special case of affine fractal interpolation functions (ie, d := 2). Then λ is uniquely determined by in the sense that there exists a linear isomorphism N

RN+1  y := (y0 , y1 , . . . , yN )t → λ = (λ1 , . . . , λN ) ∈ X P2 . i=1

By Theorem 76, this extends to a linear isomorphism RN+1  y := (y0 , y1 , . . . , yN )t → f (λ) ∈ S2 [0, 1].

(5.35)

Hence f (λ) is uniquely determined by y, and one can write f (y) = f (λ). Note that this is nothing but the construction of affine fractal interpolation functions given a set of interpolation points. t N+1 can be uniquely written in the form y =  Every vector y := (y0 , y1 , . . . , yN ) ∈ Rt N+1 , with “1” being in the jth position. j∈NN0 yj ej , where ej = (0, . . . , 1, . . . , 0) ∈ R Denote by ej the unique affine fractal function generated by ej ; that is, the affine fractal interpolation function corresponding to the interpolation set j := {(xj , δjk ): k ∈ N0N },

Construction of fractal functions

165 0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

−0.2 −0.4

1.0

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.6 0.4 0.2

0.2 0.2

0.4

0.6

0.8

1.0

0.4

−0.2

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

−0.2

Fig. 5.5 An affine fractal function (top) and its three basis functions (bottom).

j ∈ N0N . The linear isomorphism (Eq. 5.35) and the uniqueness of the fixed point imply that every f ∈ Sd [0, 1] is of the form f (y) =

N 

yj ej .

(5.36)

j=0

Example 29. Let N := 2, let := {(0, 0), ( 12 , − 14 ), (1, 12 )}, and let s1 := 12 and s2 := − 34 . Then λ1 (x) := − 12 x and λ2 (x) := − 14 + 98 x. The graph of the affine fractal function and the graphs of the three basis fractal functions are shown in Fig. 5.5. In summary, we have the following results for the case d := 2 N

Proposition 47. Assume that λ ∈ X P2 . Then the linear space S2 [0, 1] has i=1

dimension N + 1. Furthermore, every element f ∈ S2 [0, 1] is the unique linear combination f =

N  j=0

yj ej ,

166

Fractal Functions, Fractal Surfaces, and Wavelets

where ej is the unique affine fractal function interpolating the set j := {(xj , δjk ): k ∈ N0N },

j ∈ N0N .

We refer to the unique affine fractal function ej interpolating the set j := {(xj , δjk ): k ∈ N0N }, j ∈ N0N , as an affine fractal function of Lagrange type. The basis B := {ej : j ∈ N0N } ⊂ S2 [0, 1] may be orthonormalized with use of the Gram orthonormalization procedure. To do this, one has to derive formulae for the integral, the moments, and the inner product of affine fractal functions. This will be done in Theorem 84 and Proposition 56. The basis representation introduced in Proposition 47 will become important in Chapter 8 when we construct fractal function wavelets.

4 Recurrent sets as fractal functions The recurrent set formalism of Dekking can also be used to construct fractal functions. A specific example will illustrate how this may be done in general (see Eq. 4.3 in Ref. [4]). Let X := {a, b} and let S[X] be the free semigroup generated by X. Define a free semigroup endomorphism ϑ: S[X] → S[X] by ϑ(a) := a b b b,

ϑ(b) := b a a a.

Let f : S[X] → R2 be defined by f (a) := (1, 1),

f (b) := (1, −1).

Since RX ∼ = R2 , Lϑ is the full representation of ϑ and is given by Lϑ =

4 0

0 −2

.

Finally, define K: S[X] → R2 by K(x) := {tf (x): t ∈ [0, 1]}. Then the recurrent set Kϑ (a) is the graph of a continuous fractal function passing through the points (0, 0), ( 14 , − 12 ), ( 12 , 0), ( 34 , 12 ), and (1, 1). This function is known as Kiesswetter’s fractal function or Kiesswetter’s curve [87]. The graph of Kiesswetter’s fractal function is displayed in Fig. 5.6. Now it is shown how univariate Rn -valued fractal functions can be generated with the recurrent set formalism. But first we require some definitions.

Construction of fractal functions

167

1

0.5

0.2

0.4

0.6

0.8

1

−0.5

−1

Fig. 5.6 Kiesswetter’s fractal function.

Definition 106. Let k: [0, 1] → Rn be a continuous mapping. The image of k in Rn is called a Jordan curve. Definition 107. A free group endomorphism ϑ: F[X] → F[X] is called null free iff for all x ∈ X, ϑ(x) = e, where e denotes the empty word. The next result is proven in Ref. [4]. Theorem 79. Let ϑ be a null-free endomorphism of S[X] and let Lϑ be an expansive representation of ϑ. Let f : S[X] → Rn , K: S[X] → H(Rn ), and Kν (ϑ): S[X] → H(Rn ) be defined as in Eqs. (2.52), (2.54)–(2.56). Then for any nonempty word w ∈ S[X], there exists a set Kϑ (w) ∈ H(Rn ) such that Lϑ−ν Kν (ϑ)(w) −→ Kϑ (w) h

as ν → ∞. Moreover, the set Kϑ (·) does not depend on the choice of the mapping K and it is a curve. Proof. The existence of the set Kϑ (w) follows from Theorem 41. It remains to be shown that Kϑ (w) is a curve. Let w ∈ S[X], and for all ν ∈ N let ϑ ν := wν 1 , . . . , wνμ(ν) . For κ ∈ {1, . . . , μ(ν)} define line segments Iν,κ := [(κ − 1)/μ(ν), κ/μ(ν)]. Since Kν (w) is a curve, there exist functions kν : [0, 1] → Rn such that

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Fractal Functions, Fractal Surfaces, and Wavelets

kν (Iν,κ ) = Lϑ−ν K(wνκ ) + f (wν 1 . . . wνκ−1 ). For every t ∈ [0, 1] choose a t = t (t) ∈ Iν,κ with the property that the Euclidean distance dE (kν+1 (t), kν (t )) is minimized. Then dE (kν+1 (t), kν (t)) ≤ dE (kν+1 (t), kν (t )) + dE (kν (t ), kν (t )) ≤ h(Lϑ−ν K(wνκ ), Lϑ−(ν+1) K(wνκ )) + dE (kν (κ/μ(ν)), kν ((κ − 1)/μ(ν))) ≤ max h(Lϑ−ν K(x), Lϑ−(ν+1) K(x)) + max Lϑ−ν (f (x)) x∈X x∈X

−ν ≤ c δ + max f (x) λ . x∈X

Here we used Lemma 2 as well as the notation introduced in the proof of Theorem 41. Hence the functions kν converge uniformly to a continuous function k: [0, 1] → Rn .  Since [0, 1] = Iν,κ , it is easy to see that k[0, 1] is indeed Kϑ (w). Whence Kϑ (w) is a κ,ν

Jordan curve in Rn . To show the independence of Kϑ (w) on the mapping K, assume that K and K are two mappings defined by Eq. (2.54). Let c and λ be as in Lemma 2. Then, by the properties of the Hausdorff metric h, h(Lϑ−ν K(ϑ ν (w)), Lϑ−ν K (ϑ ν (w))) ≤ max h(Lϑ−ν K(x), Lϑ−ν K (x)) x∈X

≤ max c λ−ν h(K(x), K (x)). x∈X

Thus as ν → ∞, Lϑ−ν K (ϑ ν (w)) → Kϑ (w). Kiesswetter’s fractal function fK can also be generated with IFSs: Define X := [0, 1] and Y := [−1, 1]. The maps wi : X × Y → X × Y are given by w1 (x, y) =

1 4

0

0 − 12



x y

(5.37)

and wi (x, y) =

1 4

0

i−1

x 4 + i−3 1 y 2 2

0

for i = 2, 3, 4, or equivalently x i−1 + , i = 1, 2, 3, 4, 4 4 y y i−3 v1 (x, y) = − , vi (x, y) = + 2 2 2 ui (x) =

for i = 2, 3, 4, and

(5.38)

Construction of fractal functions

169

εi f ◦ u−1 i 2

K f :=

for x ∈ ui [0, 1], and f ∈ C∗ ([0, 1], [−1, 1]) := {f ∈ C([0, 1], [−1, 1]): f (0) = 0 ∧ f (1) = 1}, where  εi :=

−1 for i = 1; +1 for i = 2, 3, 4.

The graph GK of Kiesswetter’s fractal function fK is then the unique attractor of the preceding contractive IFS, or equivalently fK is the unique fixed point of the RB operator K .

5 Iterative interpolation functions In a series of articles Dubuc [75, 76, 88] introduced an iterative interpolation process for interpolation data defined on a closed discrete subgroup of Rn . For n = 1 this interpolation process defines univariate fractal functions having the interpolation property with respect to . This section is devoted to the study of this case. However, the definitions and results are stated in such a way that the generalization to Rn is immediate. But first we require some definitions. Definition 108. Let G be a group and ◦ its group operation. A topology G on the set G is said to be compatible with the group structure iff the mappings α: G × G → G, α ι (g1 , g2 ) → g1 ◦ g2 , and ι: G → G, g → g−1 , are continuous. The pair (G, G) is called a topological group iff G is a group and G is a topology on G that is compatible with the group structure. A rather simple example of a topological group is provided by the additive group (R, +) whose topology G|·| is defined by the metric | · |. Compatibility with respect to the group structure means that rn → r and sn → s imply rn + sn → r + s and −rn → −r. Definition 109. A subset Y of a topological space (X, X) is called discrete iff every y ∈ Y has a neighborhood N such that N ∩ Y = {y}. For example, the integers Z form a discrete subset of the topological group (R, +). Indeed, (Z, G|·| ) is a closed discrete topological subgroup of (R, G|·| ). For the remainder of this section it is assumed that X is a closed discrete subgroup of R whose metric topology is inherited from R and that the vector subspace vec X generated by X is dense in R. (Recall that a vector space is an abelian group over a field satisfying certain compatibility conditions.) Furthermore, assume that a set of complex-valued interpolation points over G is given; that is, = {(x, y): x ∈ X

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Fractal Functions, Fractal Surfaces, and Wavelets

∧y ∈ Y}, where Y is an |X|c -element subset of C. The set is considered as the graph of a univariate complex-valued function f : X → Y. The basic idea of Dubuc’s interpolation process is to extend f to functions fk whose domains are subgroups Xk ⊃ Xk−1 ⊃ · · · ⊃ X such that their union is dense in R. These extensions are defined via a linear transformation T: R → R and a complex-valued weight function p satisfying the following conditions: 1. 2. 3. 4.

TX ⊃ X. dom p = TX. p(0) = 1 and p(x) = 0 for all 0 = x ∈ X. |supp p|c < ∞.

A nested increasing sequence of subgroups is defined by X0 := X and Xk := T k X for  Xk . Note that X∞ is the direct limit of the directed system all k ∈ N. Let X∞ := k∈N0

{Xk , →}, where → denotes the injection map Xk−1 → Xk : X∞ = lim Xk . − → Proposition 48. There exists a unique function g: X∞ → C such that g ≡ f on X and for any k ∈ N g(T k+1 ·) =



p(T · −x ) g(T k x ).

(5.39)

x ∈X

Remark 47. The function g is called the iterative interpolation of f with respect to T and p and the quadruple (f , X, T, p) is called an iterative interpolation process. Proof. Since vec X is dense in R and TX ⊃ X, it follows that TR = R, and hence that T is injective. A sequence of functions fk : Xk → C is defined by f0 := f and fk (T k x) :=



p(Tx − x )fk−1 (T k−1 x )

(5.40)

x ∈X

for all x ∈ X. The injectivity of T implies that fk is well defined on Xk . Note that the finiteness of the support of p implies that the sum in Eq. (5.40) is finite. Clearly fk is an extension of fk−1 . Thus there is a unique function g: X∞ → C satisfying g ≡ fk on Xk for all k ∈ N. It is worthwhile noting some simple properties of the iterative interpolation process: Linearity: Let f1 and f2 be functions defined on X and let g1 and g2 be their respective iterative interpolations. If f := f1 + f2 and if g is its iterative interpolation, then g = g1 + g2 . Scalar multiplication: If f has g as its iterative interpolation and if c ∈ R, then c g is the iterative interpolation of c f .

Construction of fractal functions

171

Translation: Let y ∈ X and let g be the iterative interpolation of a function f . Then g(· + y) is the iterative interpolation of f (· + y). The following two examples illustrate the previously defined interpolation process. Example 30 (Dyadic Interpolation Process). Let X := Z, and define a linear transformation T by Tx := 2x and a weight function p: Z/2 → R by ⎧ 1 ⎪ ⎪ ⎪ ⎨9 p(x) :=

16

⎪ −1 ⎪ ⎪ ⎩ 16 0

for x = 0; for x = ± 12 ; for x = ± 32 ; otherwise.

This interpolation process is referred to as dyadic interpolation [75]. It will be seen shortly that the dyadic interpolation process can be interpreted as an IFS on [−3, 3]. The next example shows how the famous Koch curve can be obtained from iterative interpolation. Example 31 (Koch Curve). Again let X := Z. Define a linear transformation T: R → R by Tx := 4x and a weight function p: Z/4 → C by ⎧ 1 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ 3 ⎪ √ ⎪ ⎨1 + i 3 2 6 p(x) := 2 ⎪ ⎪3 ⎪ √ ⎪ ⎪ ⎪1 − i 3 ⎪ 6 ⎪ ⎩2 0

for x = 0; for x = ± 34 ; for x = − 12 ; for x = ± 14 ; for x = 12 ; otherwise.

For f one may use any function as long as f (0) = 0 and f (1) = 1. The continuous extension of g to the unit interval yields the Koch curve; see Fig. 5.7.

Fig. 5.7 The Koch curve.

Conditions guaranteeing the density of X∞ in R and the continuity of the iterative interpolation function g are derived next. This is best done by our introducing the fundamental interpolating function F.

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Fractal Functions, Fractal Surfaces, and Wavelets

Definition 110. Let (f , X, T, p) be an iterative interpolation process. Suppose that the function f is defined as follows:  f (x) :=

1 if x = 0; 0 otherwise.

The iterative interpolation of f is called the fundamental interpolating function F (with respect to T and p). Remark 48. Setting k = 0 in Eq. (5.39) implies F ≡ p on TX. The next proposition justifies the use of the term fundamental. Proposition 49. Let (f , X, T, p) be an iterative interpolation process and g the extension of f . Then g(x) =



f (x )F(x − x ).

(5.41)

x ∈X

For the proof of this proposition a lemma is needed.  j Lemma 6. Let k ∈ N0 and suppose x ∈ T k X \ j∈N0k T (supp p). Then F(x) = 0. Proof. The proof uses induction on k. The case k = 1 is obvious. Suppose that x ∈ X is such that F(T k+1 x) = 0. By Eq. (5.39) there exists an x ∈ X such that p(Tx−x ) = 0 k x) = 0. If x

:= Tx − x , then x

∈ supp p. Thus, by the induction hypothesis, and F(T k k

j k+1 x = T k x − T k x

, T k+1 x ∈ j T x ∈ k−1 j=0 T (supp p). j=0 T (supp p). Since T 

Proof of Proposition 49. By the preceding lemma, x ∈X f (x )F(x − x ) is finite for every x ∈ Xk . The linearity and  the translation property of the iterative interpolation process imply that the series x ∈X f (x )F(x − x ) is indeed  an interpolation function g for f . Since this interpolation function is unique, g ≡ x ∈X f (x )F(· − x ). Taking f (x) = F(Tx) in Proposition 49, using the linearity property of iterative interpolation, and the fact that F ≡ p on TX, one obtains the following useful functional equation for F: F(Tx) =



p(Tx )F(x − x ),

x ∈ X∞ .

(5.42)

x ∈X

The next result—which follows directly from Lemma 6—gives an upper bound for the support of the fundamental interpolation function F. Proposition 50. Assume that (f , X, T, p) is an iterative interpolation process and that T has spectral radius less than 1. Suppose that ρ is the length of the smallest interval centered at 0 containing the support of the weight function p. Then the support of F is contained in a closed interval of length ρ · k∈N0 T k  centered at 0. More can be said about the support of F in a special case. Proposition 51. Assume that (f , X, T, p) is an iterative interpolation process for which Tx = a−1 x for some 1 = a ∈ Z and X = Z. If the weights pn = F(a−1 n)

Construction of fractal functions

173

are real, then F vanishes outside the interval (M1 /(a − 1), M2 /(a − 1)), where M1 denotes the smallest index for which pn = 0 and M2 denotes the largest index for which pn = 0. Proof. Define a sequence of real numbers {tn } by t0 := 0,

tn = tn−1 + M1 b−n , n ∈ N.

Clearly {tn } is decreasing and converges to M1 /(a − 1). It now follows from Eq. (5.42) that F Xn ≡ 0 on (−∞, tn ) for all n ∈ N. Hence for every a-adique number t ≤ M1 /(a − 1), F(t) = 0. Similarly, one shows that F(t) = 0 for t ≥ M2 /(a − 1). At this point in the development the iterative interpolation scheme can be related to IFSs. This is best done by our considering the following simple example, which nevertheless contains all the ingredients necessary to understand this relation. Example 32. Let X := Z, and let T: Z → Z/2 be defined by Tx := 2x . Define the weight function p: Z/2 → C by p(0) := 1, p( 12 ) = p(− 12 ) := 34 , and p(x) := 0 for all x ∈ Z/2 \ {0, ± 12 }. Then by Proposition 51, the iterative interpolation function g vanishes outside (−1, 1). If f is chosen to be any function interpolating the set := {(n, F(n)): n ∈ {−1, 0, 1}}, then graph g is the attractor of the contractive IFS ([−1, 1], w) with w: [0, 1] × [0, 1] → [0, 1] × [0, 1] given by wi (x, y) =

1 2

1 i−1 2 (−1)

0 si

1

x (−1)i 2 + 1 y 2

for i = 1, 2. The contractivity constants are given by s1 = p(− 12 ) = 34 and s2 = p( 12 ) = 34 . The graph of g is depicted in Fig. 5.8. Now it should be clear how the dyadic interpolation process is interpreted as an IFS: The graph of the dyadic iterative interpolation function g is the unique attractor of a contractive IFS defined on [−3, 3] × [−3, 3] consisting of four maps wi which all scale 1 9 , s2 = p(− 12 ) = 16 , s3 = horizontally by 12 and vertically by s1 = p(− 32 ) = − 16 1 9 3 1 p( 2 ) = 16 , and s4 = p( 2 ) = 16 . For f , any function interpolating = {(n, F(n)): n ∈ {−3, −2, −1, 0, 1, 2, 3}} can be taken. Necessary and sufficient conditions are given next for the fundamental interpolation function to be continuous. Our presentation follows closely that in Ref. [76]. Definition 111. The iterative interpolation process (f , X, T, p) is called continuous iff its fundamental interpolation function F is uniformly continuous. Remark 49. The preceding definition is based on the following well-known fact from analysis: If F: D ⊆ Rn → R is uniformly continuous on its domain D, then it can be continuously extended to the closure of D. Therefore if (f , X, T, p) is continuous, then the iterative interpolation g of f is continuous. At this point it is natural to ask under what conditions X∞ is dense in R. The following proposition gives sufficient conditions. Proposition 52. Suppose that X is a subgroup of Rn and that vec X = Rn . If a linear transformation T: Rn → Rn is such that (1) X ⊂ TX and (2) for all λ ∈ σ (T): |λ| < 1, then X∞ is dense in Rn .

174

Fractal Functions, Fractal Surfaces, and Wavelets

2.5

2.0

1.5

1.0

0.5

−1.0

−0.5

0.5

1.0

Fig. 5.8 The iterative interpolation function g in Example 32.

Proof. Let ε > 0 be given. Then there exists an integer n such that T n  < ε. By the surjectivity of T, one can find for an arbitrary y ∈ R a z ∈ R such that T n z = y. Since vec X = Rn , there is an x ∈ X such that dE (x, z) < δ for some δ > 0. Thus dE (T n x, y) ≤ δ ε. If n = 1, it is easy to see that T has to be a contractive linear transformation on R for X∞ to be dense in R. Thus for such a T the continuous iterative interpolation process (f , X, T, p) defines a fundamental interpolation function F that can be continuously extended to all of R. In what follows, no distinction is made between F and its continuous extension to R. Definition 112. A function g: X∞ → C is called an interpolation function iff there exists a function f : X → C such that g is the iterative interpolation for an iterative interpolation process (f , X, T, p). Two types of modulus of continuity have to be considered for the study of continuity of an iterative interpolation process. Definition 113. Let (f , X, T, p) be an iterative interpolation process and F its fundamental interpolation function. Let g: X∞ → C and η ∈ R+ . Then for k ∈ N, define

Construction of fractal functions

 Ck (η) := max

x,x ∈X



175



k



|F(T x − x ) − F(T x − x )|: |x − x | ≤ η k

(5.43)

x

∈X

and ωk (g, η) := sup{|g(T k x) − g(T k x )|: x, x ∈ X ∧ |x − x | ≤ η}.

(5.44)

Some more notation needs to be introduced: let Sk := {x ∈ Xk : F(x) = 0} and let ρk denote the radius of the smallest ball centered at 0 containing Sk . The next three lemmas are required to prove one of the main theorems. Lemma 7. Assume that T is contractive and η ≥ ρ1 /(1 − T). Furthermore, assume that ξ ∈ R and x ∈ X are such that |x − ξ | ≤ η. Then if g is an interpolation function, g(Tx) =



F(Tx − x )g(x ).

x ∈X |x −Tξ |≤η

 Proof. Note that Eq. (5.41) implies g(Tx) = x ∈X F(Tx − x )g(x ). Now suppose that x ∈ X and x ∈ X are such that F(Tx − x ) = 0. Then we have x − Tξ = T(x − ξ ) − (Tx − x ). But since Tx − x ∈ S1 , |x − Tξ | ≤ Tη + ρ1 ≤ η. that g is an interpolation function. Assume that for any x ∈ X1 ,  Lemma 8. Suppose

x ∈X F(x − x ) = 1. If T < 1 and η ≥ 2ρ1 /(1 − T), then ωk (g, η) ≤ 2−k C1 (η)k ω0 (g, η). Proof. First consider the case k = 1. For a given η > 0, let x1 , x2 ∈ X be such that |x1 − x2 | ≤ η. Setting ξ = (x1 + x2 )/2 implies |x1 − ξ | = |x2 − ξ | ≤ η2 . Hence by Lemma 7,  F(Txi − x )g(x ) g(Txi ) = x ∈X |x −Tξ |≤η

for i = 1, 2. Now let m := min{g(x ): x ∈ X ∧ |x − Tξ | < η2 }, M := max{g(x ): x ∈ η 1

X ∧ |x − Tξ | < 2 }, and c := 2 (m + M). Then, with use of the hypothesis that x ∈X F(x − x ) = 1 for all x ∈ X1 , it follows that, for i = 1, 2, g(Txi ) − c =

 x ∈X η |x −Tξ |≤ 2

F(Txi − x )(g(x ) − c).

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Fractal Functions, Fractal Surfaces, and Wavelets

Thus |g(Tx1 ) − g(Tx2 )| ≤



|F(Tx1 − x ) − F(Tx2 − x )| |g(x ) − c|,

x ∈X η |x −Tξ |≤ 2

and therefore |g(Tx1 ) − g(Tx2 )| ≤



|F(Tx1 − x ) − F(Tx2 − x )|

x ∈X η |x −Tξ |≤ 2



M−m . 2

This, however, implies that ω1 (g, η) ≤ 12 C1 (η) ω0 (g, η). Induction on k now yields the general result. Lemma 9. Assume that for a given η > 0 the subgroup Y[G] of X generated by G := {x ∈ X: |x| ≤ η} is all of X, and that η > 0. Then there exists a positive integer K such that for every k ∈ N and every function g: Gk → C, ωk (g, η ) ≤ Kω(g, η). (K depends on η and η , but not on g and k.) Proof. Let G := {x ∈ X: |x| ≤ η }. Then every element of G can be expressed as a sum of elements of G. Thus for sufficiently large K, every element of G can be written as a sum of K elements of G. If x, x ∈ X satisfy |x − x | ≤ η , then K + 1 elements {x0 , x1 , . . . , xK } ⊂ X can be found such that x0 = x, xK = x , and |x − x−1 | ≤ η for  = 1, . . . , K. Hence k

g(T x) − g(T x ) = k

K 

g(T k x−1 ) − g(T k x ),

=1

and thus |g(T k x) − g(T k x )| ≤ Kωk (g, η).

Now a theorem that gives sufficient conditions for an iterative interpolation process to be continuous can be established. Theorem 80. Let (f , X, T, p) be an iterative interpolation process.  Assume that T is contractive and the fundamental interpolation function F satisfies x ∈X F(x − x ) = 1 for all x ∈ TX. If there exists an η > 0 such that η ≥ 2ρ1 /(1 − T), C1 (η) < 2, and if Y[G] = X, then (f , X, T, p) is continuous. Proof. Note that by Lemmas 8 and 9 one can find a large enough integer n so that  2 k>n ωk (F, T−1 ρ1 ) + ωn (F, 4η) < ε for any given ε > 0. Let δ := 2η/T −n . It suffices to be shown that if x, x ∈ X∞ satisfy |x − x | < δ, then |F(x) − F(x )| < ε. To this end, let x, x ∈ X∞ be such that |x − x | < δ with δ := 2η/T −n . Since X∞ is a direct limit, there exits an integer m which can be chosen larger than n so that x, x ∈ Xm . A sequence {xk }n≤k≤m is constructed recursively as follows: For k = m, let

Construction of fractal functions

177

xm := x. Since for k ≤m, xk ∈ Xk , there exists a yk ∈ X with T k yk = xk . With use of the assumption that x ∈X F(x − x ) = 1 for all x ∈ TX, there is a z ∈ X such that Tyk − z ∈ S1 . Let yk−1 := z and xk−1 := T k−1 yk−1 . Note that xk−1 ∈ Xk−1 and—by definition of ρ1 —|Tyk − yk−1 | ≤ ρ1 . Therefore Eq. (5.44) implies that |F(xk ) − F(xk−1 )| = |F(T k yk ) − F(T k−1 yk−1 )| ≤ ωk (F, T −1 ρ1 ).  Hence |F(x) − F(xn )| ≤ k>n ωk (F, T−1 ρ1 ). In the same way one can define a sequence {xk }n≤k≤m for x , yielding |F(x ) −  F(xn )| ≤ k>n ωk (F, T−1 ρ1 ). Note that if y k is defined for x as yk is for x, then |yn − y n |

≤

m−n−1 

|T k yn+k − T k+1 yn+k+1 | + |T m−n ym − T m−n y m |

k=0

+

m−n−1 

|T k y n+k − T k+1 y n+k+1 |,

k=0

|yn − y n | ≤

m−n−1 

T k |yn+k − Tyn+k+1 | + |T −n x − T −n x |

k=0

+

m−n−1 

T k |y n+k − Ty n+k+1 |,

k=0

|yn − y n | ≤ 2

m−n−1 

T k ρ1 + |T −n x − T −n x |,

k=0

|yn − y n | ≤ 2η + T −n |x − x | ≤ 4η. So |F(xn ) − F(xn )| ≤ ωn (F, 4η). Thus combining the results found gives |F(x) − F(x )| ≤ 2



ωk (F, T −1 ρ1 ) + ωn (F, 4η) < ε.

k>n

Setting  T := T n and  p(x) := F(x) for x ∈ T n X and applying the preceding theorem yields the following corollary.  Corollary 6. Suppose the fundamental interpolation function F satisfies + x ∈X F(x − x ) = 1 for all x ∈ TX. If there exists an n ∈ N and an η ∈ R n n such that T  < 1, η ≥ 2ρn /(1 − T ), Cn (η) < 2, and Y[G] = X, then (f , X, T, p) is continuous. The next goal is to obtain necessary conditions for the continuity of an iterative interpolation process (f , X, T, p). But first we need a lemma. Lemma 10. Suppose the iterative interpolation process (f , X, T, p) is continuous  and that r(T) < 1. Then for all x ∈ TX, x ∈X F(x − x ) = 1.

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Fractal Functions, Fractal Surfaces, and Wavelets

Proof. In the functional Eq. (5.42), x is taken in TX and k → ∞. This yields 1 = lim F(T k x) = lim k→∞

k→∞



F(T k x )F(x − y) =

x ∈X



F(x − y).

x ∈X

(Here we used the fact that F(T k x ) → F(0) = 1 and that, because of finite support of F, the sum is independent of k.) Finally, we can give necessary conditions for the continuity of an iterative interpolation process. Theorem 81. Assume that the iterative interpolation process (f , X, T, p) is continuous and that r(T) < 1. Then there exists a k ∈ N and an η ∈ R+ such that η ≥ 2ρk /(1 − T k ), Ck (η) < 2, and the subgroup Y[G] of X generated by G is equal to X. Proof. Since the spectral radius of T is less than 1, the support of F is finite and the sequence ρk bounded. Then there exists a real number η such that Y[G] = X and that for all k ∈ N, η ≥ 2ρk /(1 − T k ). It remains to be shown that Ck (η) < 2 for all k ∈ N. Consider the sets Mξ := {x ∈ X: F(ξ − x) = 0}, where ξ ∈ R. Let M := max{|Mξ |c : ξ ∈ R}. Then by the uniform continuity of F, there exists a δ > 0 such that whenever x, x ∈ X∞ and |x − x | < δ, |F(x) − F(x )| < 1/M. Since r(T) < 1, there is a k ∈ N such that T k η < δ. But then Ck (η) < 2. Note that together with Lemma 10 and Theorem 81, Corollary 6 gives necessary and sufficient conditions for the continuity of an iterative interpolation process. Examples of continuous iterative interpolation processes have already been encountered: dyadic interpolation and the Koch curve. Using Corollary 6, one sets in the 1 3 former case k := 3. Then ρ3 = 21 8 and T  = 8 . The number η is chosen so that 3 2ρ3 /(1 − T ) = 6. Then after some straightforward calculations, C3 (η) = 74 . In the latter case, k is taken to be 1. Then ρ1 = 34 , T = 14 , η = 2ρ1 /(1 − T) = 2, and √ C1 (η) = 2/ 3 [76].

6 Recurrent fractal functions It is natural to extend the definition of fractal functions from IFSs to recurrent IFSs. This is the objective of this section. The terminology and notation introduced in Section 2.3 will be used. Let (X, dX ) and (Y, dY ) be complete metric spaces, and let X := {Xi }i∈NN be a partition of X. Let X := {Xk }k∈NN be a partition of X with the property that each Xk is a union of elements of X . Define functions ui : X → X so that the associated set-valued map ui : H(X) → H(X), ui E = {ui (x): x ∈ E}, maps elements of X onto elements of X . The ui are then contractive with constants |ai | < 1, i ∈ NN . Now let v(x, ·): Y → Y be defined as in Eq. (5.4) and require that conditions (B0 ) and (C) hold. As in Section 1, the maximum of the ai is denoted by a and the uniform Lipschitz constant of the vi (x, ·) is denoted by L. If we

Construction of fractal functions

179

endow X × Y with the metric d (see Eq. 5.13), it is seen that the mappings wi : X × Y → X × Y, wi (x, y) := (ui (x), vi (x, y)), become contractive with contractivity max{ 12 (1 + a), s} < 1. To obtain a recurrent structure, a connection matrix C = (cki ) is defined by  cki :=

1 if Xi ⊂ Xk ; 0 otherwise.

Then if we define a mapping Xw: XH(X) → XH(X) as in Section 2.3, Eq. (2.45), and apply Corollary 1, the existence of a set XA such that Xw(XA) = XA is inferred. Arguments analogous to those given in Section 1 show that XA is the graph of a continuous function f : X → Y. This function is called a recurrent fractal function. The class of all such recurrent fractal functions is denoted by RF (X, Y). Next it is shown that recurrent fractal functions also possess the interpolation property. This is done for X = [0, 1] and Y = R and a special class of recurrent fractal functions. The general case can then easily be inferred from this special case. Definition 114. Let X and Y be metric spaces. A mapping α: X → Y is called affine iff α = L + v, where L ∈ L(X, Y) and v ∈ Y. The linear space of all affine mappings from X into Y is denoted by Aff(X, Y). Definition 115. A fractal function f ∈ F (X, Y) is called affine iff for all i ∈ NN , ui ∈ Aff(X, X), and vi ∈ Aff(X × Y, Y). Given an interpolation set = {(xj , yj ): j ∈ N0N } ⊆ [0, 1]×R, note that {xj : j ∈ N0N } induces a partition on [0, 1]. Let Xi := [xi−1 , xi ] and let Xk := [xk−1 , xk ] be the union of some of the Xi . Next we define mappings wi ∈ Aff(X × Y, X × Y) by wi (x, y) :=

ai ci

0 si



x b + i , di y

(5.45)

where the ai , bi , ci , and di are uniquely determined by the conditions wi (xk−1 , yk−1 ) = (xi−1 , yi−1 ) and wi (xk , yk ) = (xi , yi ) (or wi (xk−1 , yk−1 ) = (xi , yi ) and wi (xk , yk ) = (xi−1 , yi−1 )) and the si ∈ (−1, 1) are free parameters. The recurrent fractal function f then interpolates . We denote the family of recurrrent fractal functions that interpolate the set by RF | ([0, 1], R). Example 33. Let := {(x0 , y0 ), (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), (x4 , y4 )}. Let X1 := [0, 12 ] and let X2 := [ 12 , 1]. Define w1 (x0 , y0 ) = (x0 , y0 ), w2 (x2 , y2 ) = (x1 , y1 ), w3 (x0 , y0 ) = (x3 , y3 ), w4 (x2 , y2 ) = (x3 , y3 ),

w1 (x2 , y2 ) = (x1 , y1 ), w2 (x4 , y4 ) = (x2 , y2 ), w3 (x2 , y2 ) = (x4 , y4 ), w4 (x4 , y4 ) = (x4 , y4 ).

The connection matrix is then given by

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Fractal Functions, Fractal Surfaces, and Wavelets

⎛

1 ⎜0 ⎜ C=⎝ 1 0

1 0 1 0

0 1 0 1

⎞ 0 1⎟ ⎟. 0⎠ 1

The attractor XA = (A1 , A2 , A3 , A4 ) then satisfies (see also Corollary 1) A1 = w1 (A1 ) ∪ w1 (A2 ), A3 = w3 (A1 ) ∪ w3 (A2 ),

A2 = w2 (A3 ) ∪ w2 (A4 ), A4 = w4 (A3 ) ∪ w4 (A4 ).

In Fig. 5.9 it is shown how the recurrent fractal function interpolating is constructed; see also Ref. [37].

w2

w6

w5

w1

w4

w3

Fig. 5.9 A recurrent fractal function with interpolation property.

7 Hidden-variable fractal functions In this section a new class of fractal functions called hidden-variable fractal functions is introduced. These fractal functions are the projections of a continuous function F whose graph is the attractor of a contractive IFS. The class of hidden-variable fractal functions is more diverse than F (X, Y), as their values depend continuously on all the “hidden” variables determining F, thus making them more appealing as interpolants. Suppose that (X, dX ) and (Y, dY ) are complete metric spaces. On the Cartesian product X × Y a metric dθ = dX + θ dY is introduced, where θ > 0 has yet to be specified. Let u: X → Y and v: X × Y → Y be mappings required to satisfy u ∈ Lip(≤s1 ) (X), where s1 ∈ [0, 1), ∀ y ∈ Y: v(·, y) ∈ Lip(X),

(5.46) (5.47)

∀ x ∈ X: v(x, ·) ∈ Lip(≤s2 ) (Y), where s2 ∈ [0, 1).

(5.48)

and

Construction of fractal functions

181

If w: X × Y → X × Y is defined by w(x, y) := (u(x), v(x, y)), then w is a contraction on the complete metric space (X × Y, dθ ) with θ := (1 − s1 )/2L, where L denotes the uniform Lipschitz constant of {v(·, y)| y ∈ Y} (see the proof of Theorem 74). Now let wi : X × Y → X × Y be maps of the form wi (x, y) = (ui (x), vi (x, y)), where ui and vi satisfy Eqs. (5.46)–(5.48) for all i ∈ NN . If θ is chosen as before, it can seen that (X × Y, w) is a contractive IFS on X × Y. Denote its attractor by A(X × Y). It follows directly from Eq. (5.46) that (X, u) is also a contractive IFS; its attractor is denoted by A(X). Note that projX A(X ×Y) = A(X). The idea is to impose conditions on wi so that A(X × Y) is the graph of a continuous function F: A(X) ⊆ X → Y. Theorem 82. Let (X × Y, w) be the IFS defined above. Let := {(xj , yj ): j ∈ N0N } be an interpolation set in X × Y and suppose that ∀ i ∈ NN : wi (x0 , y0 ) = (xi−1 , yi−1 ) = wi−1 (xN , yN ).

(5.49)

Furthermore, assume that ∀ i ∈ NN : ui is invertible on ui A(X), ∀ i ∈ NN : ui A(X) ∩ ui+1 A(X) = {xi },

(5.50) (5.51)

∀ |i − k| ∈ / {0, 1}: ui A(X) ∩ uk A(X) = ∅.

(5.52)

and

Then A(X × Y) = graph F , where F ∈ F | (A(X), Y). Proof. First it is shown that F is the unique fixed point of an appropriate RB operator . Let C∗ (A(X), Y) := {f ∈ C(A(X), Y): f (xj ) = yj , ∀ j ∈ N0N }. Then C∗ (A(X), Y) is a complete metric space in the supremum norm. Note that A(X) is a complete metric space in the relative strong topology. Define : C∗ (A(X), Y) → Y A(X) by f (x) :=

N 

−1 vi (u−1 i (x), f ◦ ui (x))χXi (x),

i=1

where Xi := ui X, i ∈ NN . Then Eqs. (5.51), (5.52) show that f is well defined on A(X) \ {x0 , . . . , xN } and continuous on Xi . Moreover, −1 lim f (x) = vi (u−1 i (xi ), f ◦ ui (xi ))

x→xi−

= vi (xN , f (xN )) = vi (xN , yN )

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Fractal Functions, Fractal Surfaces, and Wavelets

= vi+1 (x0 , y0 ) = vi+1 (x0 , f (x0 )) −1 = vi+1 (u−1 i+1 (xi ), f ◦ ui+1 (xi )) = lim f (x). x→xi+

A similar calculation shows that f (x0 ) = y0 and f (xN ) = yN . Therefore f ∈ C∗ (A(X), Y). The contractivity of  follows as in the proof of Theorem 73. Thus the unique fixed point of  is a continuous fractal function F : A(X) → Y; that is, an element of F (A(X), Y). To show that F also belongs to F | (A(X), Y), note that ⊂ A(X), since (x0 , y0 ) is the fixed point of w1 and (xN , yN ) is the fixed point of wN , and (xi , yi ) ∈ A(X), for i = 2, 3, . . . , N − 1, as wi (xN , yN ) = (xi , yi ). Now suppose that X is also an R-vector space and a finite direct sum of linear K  subspaces: X = Xk . The orthogonal projection of graph F onto Xk × Y yields the k=1

graph of a continuous function f,k : Xk → Y. Definition 116. The function f,k : Xk → Y whose graph is the orthogonal projection of graph F ⊂ X × Y onto Xk × Y, k = 1, 2, is called a hidden-variable fractal (interpolation) function. Remark 50 1. The term hidden variable was first introduced in Refs. [70, 89]. It has its origin in the fact that graph f,k depends continuously on all the parameters that generate F. This implies that, in general, graph f,k is not made up of finitely many copies of itself. (k)

2. The function f,k interpolates the data set k := {(xj , yj ): j ∈ N0N }, where X  x = (x(1) , x(2) , . . . , x(K) ). The set of all hidden-variable fractal functions f,k : Xk → Y is denoted by HF(Xk , Y) and, if the interpolation property of f,k needs to be emphasized, by f,k ∈ HF| k (Xk , Y).

There exists a natural way of defining hidden-variable fractal functions: Begin with an attractor A(X) of a certain contractive IFS (X, w) and choose Y homeomorphic to the code space associated with (X, w). More precisely, let X again be a complete metric space and let (X, w) be a contractive IFS whose maps wi , i ∈ NN , satisfy the following conditions: (C1) ∀ i ∈ NN : wi ∈ Lip(≤1) (X). (C2) Let x0 , . . . , xN be N + 1 distinct points in X. Then w1 (x0 ) = x0 , wN (xN ) = xN , and ∀ k ∈ NN−1 : wk+1 (x0 ) = xk = wk (xN ). (C3) The polygon joining the points x0 , . . . , xN is the image of [0, 1] under the homeomorphism h with h(0) = x0 and h(1) = xN . (C4) There exists a closed ball B ⊆ X such that: (a) B contains  the attractor A of (X, w); wi B; (b) B ⊆ i∈NN

(c) x0 , xN ∈ ∂B ∧ ∀ k ∈ NN−1 : wk B ∩ wk+1 B = {xk+1 }.

Let (, dF ) be the code space associated with the IFS (X, w). Let I denote the unit interval [0, 1]. Define a partition P on I by P := {t0 , . . . , tN }, with tj := Nj , j ∈ N0N , and its iterates by

Construction of fractal functions

183

P 2 := P ∨ P := {t00 , . . . , t0,N−1 , . . . , tN−1,0 , . . . , tN−1,N , tN,N },

where tj1 j2 := j1 /N + j2 /N 2 , j1 , j2 ∈ N0N−1 , and P n :=

n 

P,

m=1

with tj1 ...jn = j1 /N + j2 /N 2 + · · · + jn /N n for all j1 , j2 , . . . , jn ∈ N0N−1 , n ∈ N. The partitions {P n }n∈N define the N-ary expansion of a point t ∈ I which is unique except when t = q/N n for q ∈ NN−1 , n ∈ N, in which case there are exactly two representations. The code j := (j1 j2 . . . jn . . .) is called the representation of t = tj1 j2 ...jn ... ∈ I. On the collection J of all such codes j an operator σ+ : J → J is defined by σ+ (j1 j2 . . . jn ) := (j1 + 1j2 + 1 . . . jn + 1). The next result is straightforward to prove. Proposition 53. The mapping ξ : I →  defined by  σ+ j ξ(t) := σ+ (N − 1)

if 1 = t ∈ [tj1 j2 ...jn ... , tj1 j2 ...jn +1... ]; if t = 1,

where N − 1 := (N − 1 N − 1 . . . N − 1 . . .) is a homeomorphism. Recall that there exists a continuous surjection γ :   A which is defined by γ (i) = limn→∞ wi(n) (x), where the limit is independent of x ∈ A (see Theorem 37). Define a map ψ: I → A by ψ := γ ◦ ξ . This then induces a mapping F: I → X via F(t) := x, where x = ψ(t) ∈ A. Hence the following result holds. Proposition 54. The map F as defined above is an element of F | (I, X), where = {(tj , xj ): j ∈ N0N }. Furthermore, projX graph F = A. Proof. Since every code in  determines exactly one point on A, the mapping F is well defined and a function. It is easy to show that if j1 and j2 are the codes giving the two representations of tj = Nj ∈ I, F(tj ) = xj with no ambiguity. Moreover, conditions (C1)–(C4) imply that the set G := {(t, x): F(t) = x ∧ t ∈ I} is connected  and that G j−1 ,tj ] ∩ G j ,tj+1 ] = {(tj , xj )} for all j ∈ N0N . The proof of continuity of F is essentially given in Theorem 37. The foregoing arguments also indicate that graph F is the attractor of a contractive IFS on I × X. Proposition 55. Let (X, w) be the IFS and F ∈ F | (I, X) the fractal function as defined above. Then graph F is the unique attractor of the contractive IFS (I × X, W), where W := {Wi : I × X → I × X}i∈NN , with 1 Wi (t, x) :=

N (t

+ i − 1) . wi (x)

Proof. It is not difficult to see that (I × X, W) is an IFS. To show contractivity, we introduce a metric L: (I × X) × (I × X) → R+ 0 by

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Fractal Functions, Fractal Surfaces, and Wavelets

L((t, x), (t , x )) := |t − t | + dX (x, x ). Then   1  L(Wi (t, x), Wi (t , x )) =  (t − t ) + dX (wi (x), wi (x )) N ≤ s L((t, x), (t , x )), with s := max{ N1 , s1 , . . . , sN }, where si denotes the Lipschitz constant of wi . The mappings Wi induce an RB operator  on C∗ (I, X) via g (t) :=

N 

wi ◦ g(Nt − i + 1)χ[ti−1 ,ti ] (t).

i=1

Note that condition (C2) implies that  is well defined. Also, since x0 is the fixed point of w1 and xN is the fixed point of wN , we have that g ∈ C∗ (I, X). Let us now show that  is contractive on C∗ (I, X): Let i ∈ NN and choose t ∈ [ti−1 , ti ] and g1 , g2 ∈ C∗ (I, X). Then dX (g1 (t), g2 (t)) ≤ si dX (g1 (Nt − i + 1), g2 (Nt − i + 1)) ≤ s g1 − g2 ∞ , where  · ∞ denotes the supremum norm on C∗ (I, X) induced by dX , and g ∈ C∗ (I, X). Denote s := max{si : i ∈ NN }. Hence  has a unique fixed point  the graph of  g by  G. The results in the previous section imply that w( G) =  G, where w  is the set-valued map associated with the collection W. Thus G is the unique attractor  of the contractive IFS (I × X, W). It remains ⎧ to prove that G =⎫graph F. To this end, ⎬ ⎨  ! m ⊆ I. As E is dense notice that F and  g agree on the set E := n : n ∈ N N ⎭ ⎩m∈N0 Nn

in I and both F and  g map into a Hausdorff space, F ≡  g. The function F can now be used to construct hidden-variable fractal functions. Suppose that X is an R-vector space and a finite direct sum of linear subspaces: Xk . Then the orthogonal projection of graph F induces a function-valued X = k∈NK

projection F → fk , where fk : I → Xk is a hidden-variable fractal function. The next two examples show such hidden-variable fractal functions. Example 34. Let X := R2 and choose as maps wi those that generate the Sierpi´nski triangle S. The maps for the Sierpi´nski triangle are given by w1 (x) =

1

w3 (x) =

3 8

4 1 2



− 14 1 4

− 12

− 38 − 14

x,

w2 (x) =

x+

3

4 1 2

.

1 2

0

0 1 2

x+

1

4 1 2

,

Construction of fractal functions

185

Fig. 5.10 shows the hidden-variable functions f,1 (projection of f onto the (y, t)-plane) and f,2 (projection of f onto the (x, t)-plane); see also Refs. [70, 89].

Fig. 5.10 The graphs of the hidden-variable fractal functions defined in Example 34. Projection onto the (y, t)-plane (bottom left) and onto the (x, t)-plane (bottom right).

Example 35. Let N = 2, let X := I × I, and let wi (x) :=

1 2

1 i−1 2 (−1)

1 i−1 2 (−1) 1 −2

x+

1



2 (i − 1) 1 2 (i − 1)

.

The attractor A of the IFS (X, w) is a so-called Peano curve. Fig. 5.11 shows A and the graphs of the hidden-variable fractal functions; see also Ref. [89]. Remark 51. The proof of Proposition 55 indicates that a slightly more general setup can be used. Instead of using ui (t) = N1 (t + i − 1) in Wi (t, x) = (ui (t), wi (x)), we can use any collection of contractive bijections ui : I → I with uI = I. This then also defines a slightly more general class of hidden-variable fractal functions.

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Fractal Functions, Fractal Surfaces, and Wavelets

Fig. 5.11 A Peano curve and the graphs of the hidden-variable fractal functions. Projection onto the (y, t)-plane (bottom left) and onto the (x, t)-plane (bottom right).

8 Properties of fractal functions The objective of this section is to investigate some of the properties of fractal functions. Most of these properties are a direct consequence of their method of construction. This allows us, for instance, to derive recursive formulae for the moments and linear transforms of fractal functions. It is also shown that affine fractal functions belong to a certain Lipschitz class. The location and uniqueness of the extreme values of fractal functions are studied as well. The notation and terminology that are used for the remainder of this section are introduced first. It is assumed that all fractal functions considered in this section are the unique fixed points of an RB operator  associated with functions u: [0, 1] → [0, 1] and v(x, ·): R → R as defined in Eqs. (5.3), (5.4). Furthermore, it is assumed that conditions (A), (B0 ), and (C) are satisfied, and that the ui are contractive with contractivity constants ai < 1. As usual, the constant of contractivity of vi (x, ·) is denoted by si , i ∈ NN .

Construction of fractal functions

187

8.1 Moment theory of fractal functions The moment theory of IFSs was considered in Section 2.2, and the notation introduced there will be used again here. Let ([0, 1] × R, w) be the contractive IFS associated with a fractal function f : [0, 1] → R interpolating the set := {(xj , yj ) ∈ [0, 1] × R: 0 = x0 < x1 < · · · < xN = 1}. Denote the p-balanced measure of ([0, 1], w) by μ. Note that the IFS ([0, 1], u), whose attractor is [0, 1], also admits a unique p-balanced measure  μ with supp  μ = [0, 1]. If the measure spaces of [0, 1] and G := graph f are denoted by M[0, 1] and M(G), respectively, and a homeomorphism h: [0, 1] → G is defined by h(x) := (x, f (x)), then the following relation between measures on [0, 1] and on G holds. Theorem 83. The homeomorphism h: [0, 1] → G as defined above induces a contravariant homeomorphism M(h): M(G) → M[0, 1]. Moreover, for all  μ E = μ h( E). Also, if g ∈ L1 (G, μ), then E ∈ B1 [0, 1] one has  % % % g dμ = g ◦ h d μ= g d μ. h−1 [0,1]

[0,1]

G

Proof. Define M(h): M(G) → M[0, 1] by M(h)(μ)( E) := μh( E)

for all  E ∈ B1 [0, 1] and all μ ∈ M(G). Then M(h) is a contravariant homeomorphism. Now suppose that μ is any p-balanced measure in M(G). Denoting the probabilities in the IFS by {pi }i∈NN , and using the properties of p-balanced measures, one obtains       μ(E) = pi μ(w−1 pi μ(h ◦ u−1 pi  μ(u−1 i ◦ h)E = i )E = i E). i

i

i

However, since  μ is the stationary measure for the IFS ([0, 1], u),   pi  μ(u−1  μ E= i E) i

for all  E ∈ M[0, 1]. The uniqueness of the p-balanced measure now implies that M(μ) =  μ. The integral identity now follows easily. If the one-dimensional Lebesgue measure on R is denoted by λ, and pi := ai , then the preceding integral identity gives Corollary 7. Corollary 7. Suppose the probabilities are chosen as above. Then for all g ∈ L1 (G, μ), %

% g dμ = G

[0,1]

g ◦ h dλ.

(5.53)

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Fractal Functions, Fractal Surfaces, and Wavelets

The result in Corollary 7 together with the stationarity of the p-balanced measure μ can be used to obtain a recursion relation for the moments of fractal functions. To this end, let g ∈ L1 (G, μ) and let pi := ai , i ∈ NN . Then % g(x, y) dμ(x, y) = G

=

N  i=1 N 

% g ◦ wi (x, y) dμ(x, y)

ai G

% ai

g(ui (x), vi (x, y)) dμ(x, y) G

i=1

% =

[0,1]

g(x, f (x)) dλ.

However, % g(x, y) dμ(x, y) = G

N 

% ai

i=1

[0,1]

g(ui (x), vi (x, f (x))) dλ,

and thus % [0,1]

g(x, f (x)) dλ =

N  i=1

% ai

[0,1]

g(ui (x), vi (x, f (x))) dλ.

(5.54)

Definition 117. Let f : X ⊂ Rn → R be an integrable function. Let α ∈ Nn0 be a multiindex and let β ∈ N0 . The αth moment of f with respect to the n-dimensional Lebesgue measure λ is defined as % M α f :=

xα f (x) dλ,

(5.55)

X

and the generalized moment of f with respect to λ is defined as % M

α,β

xα f (x)β dλ.

f :=

(5.56)

X

Now the moment theorem for affine fractal functions can be stated (see also Ref. [8]). Theorem 84. Let G be the graph of an affine fractal function f : [0, 1] → R. Then the moments M α f and M α,β f are uniquely and recursively determined by the lower-order moments, the interpolation set , and the contractivity constants {si }i∈NN . Proof. Note that since f is an affine fractal function, ui (x) = ai x + bi and vi (x, y) = ci x + si y + di , where the ai , bi , ci , and di are uniquely determined by . By Eq. (5.54) we have that

Construction of fractal functions

Mα f = = =

189

%

N 

ai

i=1 N 

[0,1]

%

(ai x + bi )α (ci x + si f (x) + di ) dλ si f (x) + uαi (x)(ci x + di ) aki xk bα−k i

ai [0,1] k=0 k i=1 % N  α

 α k+1 α−k xk f (x) dλ + Q(α), ai bi si k [0,1] i=1 k=0

where Q(α) :=

 i∈NN

& (M f ) 1 − α

N 

ai

(

[0,1] (ai x+bi )

' aα+1 si i

=

i=1

As

'

& α

 α

α−1  k=0

dλ

α (c x+d ) dλ. Transposing the αth term yields i i

 N α ak+1 bα−k si + Q(α). fλ;k i i k i=1



ai = 1 and |si | < 1, the expression in brackets is always nonzero. Therefore α−1 )α * k N k+1 α−k M f i=1 ai bi si + Q(α) α . (5.57) M f = k=0 k  α+1 1− N si i=1 ai i∈NN

Similarly, one computes M

α,β

f =

N  i=1

=

% ai

[0,1]

β

α  N   α β i=1 k=0 =0

=

(ai x + bi )α (ci x + si f (x) + di )β dλ

β α+β−   =0

k



% si aki bα−k i

[0,1]

xk (ci x + di )β− f  (x) dλ

%

Q(α, β, , n)

n=0

[0,1]

xn f  (x) dλ

for an appropriate polynomial Q(α, β, , n). If  = α and n = β, then |Q(α, β, α, β)| =  β si | < 1, and so | i∈NN aα+1 i α−1 )α * M

α,β

f =

n=0 n

M n,β f

N

β−1 α+β− n+1 si bα−n + =0 m=0 Q(α, β, , n)M α,m f i=1 ai i  α+1 β 1− N si i=1 ai

8.2 Integral transforms of fractal functions The moments M α f are examples of a general integral transform of the form % K(x, y)f (x) dμ(x), f −→ [0,1]

.

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Fractal Functions, Fractal Surfaces, and Wavelets

where the kernel K: K × K → K is assumed to be continuous on nonempty compact subsets of K × K. (Recall that K is a subfield of C invariant under the involuntary automorphism z → z.) Such an integral transform of a function f is denoted by  f . If f is a fractal function of the form considered earlier, Eq. (5.54) yields  f (y) =

N 

% ai

i=1

[0,1]

K(ui (x), y)vi (x, f (x)) dλ(x).

(5.58)

In particular, if f is affine, then  f (y) =

N  i=1

% ai si

[0,1]

K(ui (x), y)f (x) dλ(x) +  R(y),

(5.59)

with R(x) := ci u−1 i (x) + di . As an example the Fourier transform of an affine fractal function f : [0, 1] → K is considered. The kernel K is the Fourier kernel K(x, y) := e−ixy . One readily obtains  f (y) =

N 

f (ak y) +  ak sk e−ibk y R(y),

k=1

where  R(y) =

+ ,

N  2e−i((ak /2)−bk )y ck − 1 sin(ak y/2) + ck e−iak /2 . iy ak y k=1

For more details and examples, see Ref. [78]. Note that any kernel of the form K(x, y) = K(x − y) or K(x, y) = K(xy) allows us to express  f (·) is terms of  f (ak ·). The moments M α f also play a role in the calculation of the inner product between two affine fractal functions. To this end, suppose that f and  f are two affine fractal functions interpolating the sets := {(xj , yj ) ∈ [0, 1] × R: 0 = x0 < x1 < · · · < xN = 1} and  := {(xj , yj ) ∈ [0, 1] × R: 0 = x0 < x1 < · · · < xN = 1}, respectively. The fixed-point property of f and  f under the corresponding RB operators can be expressed as

Construction of fractal functions

191

−1 f (x) = ci u−1 i (x) + di + si f (ui (x)),

x ∈ ui [0, 1],

and similarly for  f. , Proposition 56. Let f and  f be affine fractal functions interpolating and respectively. Then ⎤ ⎡N     d f f a [(s  c f + s f + s c + s d ) i i i 1 i i 0 i i 1 i i 0 ⎦ ⎣ i=1 % 1 1  + 3 (ci + di ] ci ) + 2 (ci di + di ci ) + di f (x) f (x) dλ = , N  [0,1] 1− ai si si i=1

where N 

 ai

i=1

f0 =

1−

1 2 ci N 

+ di



ai si

i=1

and N  i=1

f1 =

  ai bi si f0 + 13 (ai ci ) + 12 (bi ci + ai di ) + bi di N 

1−

i=1

, a2i si

and similarly for  fk , k = 0, 1. Proof. The remark immediately preceding the proposition implies that % [0,1]

f (x) f (x) dλ(x) = =

N % 

xi

i=1 xi−1 N % xi  i=1

xi−1

f (x) f (x) dλ(x) −1  −1 vi (u−1 vi (u−1 i (x), f ◦ ui (x)) i (x), f ◦ ui (x)) dλ(x).

If we let ξi := u−1 i (x), the preceding integral reduces to N %  i=1

[0,1]

vi (ξi , f (ξi )) vi (ξi , f (ξi ))ai dξi .

After some considerable algebra, the given formula is obtained.

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Fractal Functions, Fractal Surfaces, and Wavelets

8.3 Lipschitz continuity of fractal functions In this section we show that affine fractal functions belong to a certain Lipschitz class Lipα ([0, 1], R). To do this, a dimension result that will be proven in Chapter 7 is needed. Theorem 85. Let f : [0, 1] → R be an affine fractal  function and G its graph. Suppose that the set w[0, 1] is not collinear and that i∈NN |si | > 1. Then the box dimension dimb G of G is the unique positive solution of N 

|si | ad−1 = 1; i

(5.60)

i=1

otherwise dimb G = 1. The following theorem relates the box dimension of G to the Hölder or Lipschitz exponent α of f . Theorem 86. Let f be an affine fractal function and let G be its graph. Then for any 0 < h < 1 and all x ∈ [0, 1], |f (x + h) − f (x)| ≤ c hα ,

(5.61)

where c > 0 and α = 2 − dimb G. In other words, f ∈ Lip2−dimb G ([0, 1], R). Proof. Let h ∈ (0, 1) be given, and let 0 < h ≤ ε < 1. Then if G is covered by squares of side ε, the proof of Theorem 85 implies that |f (x + h) − f (x)| ≤ Nε (G)ε2 , where Nε (G) denotes the minimum number of such squares needed to cover G. It can also be shown that there exists a positive constant c such that Nε (G) ≥ cε−d , where d := dimb G. Hence log |f (x + h) − f (x)| log Nε (G)ε2 log cε 2−d log c ≥ ≥ ≥ + (2 − d), log h log h log h log h since h ≤ ε. A converse of the preceding theorem can also be proven. Theorem 87. Let f : [0, 1] → R be a continuous function and G its graph. If there exists a 1 ≤ d ≤ 2 such that f ∈ Lip2−d ([0, 1], R), then dimH G ≤ dimb G ≤ d. Moreover, Hd (G) < ∞. Proof. Let 0 < ε < 1 be given and let Cε (G) be a minimal cover of G by squares of side ε. Denote the cardinality of such a minimal cover by Nε (G). Let n be the smallest integer greater than or equal to ε −1 . Then by the continuity of f , we have Nε (G) ≤

n−1 

(2 + ε−1 ω(f ; [kε, (k + 1)ε])).

k=0

Since f ∈ Lip2−d ([0, 1], R), ω(f ; [kε, (k + 1)ε]) ≤ c ε2−d for all k ∈ N0n−1 and some c > 0 independent of k. Observing that n < 1 + ε−1 , one thus obtains

Construction of fractal functions

193

Nε (G) ≤ (1 + ε−1 )(cε−1 ε2−d ) ≤ c ε−d ,

with c > 0 independent of ε. Therefore using the definitions of the Hausdorff dimension and the box dimension, we obtain the first statement of the theorem. That Hd (G) < ∞ follows from the fact that Hεd (G) ≤ Nε (G)ε−d .

8.4 Extrema of fractal functions This section deals with the location of the extrema of a class of affine fractal functions. This class consists of all nonconstant affine fractal functions interpolating := {( Nj , yj ) ∈ [0, 1] × R: j ∈ N0N } and satisfying the fixed-point equation f (x) =

yi−1 + si f ◦ u−1 i (x),

+ x ∈ Ii :=

, i−1 i , , N N

where si := yi − yi−1 ∈ (−1, 1), and i ∈ NN . This class of affine fractal functions will be denoted by E := E ([0, 1], R). Observe that these conditions imply that all ai are equal to N1 . Most of the results presented here can be found in Ref. [90]. Our presentation follows closely the arguments given there. Let f ∈ E , let m := min{f (x): x ∈ [0, 1]}, and let M := max{f (x): x ∈ [0, 1]}. The following lemma relates the quantities m and M to the yj and the differences si . Lemma 11. Let f ∈ E and assume that f takes its minimum value m on Ii , where i ∈ NN . If si ≥ 0, then m = yi−1 + si m. If si < 0, then m = yi−1 + si M. Proof. Assume first that si ≥ 0. As m = min{f (x): x ∈ Ii } and since on the interval −1 Ii , f (x) = si f ◦ u−1 i (x) + yi−1 , it follows that m = yi−1 + si min{f ◦ ui (x): x ∈ Ii }. But this last minimum equals m. This proves the first statement. Now suppose si < 0. Then employing a similar reasoning, one shows that m = yi−1 + si max{f ◦ u−1 i (x): x ∈ Ii } = yi−1 + si M. Notice that Lemma 11 also applies to the maximum value M. Now suppose that m is attained on Ii and M is attained on Ik . It is an immediate consequence of Lemma 11 that i = k. If i = k and si ≥ 0, then m = yi−1 + si m and M = yi−1 + si M, which implies that m = M. On the other hand, if si < 0 then, again by Lemma 11, m = yi−1 + si M and M = yi−1 + si m, also implying that m = M. According to the sign of si and sk , there are now three possible ways of calculating m: 1. si ≥ 0. Then m = yi−1 + si m; that is, m = yi−1 /(1 − si ). 2. si < 0 and sk ≥ 0. Then m and M satisfy m = yi−1 + si M and M = yk−1 + sk m. Thus m = yi−1 + si yk−1 /(1 − sk ). 3. si ≥ 0 and sk < 0. Then m = yi−1 + si M and M = yk−1 + sk m, implying that m = (yi−1 + si yk−1 )/(1 − si sk ). (1)

For i ∈ NN let xi (3)

(2)

:= (i − 1)/(N − 1), xi,k := [(N − 1)(i − 1) + (k − 1)]/N(N − 1), and

xi,k := [N(i − 1) + (k − 1)]/(N 2 − 1). The straightforward proof of the next lemma is left to the reader (see also Ref. [90]).

194

Fractal Functions, Fractal Surfaces, and Wavelets (1)

(2)

Lemma 12. Let f ∈ E . Then f (xi ) = yi−1 /(1−si ), f (xi,k ) = yi−1 +si yk−1 /(1−sk ), (3)

and f (xi,k ) = (yi−1 + si yk−1 )/(1 − si sk ). The lemma and the arguments preceding it suggest the introduction of an N × N matrix Y whose rows are given by ⎛ ⎞ mi ⎜0⎟ ⎜ ⎟ ⎜ .. ⎟ ⎝.⎠

⎞ mi,1 ⎜ mi,2 ⎟ ⎟ ⎜ ⎜ .. ⎟ , ⎝ . ⎠ ⎛

and

mi,N

0

(1)

according to whether si ≥ 0 or si < 0. In the former case, let mi := f (xi ), and in (2) (3) the latter case, let mi,k := f (xi,k ) if sk ≥ 0 and mi,k := f (xi,k ) if sk < 0. In a similar fashion one defines an N × N matrix X whose rows are given by ⎛

0

⎟ ⎟ ⎟ ⎠

⎞ () xi,1 ⎜ ⎟ ⎜ xi,2 ⎟ ⎜ . ⎟, ⎜ . ⎟ ⎝ . ⎠ ⎛

(1) ⎞

xi ⎜ 0 ⎜ ⎜ . ⎝ ..

and

() xi,N

according to whether si ≥ 0 or si < 0. As before, if sk ≥ 0, then  := 2, and if sk < 0, then  := 3. Extending f to a function on N × N matrices A = (aik ) via f ((aik )) := (f (aik )), we can express the results in Lemma 12 as f (X) = Y. Combining Lemmas 11 and 12 and the preceding characterization in terms of the matrices X and Y gives the next theorem. Theorem 88. Let f ∈ E . The minimum value of f is the smallest entry in the matrix Y, and the corresponding entry in X is one of the locations where f attains this minimum value. Analogously, the maximum value of f is the largest entry in X, and the corresponding entry in X is one of the locations where f attains this maximum value. (1) Proof. Lemma 11 shows that m is an entry in X and Lemma 12 gives mi = f (xi ) ≥ () m and mi,k = f (xi,k ) ≥ m. This proves the result for the minimum value. The statement for the maximum value is proven similarly. After having established the numerical value of the extremum of a fractal function f ∈ E , we need to investigate the location of this extremum and its uniqueness. For this purpose the following two sets are introduced: Im := {i ∈ NN : f (x) = m on Ii } and IM := {i ∈ NN : f (x) = M on Ii }.

Construction of fractal functions

195

Lemma 11 allows us to characterize the sets Im and IM even further: Im = {i: (si ≥ 0 ∧ m = yi−1 + si m) ∨ (si < 0 ∧ m = yi−1 + si M)} and IM = {i: (si ≥ 0 ∧ M = yi−1 + si M) ∨ (si < 0 ∧ M = yi−1 + si m)}. With these sets a directed graph G = (V, E) is associated in the following manner: the vertex set V := NN and the edge set E := {(i, k) ∈ V × V: (i ∈ Im ∧ si ≥ 0 ⇒ k ∈ Im ) ∨ (i ∈ Im ∧ si < 0 ⇒ k ∈ IM ) ∨ (i ∈ IM ∧ si ≥ 0 ⇒ k ∈ IM ) ∨ (i ∈ IM ∧ si < 0 ⇒ k ∈ Im )}. Note that since f cannot attain its minimum value and maximum value on the same interval, the ∨ symbols appearing in the definition of the edge set are exclusive. To describe the location of an extremum of a fractal function f ∈ E , every x ∈ [0, 1] needs to be expressed in an N-ary expansion: x = (i1 − 1)/N + (i2 − 1)/N 2 + · · · + (in − 1)/N n + · · · , with i ∈ NN . A more compact and succinct notation for the N-ary 0 N expansion of x ∈ [0, 1] is given by x = σ− i. Here the operator σ− : NN N → (NN−1 ) is defined by σ− (i1 i2 . . . in . . .) := (i1 − 1 i2 − 1 . . . in − 1 . . .). The next theorem gives necessary and sufficient conditions for x ∈ [0, 1] to yield the minimum value of f . Theorem 89. A number x ∈ [0, 1) yields the minimum value of a fractal function f ∈ E iff its N-ary expansion σ− (i1 i2 . . . in . . .) satisfies the following two conditions: 1. i1 ∈ Im . 2. If in follows im in σ− (i1 i2 . . . in . . .) and if sim = 0, then (im , in ) ∈ E.

Proof. Necessity is shown first. For this purpose, assume that f attains its minimum value at x ∈ [0, 1). Then write Nx = Nx + ξ , where ·: R → Z denotes the floor function: a := max{n ∈ Z: n ≤ a}. Let i1 := Nx + 1. Then x ∈ [(i − 1)/N, i/N), and thus f attains its minimum on Ii . Therefore i1 ∈ Im . To prove that the second condition also holds, note that the fixed-point property of f implies that m = f (x) = yi−1 + si f (ξ ). Hence f (ξ ) = m if si ≥ 0 and f (ξ ) = M is si < 0. If we define i2 := Nξ  + 1, it follows that i2 ∈ Im if si1 ≥ 0 and i2 ∈ IM if si1 < 0. Thus the second condition is satisfied for the first two consecutive integers in the N-ary expansion of x. Applying the preceding argumentation to ξ instead of x, one verifies the second condition for the next pair of consecutive integers in the N-ary expansion of x. Proceeding inductively yields the validity of the second condition in general.

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Fractal Functions, Fractal Surfaces, and Wavelets

For the sufficiency part suppose that the N-ary expansion of x ∈ [0, 1) satisfies the first and second conditions. As before, denote by i1 − 1 and ξ1 the integer part and the fractional part of x, respectively. Again, by the fixed-point property of f , f (x) = yi1 −1 +si1 f (ξ1 ). Now assume that si1 ≥ 0. Then using Lemma 11 and the first condition, we can rewrite the preceding fixed-point property so that f (x) − m = si1 (f (ξ1 ) − m). In the same manner one obtains f (x) − m = si1 (f (ξ1 ) − M) if si1 < 0. Applying the same reasoning to ξ1 , we see—depending on the sign of si2 —that the integer part i2 − 1 and the fractional part ξ2 of Nξ1 satisfy either f (ξ1 ) − m = si2 (f (ξ2 ) − m) or f (ξ1 ) − m = si2 (f (ξ2 ) − M). However, f (x) − m = si1 si2 (f (ξ2 ) − m) if si1 si2 ≥ 0, or f (x)−m = si1 si2 (f (ξ2 )−M) if si1 si2 < 0. Induction yields after n steps that either f (x)− m = si1 si2 . . . sin (f (ξn ) − m), if si1 si2 . . . sin ≥ 0, or f (x) − m = si1 si2 . . . sin (f (ξn ) − M), if si1 si2 . . . sin < 0. Since, by the continuity of f the set {f (ξ1 ), f (ξ2 ), . . . , f (ξn ), . . .} is uniformly bounded and since si1 si2 · · · sin → 0 as n → ∞, f (x) − m → 0. Now we can state necessary and sufficient conditions for the existence of a unique extremum value for f ∈ E . Theorem 90. The function f ∈ E attains a unique minimum iff the following two conditions hold: 1. |Im |c = 1. 2. If i is the unique element in Im and if si < 0, then |IM |c = 1.

Proof. Necessity: If i ∈ Im , the graph G can be traversed along an infinite path starting at vertex i. This infinite path yields the N-ary expansion of a point x ∈ [0, 1] which, by the preceding theorem, satisfies f (x) = m. Hence there exists a point x yielding the minimum of f whose N-ary expansion begins with i. Now suppose that f has a unique minimum value. Then, by the preceding arguments, Im consists of only one element, say, i. If si < 0, then the uniqueness of the minimum also forces the set IM to have cardinality 1. Sufficiency: Assuming that the first and second conditions hold, one concludes the existence of a unique path that originates at the vertex of Im . The preceding theorem then gives desired the result. To illustrate the foregoing theoretical considerations, consider a specific example— namely, Kiesswetter’s fractal function fK (see Section 4). Example 36. Clearly, fK ∈ E . Recall that the y-values of the interpolating set are given by y0 = 0 = y2 , y1 = − 12 , y3 = 12 , and y4 = 1. This then gives s1 = − 12 and s2 = s3 = s4 = 12 . The matrices X and Y are as follows: ⎞ ⎛5 1 2 ⎞ ⎛ 0 −1 0 1 24 3 3 1 ⎜ 1 0 0 0⎟ ⎜ 1 0 0 0⎟ 4 ⎟. ⎟, Y = ⎜ 2 X=⎜ 3 ⎝ ⎠ ⎝ 0 0 0 0⎠ 0 0 0 10 7 − 12 0 0 0 0 0 0 20 The minimum and maximum values of fK are then −1 and 1, respectively. Also, Im = {2} and IM = {4}. Thus by Theorem 90 the unique minimum and the unique maximum are attained at x = 13 and x = 1, respectively.

Construction of fractal functions

197

9 Peano curves Space-filling curves have been known in mathematics since the latter part of the 19th century, when Peano [91] showed that it is possible to map the interval [0, 1] continuously onto a compact subset of X ⊂ R2 . It came as a surprise that such a continuous vector-valued function p: [0, 1] → X exists and it proved that continuity in the component functions of p is not enough to ensure bijectivity. Hilbert [92] replaced Peano’s arithmetic definition of p by a simple geometric construction. Several other geometric constructions appeared in the literature, until Knopp [93] showed that all Peano curves can be generated with a simple geometric principle. In this section we establish that the known Peano curves may be generated— on occasion with a slight modification—with IFSs and that they are projections of isodimensional fractal functions. For a list of Peano curves, see Refs. [93–97]. Some of the material presented here can also be found in Ref. [95]. Throughout this section let I denote the unit interval [0, 1] ⊂ R. We remark that if in the construction of univariate fractal functions on I having the interpolation property with respect to := {(xj , yj ) ∈ I × R: j ∈ N0N } one does not insist on ordering the xj linearly, a fractal curve instead of a fractal function is obtained. Definition 118. If the attractor A of a contractive IFS (I × R, w) is the graph of a continuous fractal curve p: I → R2 such that the p(I) has positive two-dimensional Jordan content in R2 , then p is called a Peano or space-filling curve. Now suppose that  :=

j (xj , yj ) ∈ I × R: y0 = yN = 0 ∧ xj = , ∀ j ∈ N0N N



is an interpolation set and f : I → R the unique fractal function interpolating . It is also assumed that graph f is the unique attractor of the contractive IFS (I × R, w), where wi ∈ Aff(I × R, I × R) or wi ∈ Sim0 (I × R, I × R) for all ∈ NN . More precisely, for some s ∈ (−1, 1), wi (x, y) = or

1 N

yi − yi−1

0 s

i−1

x + N , y yi−1



x , wi (x, y) = Hs ◦ τv ◦ O y

where Hs : R2 → R2 , Hs (x) := sx, is a homothety, τv : R2 → R2 , τv x := x + v, a translation, and O: R2 → R2 an orthonormal operator on R2 (see also Remark 32). Denote by w the associated set-valued mapping w: H(I × R) → H(I × R). Note that w[0, 1] is a piecewise linear function  θ : I → R interpolating . The function  θ is extended to a piecewise linear function θ : R → R by our setting

198

Fractal Functions, Fractal Surfaces, and Wavelets

  θ (x) θ (x) :=  θ (−x) =  θ (x + 1)

if x ∈ I; otherwise.

(5.62)

This function is called the periodic extension of w[0, 1] to R. Using θ, we can express the fractal function f ∈ F | (I, R) in terms of the following infinite series: f (x) =

∞ 

sk θ (N k x),

x ∈ [0, 1].

(5.63)

k=0

 k k The partial sums pn (x) := k∈N0n s θ (N x), x ∈ [0, 1] and n ∈ N, are called the nth polygonal approximations to graph f . Then pn → f in the supremum norm as n → ∞. In what follows, it is assumed without loss of generality that the Peano curves considered are contained in Q := [0, 1] × [0, 1] ⊂ R2 . (This can always be achieved by an appropriate rescaling of coordinates.) The first example is Hilbert’s construction of Peano’s original space-filling curve [92]. In many ways this is an archetypical example exhibiting the general characteristics of Peano curves and their relation to contractive IFSs. To this end, let







  1 1 1 3 3 3 1 , , , , , , := 0, 4 4 4 4 4 4 4 and define contractive mappings on Q by w1 (x, y) :=

0 − 12

1 w3 (x, y) :=

2

0



0 x + 1 , y 0 4

1

x 0 + 21 , y − 12 2 1 2



− 12 0

w2 (x, y) := w4 (x, y) :=

0 1 2

1

x + 21 , 1 y 2 2



1 1 −2 x + 1 . y 0 2 0

Thus (Q, w) is a contractive IFS satisfying conditions (C1)–(C4) in Section 7. Hence the attractor A is the graph of a continuous function p: I → Q interpolating . By Propositions 54 and 55, graph p can be viewed as the orthogonal projection of a fractal function  p ∈ F | (I, R2 ) onto Q. In Chapter 6 it is shown not only that graph p has two-dimensional Jordan content, p = dimH graph p = 2. Hence graph p is the isodimensional but that indeed dimH graph orthogonal projection of graph p. In Fig. 5.12 the generator w[0, 1] of Peano’s curve and the third approximating polygon p3 are given. Remark 52. Comparing our generator with Peano’s, the careful reader may have noticed that ours has additional “legs”: w1 [0, 1] and w4 [0, 1]. These were introduced to guarantee the connectedness of the attractor.

Construction of fractal functions

199

(1, 1)

(0, 0)

(1, 1)

(0, 0)

Fig. 5.12 The generator (left) and p3 (right) of Peano’s space-filling curve.

All the Peano curves investigated in Ref. [97] can be obtained by use of the preceding construction (slight modifications should be made in some cases to guarantee that w[0, 1] is connected). We give one more example—namely, that of a triangular Peano curve. Define contractive mappings wi : Q → Q, i = 1, 2, by 1 1 wi (x, y) := 2 (−1)i−1

(−1)i−1 1

i−1

x 2 + i−1 . y 2

The attractor of the contractive IFS (Q, w) is an isosceles triangle with vertices at (0, 0), (1, 0), and ( 12 , 12 ). The generator of the associated Peano curve is depicted in Fig. 5.13.

Fig. 5.13 The generator of a triangular Peano curve.

In 1938, Isaac Jacob Schoenberg introduced a space-filling curve that is similar to the curve introduced earlier by Henri Lebesgue. Next it will be seen how both curves can be obtained with IFSs. The construction of Lebesgue’s space-filling curve proceeds as follows: Let C be the classical Cantor set and let x ∈ C. It is well known that every point on the Cantor set has a ternary representation of the form x=

 2ωn n∈N

3n

,

ωn ∈ {0, 1}.

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Fractal Functions, Fractal Surfaces, and Wavelets

Define continuous surjections β, δ: C  C by β(x) :=

 2ω2n−1

and

3n

n∈N

δ(x) :=

 2ω2n n∈N

3n

.

These surjections are then used to define continuous fractal functions fj : [0, 1] → R, j = 1, 2, by f1 (x) :=

 ωn n∈N

f2 (x) :=

2n

 ωn n∈N

2n

for x ∈ βC, for x ∈ δC,

and for x ∈ [0, 1] \ βC and x ∈ [0, 1] \ δC, respectively, inductively by

1 1 2 and f2 (x) := 1 − 3x, x ∈ , , f1 (x) := 2 3 3 and continued in this manner on the remaining intervals. Lebesgue’s Peano curve is then the curve pL : I → Q, pL := (f1 , f2 ). To see how the nth-approximating polygon is the image under a set-valued map (1) (2) associated with a contractive IFS, functions wi : [0, 12 ] → R and wi : [0, 12 ] → R, i = 1, 2, 3, 4, have to be defined first: For i = 1, 2, w(1) maps J (1) := [0, 12 ] i 1 2 (1) homeomorphically onto 12 J (1) + i−1 × 0, and for i = 3, 4, wi maps J (1) := [0, 12 ] 2 2 1 × { 12 }. Here the notation aX + b is used to homeomorphically onto 12 J (1) + i−3 2 denote the set {ax + b: x ∈ X ⊆ Rn }, a ∈ R, b ∈ Rn . Let J (2) := J (1) × {0} ∪ J (1) × (2) { 12 } ∪ {(x, y): 2x + 2y − 1 = 0}. The maps wi are to map J (2) homeomorphically onto 1 (2) 1 (2) + (0, 12 ), 12 J (2) + ( 12 , 0), and 12 J (2) + ( 12 , 12 ), respectively. The line segment 2J , 2J (j) (j) connecting the initial point of wi+1 J (j) to the terminal point of wi J (j) will be denoted (j)

by Li (J (j) ), j = 1, 2. (j) The set-valued map associated with wi , j = 1, 2, is defined in the usual way: 4  (j) (j) : H(Q) → H(Q) be defined by w(j) (E) := wi (E), for E ∈ H(Q). Finally, let w i=1

(j) := w(j) ∪ w

3 

(j)

Lk ,

k=1 n

(j) (j) := w(j) ◦ w w

n−1

∪

3 

(j)

(j) Lk ◦ w

n−1

.

k=1

The nth-approximating polygon pLn to pL is then given by

Construction of fractal functions

pLn

201

 m (1) (J 1 ) for n = 2m − 1; w = m (2) (J 2 ) for n = 2m. w

(5.64)

The sixth- and seventh-approximating polygons for pL are shown in Fig. 5.14. ( 78 , 43 )

( 34 , 34 )

(0, 0)

(0, 0)

Fig. 5.14 p4 (left) and p5 (right) for pL .

Schoenberg’s space-filling curve pS is an example of a more elaborate construction. It is explicitly given by pS = (f , g): [0, 1] → 2Q, where f (x) :=



2−k θ (32k−2 x)

k∈N

and g(x) :=



2−k θ (32k−1 x),

k∈N

with θ : R → R defined by ⎧ ⎪ ⎪0 ⎪ ⎨3x − 1 θ (x) := ⎪1 ⎪ ⎪ ⎩ θ (−x)

for x ∈ [0, 13 ); for x ∈ [ 13 , 23 ); for x ∈ [ 23 , 1]; otherwise.

Using the periodic function θ , we can express f and g in the following way: f =

∞ 1  1 2m θ (32m ·) √ 2 2 m=0

and

∞ 1  1 2m−1 g= √ θ (32m−1 ·). √ 2 m=1 2

202

Fractal Functions, Fractal Surfaces, and Wavelets

Note that the preceding representations of f and g imply that 2f + function h: [0, 1] → R, expressed in the form Eq. (5.63):

√

2g is a fractal

∞  1 m θ (3m ·). h= √ 2 m=0 It can be seen that graph h is the unique attractor of the contractive IFS (2Q, w) with 3

4

x , w1 (x, y) = √1 y 2 3 4

1 1 0 x 3 3 , w2 (x, y) = + 1 1 √ 1− √ y 0 2 2 3 4

1 2 0 x 3 + 3 , w3 (x, y) = 1 1 √ √ − y 1 2 2 3 4

1 0 x 1 3 w4 (x, y) = + , √1 − √1 y 1 2 2 3 4

1 4 0 x 3 w5 (x, y) = + 3 , 1 1 √ √ −1 − y 1 2 2 3 4

1 5 0 x 3 3 . + w6 (x, y) = 1 1 √ −√ y 0 2 2 1 3 − √1 2

0

The set of interpolation points is given by {(0, 0), ( 13 , 0), ( 23 , 1), (1, 1), ( 43 , 1), ( 53 , 0), (2, 0)}. Fig. 5.15 shows how pS can be constructed from f and g.

(f(x), g(x))

(x, g(x))

(f(x), x)

(x, f(x))

Fig. 5.15 The construction of pS from f and g.

The nth-approximating polygon pSn of pS can be obtained from the nth-partial sums fn and gn of f and g, respectively. The vertices of pSn which are given by

Construction of fractal functions

203

(fn ( 3qn ), gn ( 3qn )), with q = 0, 1, . . . , 3n − 1, are the images of [0, 2] under the set-valued map u: H([0, 2]) → H([0, 2]) associated with ui = 13 (· + i − 1), i = 1, . . . , 6. Note that the ui are the “horizontal” components of the wi . Peano curves can also be constructed with use of the recurrent set formalism. This is done by means of a specific example which can also be found in Ref. [4], where several more examples of space-filling curves are listed. Let X := {a, b, c, d} and let S[X] be the free semigroup generated by X. Define a free semigroup endomorphism ϑ: S[X] → S[X] by ϑ(a) := a b a d,

ϑ(b) := a b a b,

ϑ(c) := c d c b,

ϑ(d) := c d c d.

Let V := {(x, y, x, y): x, y ∈ R} ⊆ R4 . Then it is readily seen that V is ϑ ∗ -invariant. Let f : S[X] → R2 be defined by f (a) := (1, 0) =: − f (c),

f (b) := (0, 1) =: − f (d).

Thus Lϑ is given by Lϑ =



2 2 . 0 2

Finally, define K: S[X] → R2 by K(x) := {tf (x): t ∈ [0, 1]}. The recurrent set Kϑ (a) is a space-filling curve whose second approximation is depicted in Fig. 5.16.

Fig. 5.16 The second approximation to Kϑ (a).

204

Fractal Functions, Fractal Surfaces, and Wavelets

10 Fractal functions of class C k In Section 1 it was shown how to construct continuous fractal functions as unique fixed points of an RB operator  acting on a set of continuous functions. It is natural to ask whether it is possible to obtain fixed points of class Ck , k ∈ N, starting with an RB operator  acting on a class of Ck functions. In this section we answer this question and present a method that allows the construction of Ck -fractal functions via integration. This latter approach was undertaken in Ref. [98]. Throughout this section X := [0, 1] and Y := R, although a more general setup could be considered. It is also assumed that the maps ui are affine contractions −1 on [0, 1] satisfying u−1 i (x) = 1 and ui+1 (x) = 0 for x ∈ ui [0, 1] ∩ ui+1 [0, 1], and that the mappings vi (x, ·) are of the form vi (x, ·) = λi (x) + si (·), with λi ∈ Lip(X) and si ∈ (−1, 1), i ∈ NN . Again these are certainly not the most general assumptions; however, for our purposes they will prove more than sufficient. For by convenience, let ui [0, 1] =: Ii and denote compositions of the form f ◦ u−1 i #f . Let k always denote a nonnegative integer. To proceed, two definitions are i needed. Definition 119. Let U and V be open subsets of Rn endowed with the supremum norm  · ∞ , and let Ck (U, V) denote the R-vector space of all functions f : U → V that possess continuous and bounded derivatives up to order k on X. The Ck -topology on Ck (U, V) is the topology induced by the norm f Ck :=

sup

{f () ∞,U },

f ∈ Ck (U, V).

∈{0,1,...,k}

Remark 53. Convergence in Ck (U, V) means uniform convergence in U not only of the sequence of functions itself but also of the sequence of derivatives up to order k. Definition 120. Let U and V be open subsets of Rn and let f ∈ Ck (U, V). The function f is said to belong to the class Lipα+k (U, V) iff f (k) ∈ Lipα (U, V), 0 < α < 1. As in Eqs. (5.3), (5.4), define u: [0, 1] → [0, 1] by u(x) =



u−1 i (x)χIi (x)

i∈NN

and v(x, ·): R → R by v(x, y) =



(λi (x) + si y)χIi (x).

i∈NN

Instead of requiring condition (B0 ) to hold, the stronger condition (Bk ) is needed:

Construction of fractal functions

205

(Bk ) There exists a family C∗k [0, 1] of Ck -functions such that for all f ∈ C∗k [0, 1], the functions λi ◦ u + f ◦ u are elements of C∗k [0, 1] and     d  # #f #λ #f (x) = d λ (x) + s (x) + s (x) i,i i i i+1,i+1 i+1 i+1 dx dx

(5.65)

for all  ∈ N0k and x ∈ Ii ∩ Ii+1 , i ∈ NN−1 .

Suppose that conditions (A) and (Bk ) are satisfied. Let ui (x) := ai x + bi , ai = 0, i ∈ NN , and define an R-vector space by  k

 :=

()

gi ∈ C [0, 1]: k

()

=

gi+1 (0) ai+1

3 1+

gi (1) ai si+1

aN − sN

3

4

1+

si

4

a1 − s1



, ∀ ∈ N0k , ∀ i ∈ NN−1 .

The RB operator associated with the functions u and v(x, ·) defined above is denoted by . Let f be the unique fixed point of this operator. In the following it is shown that f is of class Ck . Theorem 91. Let  be the RB operator defined above and let λi ∈ k for all i ∈ NN . Suppose that α := (min{ai : i ∈ NN })−k · max{|si |: i ∈ NN } < 1. Then the unique fixed point f of  is of class Ck . Proof. It is necessary to express the dependence of  on the λi and the si explicitly. Therefore  is written as (λ, s), where λ := (λ1 , . . . , λN ) and s := (s1 . . . , sN ). k [0, 1] := Ck ([0, 1], R), and let f be the unique fixed point of . Let  ∈ k. Define CR  Then for all i ∈ NN , () () −1 − −1 f() (x) = a− i λi ◦ ui (x) + ai si f ◦ ui (x),

x ∈ Ii .

With use of the fact that all λi ∈ k and that α < 1, it follows that (λi , si ) is a contraction in the C -topology with contractivity α. Hence its unique fixed point f is () of class Ck . Moreover, by the preceding equation, notice that f(λ,s) is the unique fixed point of the RB operator (a− λ() , a− s); that is, ()

f(λ,s) = f(a− λ() ,a− s) .

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Fractal Functions, Fractal Surfaces, and Wavelets

Here we used the uniqueness of the fixed point of . Above we also defined an integer p p power p of an N-tuple by ap := (a1 , . . . , aN ) and the product of two N-tuples a and b by ab := (a1 b1 , . . . , aN bN ). This theorem gives rise to a definition. Definition 121. The unique fixed point of the RB operator  is called a fractal function of class Ck . The collection of all such fractal functions is denoted by F k ([0, 1], R), or by FRk [0, 1] for short; if k = 0, then FR0 [0, 1] := FR [0, 1]. Remark 54. The proof of Theorem 91 implies that if f is a fractal function of class Ck , then its kth derivative is a fractal function as introduced in Section 1. In particular, if f is an affine fractal function and of class Ck , then f ∈ Lipα+k [0, 1], where α = 2 − dimb graph f . As an example, let us consider the case ai := N1 and si := s for all i ∈ k [0, 1] that NN , with |s| < 1. Then (k) consists of all those functions gi ∈ CR satisfy     () () () () (1 − sN  ) gi+1 (0) − gi (1) + (sN  ) g1 (0) − gN (1) = 0

(5.66)

for all i ∈ NN−1 . We remark that Eq. (5.66) has the following—later very important— interpretation: as λ = (λ1 , . . . , λN ), it can be interpreted as an element of the space 5 k [0, 1]. Here 5 denotes the direct product of linear spaces; it is the extension C i∈NN R of the direct product of the underlying abelian groups. The scalar multiplication is defined in the usual componentwise way. () 5 k ([0, 1]) → R by If for i ∈ NN−1 we define linear functionals Li : i∈NN CR     () () () () () Li λ := (1 − sN  ) λi+1 (0) − λi (1) + (sN  ) λ1 (0) − λN (1) ,

(5.67)

then the join-up conditions (5.65) are equivalent to λ∈

6 i∈NN

k ⇐⇒ λ ∈

k N−1  

()

ker Li .

=0 i=1

Hence a fractal function f generated by the special mappings defined above is of class k N−1 7 7 () ker Li . In Chapter 8 this description of a fractal Ck only if N k |s| < 1 and λ ∈ =0 i=1

function of class Ck in terms of λ will be reconsidered. Example 37. Reconsider Example 25; there N = 2, a1 = a2 = 12 , s = 14 , and λ1 (x) = x = 1 − λ2 (x). One verifies that the hypotheses of Theorem 91 are satisfied () for k = 1. Clearly |s|N < 1. The operators Li are given by

3 3 (0)  = 0 : L1 λ = 1 − (λ2 (0) − λ1 (1)) + (λ1 (0) − λ2 (1)), 4 4

Construction of fractal functions

207



1 1 (1)  = 1 : L1 λ = 1 − (λ 2 (0) − λ 1 (1)) + (λ 1 (0) − λ 2 (1)), 2 2 (0)

(1)

with λ = (λ1 , λ2 ). One quickly verifies that λ ∈ ker L1 ∩ ker L1 . Thus the unique fixed point f of the RB operator  is a fractal function of class C1 ; that is, an element of FR1 [0, 1]. Furthermore, it is seen that the value for s (ie, 14 ) is unique.

10.1 Indefinite integrals of continuous fractal functions A somewhat different approach to construct fractal functions of class Ck was undertaken in Ref. [98]. The basic idea is to take the indefinite integral of a C0 -fractal function, thus obtaining a more regular fractal function. This method is investigated in this section. Let f be a fractal function generated by an RB operator  associated with maps ui and vi (x, ·) := λi (x) + si (·) as defined at the beginning of Section 10. Denote by the −1 N + 1 distinct points (xj , yj ) ∈ R2 determined by the maps u−1 i (0), ui (1), and f (xj ), respectively. We define % x f (t) dt,  y0 ∈ R. (5.68) F(x) :=  y0 + 0

Recall that the fixed-point equation for f may be written in the form f ◦ ui (x) = λi (x) + si f (x),

x ∈ [0, 1], i ∈ NN .

Hence % F ◦ ui (x) =  y0 +

ui (x)

% f (t) dt =  y0 +

0

% =  y0 +

xi−1



%

f (t) dt + ai

0

y0 + Setting  yi−1 := 

( xi−1 0

xi−1

% f (t) dt +

0 x

ui (x)

f (t) dt

xi−1

f ◦ ui (t) dt.

0

f (t) dt and using the fixed-point equation for f yields

% yi−1 − ai si y0 + ai F ◦ ui (x) = 

x 0

Now let  yi−1 − ai si y0 + ai λi := 

%

x

λi (t) dt 0

λi (t) dt + ai si F(x).

208

Fractal Functions, Fractal Surfaces, and Wavelets

and si := ai si , i ∈ NN . One verifies that  :=   λi ◦ u−1 si (·) ◦ u−1 i + i is a contractive RB operator on C∗ [0, 1] := {f ∈ C[0, 1]: f (0) = y0 ∧ f (1) = yN } and that F is its unique fixed point. Let  yi := F(xi ), i ∈ NN . Setting x = 1 in the functional equation for F gives %

1

yi−1 − ai si y0 + ai yi = F ◦ ui (1) = 

λi (t) dt + ai si yN .

0

y0 + Therefore, since  yi =   yi =  y0 +

i 

i

yk k=1 (

− yk−1 ),



%

1

ak sk [ yN − y0 ] +

λk (t) dt .

(5.69)

0

k=1

Setting i = N in the preceding equation, we derive the following expression for  yN : (1 N ai λi (t) dt  yN =  y0 + i=1 N0 . (5.70) 1 − i=1 ai si These results are summarized in the following theorem. Theorem 92. Let f ∈ F | [0, 1] for an RB operator of the form −1  := λi ◦ u−1 i + si (·) ◦ ui .

Then the indefinite integral % x F(x) :=  y0 + f (t) dt,

 y0 ∈ R,

0

 [0, 1], where  := {(xj , yj ): j ∈ N0N } with is an element of F 1 |  yi =  y0 +

i 



(1 i=1 ai 0 λi (t) dt .  1− N i=1 ai si

λk (t) dt ,

0

N  yN =  y0 +

1

ak sk [ yN − y0 ] +

k=1

and

%

i ∈ NN−1 ,

Construction of fractal functions

209

Proof. It suffices to show that F ∈ F 1 [0, 1]. This, however, is immediate. Repeating the foregoing procedure generates fractal functions of progressively higher regularity. This is most easily done when the functions λi are polynomials. It is worthwhile mentioning that in the case of a polynomial λi and si = 0 for all i ∈ NN , the fractal functions are the classical splines. (The reader knowledgeable in nonstandard analysis may have noticed that interpolating fractal functions are indeed hyperfinite splines; see Refs. [34, 86].) We end this section with an example of the indefinite integral of an affine fractal function. Example 38. Let f be the unique affine fractal function interpolating the set := {(0, 0), ( 23 , 34 ), (1, 12 )}. Let s1 := 34 and s2 := − 12 . Then λ1 (x) = 38 x and λ2 (x) = 34 . The graph of f is shown in Fig. 5.17.

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

Fig. 5.17 The fractal function f in Example 38.

Set  y0 := 0. Then the indefinite integral F of f interpolates the set  :=



 9 2 13 , , 1, . (0, 0) , 3 32 16





Furthermore,  λ1 (x) = Fig. 5.18.

1 2 8x

and  λ2 (x) =

1 4

x+

13 32 .

The graph of F is displayed in

210

Fractal Functions, Fractal Surfaces, and Wavelets

0.6 0.5 0.4 0.3 0.2 0.1

0.2

0.4

0.6

0.8

1.0

Fig. 5.18 The indefinite integral of f .

11 Biaffine fractal functions In this section we introduce the concept of biaffine or, as it is sometimes called, bilinear fractal interpolation. The terms bilinear filtering and bilinear interpolation are found in the computer graphics field in the context of computing intermediate values for twodimensional regular grids. There the main goal is to smooth textures when they are enlarged or reduced in size. In mathematical terms, this interpolation technique tries to find a function f (x, y) of the form f (x, y) = a + bx + cy + dxy, where a, b, c, d ∈ R, that passes through prescribed data points. As textures are, in general, not smooth features but exhibit multiscale behavior, a fractal interpolation technique may be more appropriate for modeling. The material in this section is based on Ref. [99]. To this end, let := {(xj , yj ): j ∈ N0N } denote the Cartesian coordinates of a finite set of points in R2 where x0 < x1 < · · · < xN . Denote by I the interval [x0 , xN ]. For n ∈ NN , let ln : I → [xn−1 , xn ] be a continuous bijection. Let L: I → I be such that L(x) := ln−1 (x),

x ∈ [xn−1 , xn ), n ∈ Nn .

Here we tacitly understand that for n = N the interval is [xN−1 , xN ]. Let S: I → R be a bounded and piecewise continuous function, where the only possible discontinuities are finite jumps occurring at the interior points {x1 , x2 , . . . , xN−1 }. Set s := max{|S(x)|}. x∈I

Consider the complete metric space (C(I), d∞ ), where d∞ (f , g) = max{|f (x) − g(x)|}. x∈I

Construction of fractal functions

211

Let C∗ (I):= {f ∈ C(I): f (x0 ) = y0 ∧ f (xN ) = yN }, C (I):= {f ∈ C(I): f (xj ) = yj , ∀ j ∈ N0N }. Then C∗ (I) and C (I) are closed metric subspaces of C(I) with C (I) ⊂ C∗ (I) ⊂ C(I). Each function in C interpolates the data . Let b ∈ C∗ (I) and h ∈ C (I). Define an RB operator : C∗ (I) → C (I) by g := h + S · (g ◦ L − b ◦ L).

(5.71)

Theorem 93. The mapping : C∗ (I) → C (I) satisfies d∞ (g1 , g2 ) ≤ s d∞ (g1 , g2 ) for all g1 , g2 ∈ C∗ (I). In particular, if s < 1, then  is a contractive mapping and possesses a unique fixed point f ∈ C (I). Proof. The operator  is well defined: For i ∈ NN−1 , we have that g(Xi −) = h(Xi ) = g(Xi +). To show contractivity, note that d∞ (g1 , g2 ) = max{|S(x)(g1 (L(x)) − g2 (L(x)))| : x ∈ I} ≤ s max{|(g1 (ln−1 (x)) − g2 (ln−1 (x)))|: x ∈ [xn−1 , xn ], n ∈ NN } = s d∞ (g1 , g2 ). As f (C∗ (I)) ⊂ C (I) and (C (I), d∞ ) is closed, hence complete, it follows that f ∈ C (I). Note that g = H + S · g ◦ L, where H = h − S · b ◦ L. Hence the fractal function f is uniquely defined by three functions, H, S, and L, of the special forms defined above. Also, the fixed point f of  interpolates the data . One way to evaluate f is to use the Banach fixed-point Theorem 6: ⎛ f = lim k (f0 ), k→∞

⎞

⎝k =  ◦ · · · ◦ ⎠ , 8 9: ; k times

where f0 ∈ C∗ (I) is arbitrary. The proof of the Banach fixed-point theorem also gives an estimate for the rate of convergence (see Ref. [100, Theorem 5.2.3]): < < < < 0 and q: I → R. Let dq : (I × R) × (I × R) → R+ 0 be defined by dq ((x1 , y1 ) , (x2 , y2 )) := α |x1 − x2 | + β |(y1 − q(x1 )) − (y2 − q(x2 ))| for all (x1 , y1 ), (x)2 , y2 ) ∈ I* × R. Then dq is a metric on I × R. If, in addition, q is continuous, then I × R, dq is a complete metric space. Proof. The symmetry of dq is immediate: dq ((x2 , y2 ) , (x1 , y1 )) = dq ((x1 , y1 ) , (x2 , y2 )) ≥ 0. Assume that dq ((x1 , y1 ) , (x2 , y2 )) = 0. Then α |x1 − x2 | + β |(y1 − q(x1 )) − (y2 − q(x2 ))| = 0, which implies x1 = x2 . Hence |(y1 − q(x1 )) − (y2 − q(x1 ))| = 0, giving y1 = y2 . To show the triangle inequality, let (xi , yi ) ∈ I × R for i = 1, 2, 3. Write qi = q(xi ) for i = 1, 2, 3. We obtain dq ((x1 , y1 ) , (x2 , y2 )) + dq ((x2 , y2 ) , (x3 , y3 )) = α |x1 − x2 | + β |(y1 − q1 ) − (y2 − q2 )| + α |x2 − x3 | + β |(y2 − q2 ) − (y3 − q3 )| = α(|x1 − x2 | + |x2 − x3 |) + β(|(y1 − q1 )| |− (y2 − q2 )| + |(y2 − q2 ) − (y3 − q3 )|) ≥ α(|x1 − x3 |) + β(|(y1 − q1 ) − (y2 − q2 )| + |(y2 − q2 ) − (y3 − q3 )|) ≥ α(|x1 − x3 |) + β(|(y1 − q1 ) − (y3 − q3 )|) = dq ((x1 , y1 ) , (x3 , y3 )). To prove completeness in the case that q is continuous, let {(xk , yk )}k∈N denote a Cauchy sequence with respect to the metric dq . Given ε > 0, there exists an integer N(ε) such that α |xk − xm | + β |(yk − q(xk )) − (ym − q(xm ))| < ε whenever k, m > N(ε). It follows that {xk }k∈N is a Cauchy sequence with respect to the Euclidean norm, and so it converges to a limit x∗ ∈ I. As q is continuous,

Construction of fractal functions

213

{q(xk )}k∈N converges to some limit q∗ ∈ R. In turn, this implies that {yk }k∈N * ) converges to some y∗ ∈ R. Hence {(xk , yk )}k∈N converges to (x∗ , y∗ ) ∈ I × R. Thus I × R, dq is complete. Next we characterize the graph of the fixed point f of the RB operator  as an attractor of an IFS. Define wn : I × R →I × R by wn (x, y) := (ln (x), h(ln (x)) + S(ln (x))(y − b(x))),

n ∈ NN ,

and let w := {wn }n∈NN . Define an IFS by (I × R, w). We make use of the metric dq of Proposition 57 with q = f , the fixed point of . To simplify the notation, we drop the subscript  from f . Let η > 0 and let Xη := {(x, y) ∈ I × R: |y − f (x)| ≤ η} . It is readily shown that when Theorem 93 holds (ie, when s < 1), w(Xη ) ⊂ Xη . The next theorem gives conditions under which (1) the IFS (Xη , w) is contractive with respect to df and (2) the IFS (I × R, w) has a unique attractor. The following result is a substantial generalization of Theorem 74 which requires, in the present setting, that h is uniformly Lipschitz. Here we avoid this restriction by using the metric dq introduced above with q = f . Theorem 94. Let s < 1 and let f ∈ C (I) be the fixed point of the RB operator  as in Theorem 93. Let ln : I → I have uniform Lipschitz constant λl < 1; that is, |ln (x1 ) − ln (x2 )| ≤ λl |x1 − x2 | ,

∀ x1 , x2 ∈ I, ∀ n ∈ NN .

Let S: I → R have Lipschitz constant λS so that |S(x1 ) − S(x2 )| ≤ λS |x1 − x2 | ,

∀ x1 , x2 ∈ I.

Then the IFS (Xη , w) is contractive with respect to the metric df where α = 1 and 0 < β < (1 − λl ) /λl λS η. In particular, under these conditions the IFS (I × R, w) has a unique attractor A = graph f . Proof. Let (x1 , y1 ) , (x2 , y2 ) ∈ Xη . Then df (wn (x1 , y1 ) , wn (x2 , y2 )) − α |ln (x1 ) − ln (x2 )| = β|h(ln (x1 )) + S(ln (x1 ))(y1 − b(x1 )) − f (ln (x1 )) − (h(ln (x2 )) + S(ln (x2 ))(y2 − b(x2 )) − f (ln (x2 )))| = β| (S(ln (x1 ))(y1 − f (x1 ))) − (S(ln (x2 ))(y2 − f (x2 ))) | ≤ β |S(ln (x1 ))| · |(y1 − f (x1 )) − (y2 − f (x2 ))| + |S(ln (x1 )) − S(ln (x2 ))| · |(y2 − f (x2 ))| ≤ βs|(y1 − f (x1 )) − (y2 − f (x2 ))| + βλl λS η |x1 − x2 | . For the second equality, we employed the fact that f is the fixed point of Eq. (5.71).

214

Fractal Functions, Fractal Surfaces, and Wavelets

Hence df (wn (x1 , y1 ) , wn (x2 , y2 )) ≤ (αλl + βλS λl η) |x1 − x2 | + βs|(y1 − f (x1 )) − (y2 − f (x2 ))| ≤ (α + βλS η) λl |x1 − x2 | + βs|(y1 − f (x1 )) − (y2 − f (x2 ))| ≤ C · (α |x1 − x2 | + β |(y1 − f (x1 )) − (y2 − f (x2 ))|) , where C := max {s, λl + βλl λS η/α}. As λl < 1, one can choose α, β > 0 so that C < 1. One such choice is α := 1 and 0 < β < (1 − λl ) /λl λS η. It follows that the IFS (Xη , w) is contractive and thus has a unique attractor. This attractor must be graph f since a contractive IFS has a unique nonempty compact (graph f ) = graph f , where w  is the invariant set and it is readily verified that w set-valued map associated with the IFS (Xη , w). As one can choose the constant η arbitrarily large, it follows that (I × R, w) has a unique attractor—namely, graph f . Remark 55. We have not provided a metric with respect to which (I × R, w) is contractive! We mention that, as can be easily verified, graph g = w(graph g),

∀ g ∈ C∗ (I).

If S is Lipschitz continuous with Lipschitz constant s < 1, and the functions ln are contractive, the graph of the fractal interpolant f can be approximated by the “chaos game” algorithm. The interested reader may wish to consult [101, 102] for new topological viewpoints of the “chaos game” and more information. Now we consider a specific example of the preceding theory. To this end, let ln : I → [xn−1 , xn ] be given by ln (x) := xn−1 +

xn − xn−1 xN − x0

(x − x0 )

(5.73)

and S: I → R be given by S := Sn ◦ ln−1 ,

x ∈ [xn−1 , xn ],

where Sn : I → R, Sn (x) := sn−1 +

sn − sn−1 xN − x0

(x − x0 ) ,

! with sj : j ∈ N0N ⊂ (−1, 1) and n ∈ NN . Note that the sj , j ∈ N0N , need not be ordered. Then S is continuous and obeys the following estimate:

Construction of fractal functions

215

|S(x)| ≤ max{|Sn (ln−1 (x))| : x ∈ [xn−1 , xn ] ∧ n ∈ NN } 2 1  = max sj  : j ∈ N0N =: s < 1. Moreover, define b: I → R by b(x) := y0 +

yN − y0 xN − x0

(x − x0 )

(5.74)

and h: I → R by h(x) :=



N + 

yn−1 +

n=1

yn − yn−1 xn − xn−1



, (x − xn−1 ) χ[xn−1 ,xn ] (x).

(5.75)

We remark that b ∈ C∗ (I) and h ∈ C (I). Theorem 93 implies that the RB operator  has a unique fixed point f for this special choice of functions b and h. Specifically, f is the unique solution of the set of self-referential functional equations of the form f (ln (x)) − h(ln (x)) = Sn (ln (x))[f (x) − b(x)],

n ∈ NN ; x ∈ I.

(5.76)

We refer to f as a biaffine or bilinear fractal interpolant. This terminology has its origin in the fact that the functions wn of the IFS (I × R, w) take the form wn (x, y) := (ln (x), a + b x + c y + d x y), where a, b, c, and d are real constants, and functions of the form B(x, y) = a + b x + c y + d x y are called bilinear in the computer graphics literature. We note that B is for fixed x or fixed y affine in the other variable. More precisely, B((1 − t)x1 + tx2 , y) = (1 − t)B(x1 , y) + tB(x2 , y), B(x, (1 − t)y1 + ty2 ) = (1 − t)B(x, y1 ) + tB(x, y2 ) for all x1 , x2 , y1 , y2 , t ∈ R. Using the above expressions for ln , Sn , and h, we can write the functions wn in the form



xn − xn−1 yn − yn−1 (x − x0 ), yn−1 + (x − x0 ) wn (x, y) = xn−1 + xN − x0 xN − x0 ,+ ,



+ sn − sn−1 yN − y0 (x − x0 ) y − y0 − (x − x0 ) . + sn−1 + xN − x0 xN − x0 (5.77) In particular, one has

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Fractal Functions, Fractal Surfaces, and Wavelets

wn (xN , y) = (xn , yn + sn (y − yN )) and wn+1 (x0 , y) = (xn , yn + sn (y − y0 )). Hence the images of any (possibly degenerate) parallelogram with vertices at (x0 , y0 ± H) and (xN , yN ± H), H ∈ R, under the IFS fit together neatly, as illustrated in Fig. 5.19. 1.0

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

Fig. 5.19 A biaffine fractal interpolation function.

12 Local fractal functions and smoothness spaces In this section we find conditions that guarantee that a class of local fractal functions belongs to a given function space. The function spaces that will be considered here are the Lebesgue spaces Lp , 0 < p ≤ ∞, the classical smoothness spaces Cn , n ∈ N0 , the homogeneous Hölder spaces C˙ s , 0 < s < 1, the Sobolev spaces W m,p , m ∈ N0 , 1 ≤ p ≤ ∞, the Besov spaces Bsp,q , s ∈ R+ , 0 < p, q ≤ ∞, and the Triebel-Lizorkin s , s ∈ R+ , 0 < p < ∞, 0 < q ≤ ∞. spaces Fp,q

12.1 Lebesgue spaces Lp , 0 < p ≤ ∞ Recall the definition of the Lebesgue spaces from Section 4.1 and the terminology and notation used there. We first present a result that gives conditions for a local fractal function f generated by an RB operator of the form Eq. (5.30) to belong to

Construction of fractal functions

217

the quasi-Banach spaces Lp (X). (See also Ref. [42, 80] for the case n = 1. There, however, Lp , 0 < p < 1, is considered to be a complete metric space.) In this section we assume that the two n-tuples λ and S are fixed so that we may suppress them in the notation. The next theorem connects with the setting in Section 2 employed for local fractal functions. Theorem 95. Let X be a bounded domain in Rn , let Y := R, and let 0 < p ≤ ∞. In addition to satisfying condition (P) in Section 2, the family of subsets {Xi }i∈NN ∈ 2X and bijective mappings {ui }i∈NN are supposed to be such that each ui ∈ Sim(Rn ); that is, a mapping Xi → X obeys the condition |ui (x) − ui (x )| = γi |x − x |,

∀ x, x ∈ Xi ,

for some constant γi ∈ R+ . (We do not assume that all γi < 1!) Furthermore, we assume that the functions λi ∈ Lp (X) and the functions Si are bounded on Xi . Then the RB operator  defined in Eq. (5.30) maps Lp (X) into itself. Moreover, if ⎧ ⎨ N

p

n i=1 γi Si ∞,Xi

1/p

= −1 −1 Dk Si (u−1 (x)) · (f (u (x)) − g (u (x))) χui (Xi ) i i i i i

i∈NN

=

k

 k k = k−l 2 (D (fi − gi )) l i∈NN l=0 > −1 l (x)) · (D S )(u (x)) χui (Xi ) , (u−1 i i i

where we applied the Leibniz differentiation rule. Thus Dk f − Dk g∞ ≤ 2k

k

 k Dl Si ∞,Xi Dk−l (f − g)∞ . l

i∈NN l=0

Hence

Construction of fractal functions

f − gCn =

n 

223

Dk f − Dk g∞

k=0

≤2

n

k

n   k

i∈NN k=0 l=0

= 2n

l

Dl Si ∞,Xi Dk−l (f − g)∞

n  k  n−k+l l

i∈NN k=0 l=0

Dl Si ∞,Xi Dn−k (f − g)∞ .

The last equality follows from mathematical induction. Therefore 3



f − gCn ≤ 2 max max n

i∈NN k=0,1,...,n

4

k  n−k+l l=0

l

D Si ∞,Xi l

f − gCn ,

and the theorem is proved.

12.2.2 Vanishing endpoint conditions for Si Here a slightly more general setup than in the previous section is considered. To this end, suppose that X := [0, 1] and that Xi := [ai , bi ] for i ∈ NN are N different subintervals of positive length. Further assume that {0 =: x0 < x1 < · · · < xN−1 < xN := 1} is a partition of X and that an enumeration has been chosen in such a way that the mappings ui : Xi → X obey ui ([ai , bi ]) := [xi−1 , xi ],

∀ i ∈ NN .

Note that a1 = x0 , bN = xN , and ui (bi ) = xi = ui+1 (ai+1 ) for all interior knots x1 , . . . , xN−1 . It is assumed that the ui are affine functions but they may not necessarily be contractive. Suppose 2 1 := (xj , yj ): j ∈ N0N

(5.91)

is a given set of interpolation points. Define C [0, 1] := {f ∈ C[0, 1]: f (xj ) = yj , ∀ j ∈ N0N }.

(5.92)

Our objective is to construct a local fractal function that belongs to the function class C [0, 1] and which is generated by an RB operator of the form Eq. (5.30). For this purpose, one has to impose continuity conditions at the interpolation points: f (x0 ) = y0 , f (xN ) = yN , f (xi −) = yi = f (xi +), i ∈ NN−1

(5.93)

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Fractal Functions, Fractal Surfaces, and Wavelets

for any f ∈ C [0, 1]. Substitution of  into these equations and simplification produces λ1 (x0 ) + S1 (x0 ) y0 = y0 , λN (xN ) + SN (xN ) yN = yN , λi (bi ) + Si (bi ) f (bi ) = yi = λi+1 (ai+1 ) + Si+1 (ai+1 ) f (ai+1 ),

i ∈ NN−1 .

As f may not be known at the points ai and bi , we impose the following vanishing endpoint conditions on the functions Si : Si (ai ) = 0 = Si (bi ),

∀i ∈ NN .

(5.94)

Therefore the functions λi have to obey the following simpler conditions: λ1 (x0 ) = y0 ,

λN (xN ) = yN ,

λi (bi ) = yi = λi+1 (ai+1 ),

i ∈ NN−1 .

Two function tuples λ and S that satisfy Eqs. (5.93), (5.94) are said to have property (S). The class of polynomial B-splines Bn of order 2 < n ∈ N centered at the midpoint of the interval [ai , bi ] is an example of a class of functions Si for which condition (5.94) holds. Such polynomial B-splines Bn have the additional property that all derivatives up to order n − 2 vanish at the endpoints: Dk Bn (ai ) = 0 = Dk Bn (bi ) for all k ∈ N0n−2 . The above considerations imply the next theorem. Theorem 99. Let X and Xi , i ∈ NN , be as defined above. Let be as in Eq. (5.91). N

Assume that λ, S ∈ X C(Xi ) and that they possess property (S). Then the RB operator i=1

(5.30) maps C [0, 1] as given by Eq. (5.92) into itself and is well defined. Moreover, if ! max Si ∞,Xi : i ∈ NN < 1, then  is contractive on C [0, 1]. Proof. The conditions imposed on λ and S guarantee that  is well defined and maps C [0, 1] into itself. The contractivity of  under the given condition is a direct consequence of the arguments used in the proof of Theorem 95. The current setting also allows the construction of fractal functions of class C˙ s and Cn by imposing the appropriate conditions on the function tuples λ and S and choosing the correct interpolation sets. We encourage the diligent reader to obtain these conditions and establish the corresponding results.

12.3 Sobolev spaces W m,p At this point we refer to Section 4.3 and the notation and terminology used there. In this section we derive conditions so that the RB operator (5.30) maps a Sobolev space into itself.

Construction of fractal functions

225

To this end, suppose that X := (0, 1) and {Xi }i∈NN is a collection of nonempty open intervals of X. Further suppose that {x1 < · · · < xN−1 } is a partition of X and that {ui : Xi → X}i∈NN is a family of affine mappings with the property ui (Xi ) = (xi−1 , xi ),

i ∈ NN ,

where we set x0 := 0 and xN := 1. N

Theorem 100. Under the above assumptions, suppose that λ ∈ X W m,p (Xi ) and i=1

let S := (s1 , . . . , sN ) ∈ RN . Then the range of the RB operator : W m,p (X) → R(0,1) , m ∈ N0 and 1 ≤ p ≤ ∞, defined by g :=

N 

(λi ◦ u−1 i )χui (Xi ) +

N 

i=1

si (gi ◦ u−1 i )χui (Xi ) ,

i=1

is contained in W m,p and well defined. If, in addition, ⎧ ⎪ ⎨ max  ⎪ ⎩

 k∈{0,1,...,m}

i∈NN

|si | aki

i∈NN

|si |p kp−1 ai

1/p

< 1,

< 1,

1 ≤ p < ∞;

(5.95)

p = ∞,

then  is a contraction on W m,p (X). The unique fixed point f of  is referred to as a local fractal function of class W m,p . Proof. The well definedness of  and the fact that its range is contained in W m,p are consequences of the assumption on the function tuple λ and the observation that if the weak derivative of a function f exits and ui is a diffeomorphism, then the weak −1 −1 (1) derivative of f ◦ u−1 i exists and equals (D f ) ◦ ui · Dui . m,p To establish the contractivity of  on W , let g, h ∈ W m,p , k ∈ N0m . Denote the ordinary derivative of ui by ai . For 1 ≤ p < ∞, one obtains the following estimates: p

D(k) g − D(k) hLp

p   %    (k)  −1  = si (gi − hi )(ui )(x) χui (Xi ) dx D X  i∈NN %  p 1 kp   (k)  −1 p |si | dx ≤ D (gi − hi )(ui (x)) ai u (X ) i i i∈NN kp−1 %  p  1  (k)  p ≤ |si | D (gi − hi )(x) dx ai Xi i∈NN ⎞ ⎛ kp−1  1 ⎠ D(k) g − D(k) hp p . |si |p ≤⎝ L ai i∈NN

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Fractal Functions, Fractal Surfaces, and Wavelets

Summation over k ∈ N0m , and factoring out the maximum value of the expression in parentheses, yields the statement in the theorem. The condition for p = ∞ is obtained by application of the limiting process p → ∞ to the result for 1 ≤ p < ∞.

12.4 Besov and Triebel-Lizorkin spaces This section builds on the concepts of Besov and Triebel-Lizorkin spaces, which were introduced in Section 4.4. For definitions and terminology, we refer to Section 4.4. The objective is to derive conditions so that a fractal function defined by an RB operator of the form Eq. (5.30) belongs to Asp,q (X), where A = B or A = F. The conditions that will be presented correct those presented in Ref. [103], and extend and generalize the results in Refs. [78, 80, 83, 103, 104]. The material is based on Ref. [105]. For this purpose the following set of assumptions is made and required to hold throughout this section: (A1) X ∈ Rn is a bounded domain and Y := R. (A2) In addition to satisfying condition (P), the family of subsets {Xi : i ∈ NN } ∈ 2X and bijective mappings {ui : i ∈ NN } are assumed to be so that each ui ∈ Sim(Rn ); that is, a mapping Xi → X satisfying |ui (x) − ui (x )| = γi |x − x |,

∀ x, x ∈ Xi ,

for some constant γi ∈ R+ . Is not assumed that γi < 1, i ∈ NN . Each ui is then of the form ui (·) = γi Oi (·) + τi , where Oi ∈ SO(n) and τi ∈ Rn . (A3) Let 0 < p ≤ ∞ (0 < p < ∞ for the F-spaces), 0 < q ≤ ∞, and s ∈ R+ . (A4) For i ∈ NN the function λi : Xi → R belongs to Asp,q (X). (A5) For i∈ NN the functionSi : Xi → R belongs to the Zygmund space C (X), where >   max s, n 1p − 1 − s . +

(A6) The two n-tuples λ and S are fixed and we suppress them in the notation for  and its fixed points.

Finally, if a function f has support in X ⊂ Rn , we—if need be—regard it as defined on all of Rn by setting it equal to zero on Rn \ supp f .

12.4.1 Besov spaces We derive explicit conditions for the parameters γi and the functions Si , i ∈ NN , so that the associated local fractal function lies in a Besov space. Theorem 101. Assume that assumptions (A1)–(A6) hold and that s > σn,p . Then the affine RB operator given by Eq. (10.2) is well defined and maps Bsp,q (X) into itself. Let η := (η1 , . . . , ηN )t ∈ RN , where n

ηi := γi p If

−s

Si ∞,Xi ,

i ∈ NN .

(5.96)

Construction of fractal functions

227

! max ξ p , ηq < 1,

0 < p, q ≤ ∞,

(5.97)

where ξ ∈ RN is given by Eq. (5.79), the RB operator  is a contraction. Its unique fixed point f ∈ Bsp,q (X) is termed a local fractal function of class Bsp,q . Proof. Let f , g ∈ Bsp,q (X) and set φ := f −g. Then supp f ⊆ X and f Bsp,q < ∞, since all λi ∈ Bsp,q (X) and the functions Si are pointwise multipliers in Bsp,q . Hence  is well defined and maps Bsp,q (X) into itself. Suppose q < ∞. On ui (Xi ), the following holds: % Rn

|h|

−sq

 p q/p dh  M   h (φ) dm n |h|n R % %  p q/p dh  M  = |h|−sq ,  h (φ)(x ; ui (Xi )) dx |h|n ui (Xi ) ui (Xi )

%

since only if h ∈ ui (Xi ) is there any guarantee that M h (φ)(x ; ui (Xi )) = 0. Therefore if we use the fact that ui = γi Oi + τi , with γi > 0, Oi ∈ SO(n) and τi ∈ Rn , the last expression above yields

% |h|

−sq

ui (Xi )

%

 p q/p dh  M 

 h (φ)(x ; ui (Xi )) dx |h|n ui (Xi ) p q/p %  %   M dh −sq n  = |h| (Si · φi )(x; Xi ) dx γi  γi−1 O−1 n h |h| i ui (Xi ) Xi

%  % q/p p dh  M  −sq nq/p q ≤ γi |h|−sq γi Si ∞,Xi  h φi (x; Xi ) dx n |h| Xi Xi

q/p %  %  q( n −s) dh  M p q ≤ γi p Si ∞,Xi |h|−sq ,  h φ  dm |h|n Rn Rn

where we set x := ui (x). Hence |φ|B˙ s ≤

3 m 

p,q

 q

γi

41/q



n p −s

q Si ∞

|φ|B˙ s . p,q

i=1

The case q = ∞ follows from our letting q → ∞ in the above parenthetical expression. If we define a vector η ∈ Rm whose components are given by Eq. (5.96), using Eq. (5.79) and Theorem 95, we obtain ! ∀ 0 < p, q ≤ ∞, ∀s > σp : max ξ p , ηq < 1 ⇒ f ∈ Bsp,q (X).

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Fractal Functions, Fractal Surfaces, and Wavelets

Example 39. The setting of global fractal functions in R was considered in Ref. [83]. One obtains from Eqs. (5.79), (5.96) for n = 1, γi := N1 , Si := si ∈ R, i ∈ NN , s := k ∈ N0 , and p = q := 2, the condition derived in Ref. [83]: m 

|si |2 N 2k−1 < 1 ⇒ f ∈ W k,2 .

i=1

Note that for this particular setting ξi ≤ ηi for all i ∈ NN . Example 40. For the Slobodeckij spaces W s,p = Bsp,p , n := 1, 1 < p < ∞, and 0 σn,p,q . s (X) into itself. Let η ∈ RN have Then the RB operator  given by Eq. (10.2) maps Fp,q components n

ηi := γi p

−s

Si ∞,Xi

i ∈ NN .

(5.98)

If ! max ξ p , ηp < 1,

0 < p < ∞,

(5.99)

s (X). The unique where ξ ∈ RN is given by Eq. (5.79), then  is a contraction on Fp,q s (X) is called a local fractal function of class F s . fixed point f ∈ Fp,q p,q Remark 57. We note that condition (5.99) is independent of q. This is a direct consequence of the placement of the norms in the definition (1.69) of a TriebelLizorkin space.

Construction of fractal functions

229

s (X) and set φ := f − g. Then supp f ⊆ X and f  s < ∞ Proof. Let f , g ∈ Fp,q Fp,q s s . Therefore the as all λi ∈ Fp,q (X) and the functions Si are pointwise multipliers in Fp,q s (X) into itself. RB operator  maps Fp,q Now assume that 0 < q < ∞. The following estimates hold on ui (Xi ):

 q dh p/q  M |h|  h φ  dm |h|n Rn Rn % %  q dh p/q  

= |h|−sq  M (φ)(x ; u (X )) dx i i  h n |h| ui (Xi ) ui (Xi ) q 

% %  dh p/q  = γin |h|−sq  M−1 −1 Si (x) · φi (x; Xi ) dx γi Oi h |h|n Xi ui (Xi ) % %  q dh p/q  n−ps p −sq  M Si ∞,Xi |h|  h φi (x; Xi ) dx ≤ γi |h|n Xi Xi % %  q dh p/q   n−ps p ≤ γi Si ∞,Xi |h|−sq  M φ dx. h  n n |h|n R R

%

%

Thus |φ|F˙ p,q s ≤

−sq 

3 m 

p( n −s) p γi p Si ∞,Xi

41/p |φ|F˙ p,q s .

i=1

As the above expression in parentheses is independent of q, the case q = ∞ follows immediately. If we define a vector η ∈ RN by Eq. (5.98), recall the definition of ξ ∈ RN (Eq. 5.79), and combine everything with Theorem 95, we obtain for all 0 < p < ∞, for all 0 < q ≤ ∞, and for all s > σn,p,q ! s max ξ p , ηp < 1 ⇒ f ∈ Fp,q (X). Example 42. For a local fractal function to be an element of a Bessel potential s , one obtains in the particular case where n := 1, γ := 1 , and space H s,p = Fp,2 i N Si := si ∈ R, the condition N 

N ps−1 |si |p < 1

i=1

for 1 < p ≤ 2 and s > 1/p. Example 43. The local Hardy spaces are another class of function spaces that are Triebel-Lizorkin spaces for a certain range of the indices p, q, and s. These spaces are defined as follows. Let 0 < p < ∞ and let D0 := D0 (Rn ) denote the class of all

230

Fractal Functions, Fractal Surfaces, and Wavelets

C∞ -functions ϕ with compact support satisfying ϕ(0) = 1. Define ϕt (x) := ϕ(tx) for t > 0 and x ∈ Rn . Then <    < <  −1 < < (F ϕ F )f < ∞ , hp := hp (Rn ) := f ∈ Lp : < sup  < t < 0 0, is ( 2x , 2yp ). Then a fractel for 0 f (t)dt = xp+1 p+1

y is ( 2x , 2p+1 ). Employing Cauchy’s formula for repeated integration, we can extend Proposition 68 to fractional integrals. For this purpose, suppose that f , g ∈ L1 (0, b) for some 0 < b ≤ ∞. Then for α > 0, the α-fractional integral of f , J α f , is defined by

J α f (x) :=

1 (α)



x

(x − t)α−1 f (t)dt,

x ≤ b.

(6.7)

0

If w(x, y) = (sx, g(x) + cy) is a fractel for f , then an argument similar to the one employed in the proof of Proposition 68 gives w(x, y) = (sx, sα J α g(x) + sα cy) as a fractel for J α f . Note that in the case α := 12 , Eq. (6.7) is related to the Abel integral equation.

Dimension of fractal functions

7

Abstract This chapter is devoted to dimension calculations for univariate fractal functions. First the dimension theorems for those functions considered in Sections 1, 4, and 5 are proven. Then a dimension result for Rm -valued fractal functions is stated without proof. The last section gives the box dimension for a class of biaffine fractal interpolation functions. The material in this section is based on the original articles [89, 99, 106, 107, 205, 206].

1 Affine fractal functions First a dimension theorem for affine fractal functions (Example 26) is proven. The methods used in the proof of this theorem then lead the way to dimension formulae for recurrent, hidden-variable, and Rm -valued fractal functions. Several arguments presented here were originally derived in Ref. [89]. Let X := [0, 1] and let Y := R. Let f : [0, 1] → R be a continuous affine fractal function and let G denote its graph. Using the notation introduced in Section 1, in particular Example 26, we make the following two assumptions: (D1):

N 

|si | > 1;

and

(D2):  is not collinear.

i=1

The dimension theorems stated later involve expressions of the form  |si |ad−1 , i i∈NN

 where the si satisfy assumption (D1) and i∈NN ai = 1. Note that the function  d−1 is strictly increasing in d and satisfies h: R → R defined by h(d) := i∈NN |si |ai limd→−∞ h(d) = ∞ and h(2) ≤ max{|si |: i ∈ NN } < 1. Hence there exists a unique d∗ ∈ (−∞, 2) such that h(d∗ ) = 1. Furthermore, d∗ > 1 iff assumption (D1) is satisfied. The main idea behind all the dimension calculations is to define the right covers for the graphs G of the fractal functions. Now a class of such covers is defined which allows one to relate covers of different sizes. Definition 127. Let 0 < ε < 1. The collection of numbers {P :  ∈ N0K } is called an ε-partition iff: Fractal Functions, Fractal Surfaces, and Wavelets. http://dx.doi.org/10.1016/B978-0-12-804408-7.00007-2 Copyright © 2016 Elsevier Inc. All rights reserved.

240

Fractal Functions, Fractal Surfaces, and Wavelets

1. ∀ ∈ N0K : − ε/2 < P < 1; 2. ∀ ∈ N0K−1 : ε/2 < P+1 − P ≤ ε.

A cover C of G is called an ε-column cover with associated ε-partition {P } iff there exist n ∈ N0 and ξ ∈ R,  ∈ N0K , such that C = {[P , P + ε] × [ξ + (j − 1)ε, ξ + j ε]: j ∈ Nn ∧  ∈ N0K }. The class of all such covers of G is denoted by C ∗ (ε). A nonoverlapping ε-cover of G is a cover of G consisting of (ε × ε)-squares with  disjoint interiors of the form [ε, ( + 1)ε] × [y, y + ε], where  ∈ {0, 1, . . . , ε −1 } and y ∈ R. Let C ∗∗ (ε) denote the class of all nonoverlapping ε-covers of  G. Note that a cover C ∈ C ∗ (ε) consists of ∈N0 n closed (ε × ε)-squares arranged K in K + 1 columns. Let N ∗ (ε) := min{|C|c : C ∈ C ∗ (ε)}

and N ∗∗ (ε) := min{|C|c : C ∈ C ∗∗ (ε)},

and let N (ε) denote the minimum number of (ε × ε)-squares [x, x + ε] × [y, y + ε], x, y ∈ R, necessary to cover G. The next result shows that for the calculation of the box dimension of G, it suffices to consider covers from C ∗ (ε). Proposition 69. For all 0 < ε < 1: N (ε) ≤ N ∗ (ε) ≤ 2 N (ε). Proof. The definition of nonoverlapping ε-cover implies that N ∗∗ (ε) ≤ 2 N (ε). Also, if C ∈ C ∗∗ (ε), then, since f is continuous, C ∈ C ∗ (ε). Hence N ∗ (ε) ≤ N ∗∗ (ε) ≤ 2 N (ε).

For the proof of Theorem 104 the following lemmas are needed. Lemma 13. If assumptions (D1) and (D2) are satisfied, limε→0 ε N ∗ (ε) = ∞. Proof. As  is not collinear, there exists an n ∈ NN such that V := |yn − y0 − (yN − y0 )xn | > 0. Let a := min{ai : i ∈ NN }. The continuity of f implies ∗

N (ε) ≥

N  i1 =1

···

N  

|si1 · · · sik | Vε

ik =1

for all 0 < ε < ak and all k ∈ N. Thus

−1



≥

 N  i=1

k |si |

Vε−1

then

Dimension of fractal functions

241

lim ε N ∗ (ε) ≥ lim

 N 

k→∞

ε→0

k |si |

V=∞

i=1

by assumption (D1). Lemma 14. Let x0 ∈ (1, ∞) and let ξ : (0, x0 ] → R be a decreasing function. Assume that limx→∞ x ξ(x) = ∞. Furthermore, suppose that for all 0 ≤ x ≤ x0 , ξ satisfies the functional inequalities



 N N   |si | |si | x x α β ξ ξ − ≤ ξ(x) ≤ + , ai ai x ai ai x i=1

i=1



where 0 < ai , |si | < 1, i∈NN ai = 1, and α, β > 0. If (D1) holds, then there exist positive constants A and B such that ∗

∗

A x−d ≤ ξ(x) ≤ B x−d ,

(7.1)

where d∗ is the unique positive solution of N 

|si |ad−1 = 1. i

i=1



|si | ≤ 1, then there exists a B > 0 such that ξ(x) ≤ B x−1 . Proof. Assume that (D1) holds. Let a := min{ai : i ∈ NN } and a¯ := max{ai : i ∈ x > 0 small enough such that for all NN }. As limx→∞ x ξ(x) = ∞, there exists an x ∈ (0, x/a ], If

i∈NN

2α x−1 . |s | − 1 i i∈NN

ξ(x) ≥ 

Now select a k1 > 0 small enough such that ∗

x−d ≤  k1

α x−1 , i∈NN |si | − 1

and choose k2 > 0 large enough so that ξ(x) ≤

1−

β  i∈NN

|si |

x−1 + k2 x−d

∗

 for x ≤ x ≤ x/a. Recall that d∗ > 1 iff i∈NN |si | > 1. Now define upper and lower functions ξ , ξ : (0, x0 ] → R by

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Fractal Functions, Fractal Surfaces, and Wavelets

ξ (x) :=

1−

β 

|si |

i∈NN

x−1 + k2 x−d

∗

and ξ (x) := 

α ∗ x−1 + k1 x−d , |s | − 1 i∈NN i

respectively. Then the preceding arguments yield that ∀x ∈ [ x, x/a]: ξ ≤ ξ ≤ ξ . Note that by construction the functions ξ and ξ satisfy the following functional equations: ξ (x) =

 N

 |si | i=1

ai

ξ

x ai



− αx−1

and ξ (x) =

 N

 |si | i=1

ai

ξ

x ai



+ βx−1 .

If a x ≤ x ≤ x, then x ≤ x ≤ x/a and thus ξ(x) ≤

 N

 |si | i=1

ai

ξ

x ai



+ βx−1 ≤

 N

 |si | ai

i=1

ξ

x ai



+ βx−1 = ξ (x).

A similar argument shows that ξ ≤ ξ on [a x, x]. x, x], The preceding argument serves as an induction step: Suppose that for x ∈ [an x, x]. Since an → 0 as n ∈ N, ξ (x) ≤ ξ(x) ≤ ξ (x); then it must also hold for x ∈ [an+1 n → ∞, x]. ξ (x) ≤ ξ(x) ≤ ξ (x) for all x ∈ (0, Furthermore, since d∗ > 1, there exist two positive constants A and B such that ξ (x) ≥ ∗ −d∗ . This proves the lemma under assumption (D1). A x−d and ξ (x) ≤ B x Suppose then that i∈NN |si | < 1. As before, a positive constant k2 can be found such that ξ(x) ≤ ξ (x) =

1−

β  i∈NN

|si |

x−1 + k2 x−d

∗

Dimension of fractal functions

243

 for all x ∈ (0, 1]. Now, however, d∗ < 1, and since β (1 − i∈NN |si |)−1 > 0, there is −1 some B > 0 such that ξ(x)  ≤ Bx . Finally, suppose that i∈NN |si | = 1. Then define an upper function ξ by ξ (x) := 

log x β + k2 x−1 x i∈NN |si | log ai

for any k2 ∈ R. This upper function then satisfies ξ (x) =

  N

 x |si | ξ + βx−1 ai ai i=1

for all x ∈ (0, 1]. Hence, as before, a B > 0 can be found such that ξ(x) ≤ ξ (x) ≤ B x−1 . This proves the lemma. Theorem 104. Let f : [0, 1] → R be a continuous affine fractal function and G its graph. Suppose that assumptions (D1) and (D2) are satisfied. Then the box dimension dimb G is the unique positive solution d∗ of N 

|si |ad−1 = 1; i

(7.2)

i=1

otherwise dimb G = 1. Proof. Let ([0, 1] × R, w) be the contractive iterated function system (IFS) whose unique attractor is G. Let ε ∈ (0, 1) be given. Choose a minimal ε-cover C(ε) of G. By Proposition 69 it may be assumed that C(ε) ∈ C ∗ (ε). Let {P } be the associated ε-partition. For i ∈ NN , let [a, b + ε] be the smallest interval of the form [P1 , P2 + ε] that covers Ii = ui [0, 1]. Denote by Ci∗ (ε) := {C ∈ C ∗ (ε): C ⊆ [P1 , P2 + ε] × R} ∗ (ε) ∩ the “restriction” of C ∗ (ε) to Ii and denote by Ni∗ (ε) its cardinality. Note that |Ci+1 Ci∗ (ε)|c ≤ 2 and that f is uniformly bounded on [0, 1]. Therefore there exists an α1 > 0 such that for all 0 < ε < 1, N 

Ni∗ (ε) ≤ N ∗ (ε) + α1 ε−1 .

i=1

Now suppose that i ∈ NN is such that si = 0. Let R be an element of Ci∗ (ε) which is the union of n (ε × ε)-squares. Since the map wi ∈ w is invertible, the inverse image w−1 i R of R is a parallelogram of width ε/|ai |, height nε/|si |, and shear ci /(ai |si |). Thus w−1 i R can be covered by

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Fractal Functions, Fractal Surfaces, and Wavelets

  nai  ci  + +1 |si |  ai 

∗ ∗ (ε/ai × ε/ai )-squares. Hence w−1 i Ci (ε) is an element of the covering class C (ε/ai ) ∗ of G. Realizing that there are at most (2ai )/ε + 2 elements R in Ci (ε), we can find a positive constant β1 such that N  |si | i=1

β1  ∗ α1 N (ε/ai ) − Ni (ε) − ≤ ≤ N ∗ (ε). ai ε ε N

∗

(7.3)

i=1

Next an upper bound for N ∗ (ε) is obtained. Suppose that C ∈ C ∗ (ε/ai ) is a minimal cover of G and R is one of its elements. The direct image of R under wi is also a parallelogram, but of width ε, height (n|si |ε)/ai , and shear |ci |ε. (Here it is again assumed that R is covered by n (ε/ai × ε/ai )-squares.) The number of (ε × ε)-squares needed to cover wi R is then given by

 n|si | |ci | + + 1. ai ai

This way a cover Ci of wi G consisting of (ε × ε)-squares is generated. To relate the cardinality of Ci to the cardinality of C, one again makes use of the fact that C contains at most (2ai )/ε + 2 elements R and thus obtains

Ci c ≤

|si | ∗ α2 N (ε/ai ) + ai ε

for some positive constant α2 . The union

N 

Ci is a cover of G but in general is not an

i=1

element of C ∗ (ε), since the columns R in adjacent covers Ci may not properly join up. N  However, a cover in C ∗ (ε) can be constructed from Ci by replacement of at most i=1

two elements in Ci ∩ Ci+1 with at most two properly spaced ones. This then yields the required upper bound for N ∗ (ε):  N 

 N    4n   ∗ N (ε) ≤  Ci  ≤ |Ci |c + +1 .   ε i=1

c

i=1

Hence a positive constant β2 can be found such that N ∗ (ε) ≤

N  |si | i=1

ai

N ∗ (ε/ai ) +

β2 . ε

(7.4)

Dimension of fractal functions

245

Applying Lemma 14 to Eqs. (7.3), (7.4) yields the required results if assumption (D2) holds. If assumption (D2) is not true, then the attractor G of the IFS ([0, 1] × R, w) is the line segment joining (0, y0 ) and (1, yN ). Hence dimb G = 1.

2 Recurrent fractal functions Next the dimension theorem for recurrent fractal functions is stated (see also Ref. [37]). At this point the reader may want to review the notation and terminology introduced in Section 6. The case when the graph G of the recurrent fractal function is the attractor of a recurrent IFS whose maps wi are affine will be considered. First some more notation is introduced. Let D(d) denote the diagonal matrix ⎛

|s1 | |a1 |d−1 ⎜ 0 ⎜ D(d) := ⎜ ⎝ 0 0

0 |s2 | |a2 |d−1 0 0

... ... .. . ...

0 0 0 |sN | |aN |d−1

⎞ ⎟ ⎟ ⎟ ⎠

and let r(d) := r(CD(d)) denote the spectral radius of the product CD(d), where C is the irreducible N × N connection matrix associated with the recurrent IFS. It is not hard to see that there exists a number d∗ > 0 such that r(d∗ ) = 1. (Use the PerronFrobenius theorem and the properties of the function h defined at the beginning of this chapter.) Theorem 105. Let f : [0, 1] → R be a continuous recurrent fractal function and let G be its graph. If r(CD(1)) > 1 and if {(xi , yi ): xi ∈ Xk } is not collinear, then dimb G = d∗ ; otherwise dimb G = 1. Proof. The proof of this theorem relies essentially on the same type of arguments ∗ that were used in the proof  of Theorem 104. The covers needed are those in C (ε), Gi , where Gi is the portion of G above Xi , one obtains, with 0 < ε < 1. Since G = i∈NN

only minor modifications—the details of which are left for the reader—the following system of functional inequalities:



   si    si    N ∗ ε − αi ε−1 ≤ N ∗ (ε) ≤   N ∗ ε + βi ε−1 , i a  j a a  j a i i i i

j∈I(i)

(7.5)

j∈I(i)

where Ni∗ (ε) := min{|C|c : C ∈ C ∗ (ε)} and αi and βi are positive constants, i ∈ NN . However, to apply Lemma 14, Lemma 13 needs to be modified. Lemma 15. If  := r(1) > 1 and if there exists a k ∈ NN such that {(xi , yi ): xi ∈ Xk } is not collinear, then limε→0 ε Ni∗ (ε) = ∞ for all i ∈ NN . Proof. As {(xi , yi ): xi ∈ Xk } is not collinear and C is irreducible, the portion Gi of G above Xi is not a line segment for any i ∈ NN . Therefore one can find three noncollinear

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Fractal Functions, Fractal Surfaces, and Wavelets

points Pi (ξi , ηi ), Pi (ξi , ηi ), and Pi (ξi , ηi ) on the graph of f that belong to the interior of Xi × R and that satisfy ξi < ξi < ξi . Let Vi denote the vertical distance between the point Pi (ξi , ηi ) and the line segment connecting Pi (ξi , ηi ) and Pi (ξi , ηi ). A simple calculation shows that 

   η − ηi  Vi = ηi − ηi + i (ξi − ξi )  . ξ i − ξi Let δ := min{|ξi − ξi+1 |: i ∈ NN−1 }. The Perron-Frobenius theorem guarantees the existence of a positive eigenvector v = (v1 , . . . , vN ) of CD(1) with eigenvalue  such that vi ≤ Vi for all i ∈ NN . As in Lemma 13, the continuity of f implies that Ni∗ (ε) ≥ Vi ε−1 ≥ vi ε−1 .

Now leta := min{|ai |: i ∈ NN }. It follows from the recurrent structure of G—that wi (Aj )—that if ε < a δ, then is, Gi = j∈ I(i)

Ni∗ (ε) ≥ |si |



Vj ε−1 ≥ |si |

j∈I(i)



vj ε−1 = ε−1 (CD(1))i = vi ε−1 .

j∈I(i)

Applying the preceding arguments to the images of wi Aj under the recurrent IFS mappings, one obtains after n steps εNi∗ (ε) ≥ n vi for 0 < ε < δan . This proves the lemma. Proof of Theorem 105 continued. Since  = r(1) > 1, there exists a unique d∗ > 1 such that r(d∗ ) = 1. As in the proof of Lemma 15, the positive eigenvector of CD(1) is denoted by v and its eigenvalue is denoted by . The positive eigenvector u of CD(d∗ ) whose eigenvalue is equal to 1 is also needed. Next real numbers α and β are chosen such that for all i ∈ NN , αvi ≥ αi and βvi ≥ βi . With these new additions, the system of functional inequalities Eq. (7.5) now reads    



   si    si    N ∗ ε − αvi ε−1 ≤ N ∗ (ε) ≤   N ∗ ε + βvi ε−1 . (7.6) i a  j a a  j a i i i i

j∈I(i)

j∈I(i)

It is easy to verify that the functions ∗

ζi (ε; K, γ ) := Kui ε−D +

γ vi ε−1 , 1−

i ∈ NN ,

solve the following system of functional equations associated with Eq. (7.6):

Dimension of fractal functions

247

  si  ε    ζj ζi (ε) = + γ vi ε−1 , a  ai i

i ∈ NN ,

j∈I(i)

where K ∈ R is arbitrary and γ denotes either −α or β. Select K1 large enough so that ∀x ∈ [ a, 1]: Ni∗ (ε) ≤ ζi (ε; K1 , γ ),

i ∈ NN .

Let a := max{|ai |: i ∈ NN }. As in the proof of Theorem 104 the validity of the preceding inequality for all x ∈ (0, ε] needs to be established, where ε has to be chosen appropriately. For this purpose, observe that if a a ≤ ε ≤ a, then a ≤ ε/|ai | ≤ 1 and therefore, by system (7.5), Ni∗ (ε) ≤



|si /ai | Nj∗ (ε/ai ) + βvi ε−1

j∈I(i)

≤



|si /ai | ζj (ε/ai ; K1 , β) + βvi ε−1 ≤ ζi (ε; K1 , β)

j∈I(i)

for i ∈ NN . Inductively one thus obtains ∀x ∈ [a n a, 1]: Ni∗ (ε) ≤ ζi (ε; K1 , γ ),

i ∈ NN ; n ∈ N,

and since a n → 0 as n → ∞, Ni∗ (ε) ≤ ζi (ε; K1 , γ )

for all ε ∈ (0, 1]. The definition of the box dimension for an attractor of a recurrent IFS then gives an upper bound for dimb G: dimb G ≤ lim

log



ε→∞

i ζi (ε; K1 , β) = max{d∗ , 1}. − log ε

ε > 0 small enough To obtain a lower bound for dimb G, Lemma 15 is used. Choose an so that ∀ε ∈ (0, ε]: ε Ni∗ (ε) >

γ vi ε−1 , 1−

i ∈ NN .

Then it is possible to select a K2 > 0 such that Ni∗ (ε) ≥ ζi (ε; K2 , α)

for a ε ≤ ε ≤ ε and i ∈ NN . Arguments similar to those given earlier then that ∀ε ∈ (0, ε): Ni∗ (ε) ≥ ζi (ε; K2 , α),

i ∈ NN .

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Fractal Functions, Fractal Surfaces, and Wavelets

Hence dimb G ≥ max{d∗ , 1} implying that dimb G = d∗ . If  ≤ 1, then d∗ ≤ 1, and thus dimb G ≤ 1. However, since G is the graph of a continuous function, dimb G ≥ 1. Hence dimb G = 1. If {(xi , yi ): xi ∈ Xk } is collinear for all k ∈ NN , then G is the union of a finite number of line segments and hence has box dimension equal to 1.

3 Hidden-variable fractal functions In the case of hidden-variable fractal functions, there is an interesting relation between the box dimension of graph fk and the ambient space. However, before this relationship is stated and proved, the necessary background needs to be provided. The notation and terminology introduced in Section 7 are used but will be restricted to the following setup: 1. The mappings vi are independent of x: vi : Y → Y. 2. ∀i ∈ NN : ui ∈ Lip(≤si ) (X) ∧ vi ∈ Lip(≤si ) (Y) (0 < si < 1).

Now let F : A(X) → function given in Theorem 82 and let d∗ be the unique Y be the d positive solution of i∈NN si = 1. Theorem 106. Let (X × Y, w) be the IFS considered in Theorem 82 with wi ∈ S ∗ (X × Y). Then dimb graph F ≤ d∗ . Proof. The proof is rather straightforward: Let C be a cover of graph F of minimal cardinality N (ε) consisting of ε × ε balls. Denote by Ni (ε) the least number of such balls to cover wi (graph F ). Since wi maps any ε/si -ball into an ε-ball, it is seen that Ni (ε) ≤ N (ε/si ). Therefore N (ε) ≤

N  i=1

Ni (ε) ≤

N 

N (ε/si ).

i=1 ∗

This, however, is a special case of Lemma 14. Hence N (ε) ≤ B ε−d for some B > 0, which proves the theorem. Since the attractor A(X) of the contractive IFS (X, u) also has box dimension equal to d∗ , and since projX graph F = projX A(X × Y) = A(X), we obtain the following corollary. Corollary 10. dimb graph F = d∗ . Now assume that graph F is the attractor of the contractive IFS (I × X, W) defined in Proposition 55, with 0 < |si | < 1 and wi ∈ S ∗ (I × X), i ∈ NN . If si ≥ 1/N for all i ∈ NN , Corollary 10 gives immediately the box dimension of graph F. The goal is to relax this condition on the si . Theorem 107. dimb graph F = d∗ . Proof. Let n ∈ N and let Bn denote a ball of radius 1/n. Let C(n) := {[(j − 1)/N n , j/N n ] × Bn : j ∈ NN n }

Dimension of fractal functions

249

be a minimal cover of G := graph F of cardinality N (n). Denote by N  (ε) and N  (ε) the least number of ε-balls needed to cover  G and A(X), respectively. It is easy to see that N (n) ≥ N  (2/N n ). Let Gi := G [(j−1)/N n ,j/N n ]×X and let j(n) be the code corresponding to [(j − 1)/N n , j/N n ]. Let C be a minimal (N n sj(n) )−1 -cover of A(X). If the map Wj(n) is applied to {[0, 1] × B : B ∈ C }, then Gi can be covered by N  ((N n sj(n) )−1 ) sets from C(n). Hence, N (n) ≤

N 

···

j1 =1

= B N nd

N 

N  ((N n sj(n) )−1 ) ≤ B N nd

jn =1

 ∗

N 

∗

N  j1 =1

sdi

∗

···

N 

∗

sdj(n)

jn =1

∗

= B N nd .

i=1

To obtain the second inequality, the fact dimb A(X) = d∗ was used (ie, N  (ε) ≤ B ε−d for some B > 0). If ε is chosen in [2/N n , 2/N n−1 ], then

∗

∗

log N (ε) log B N nd , ≤ − log ε log N n−1 /2 and thus lim sup ε→0

log N (ε) ≤ d∗ . − log ε

However, as dimb A(X) = d∗ , dimb G ≥ d∗ . Next the dimension formula for the hidden-variable fractal functions fk ∈ HF (I, Xk ) is derived, where Xk is a summand in the finite direct sum decomposition of the linear space X. It is assumed that Xk = R. Following Remark 51, maps Wi (t, x) of the form 

ai t + bi , i ∈ NN , Wi (t, x) = wi (x) are considered, where the ai and bi are determined by the interpolation property and wi ∈ Sim0 (X). Theorem 108. Let fk : I → R be a hidden-variable fractal function defined via the preceding maps. Then the box dimension of graph fk is the unique positive solution d∗ of N  i=1

si ad−1 = 1. i

(7.7)

250

Fractal Functions, Fractal Surfaces, and Wavelets

Proof. The notation and terminology developed in Theorem 104 will be used. Let {P :  ∈ N0K } be the ε-partition associated with a minimal ε-column cover C(ε) of graph fk .  For a fixed , consider the union U of all (ε×ε)-squares covering graph fk [P ,P+1 ] . The continuity of fk implies that U is a rectangle of width ε and height h . Consider the For a pencil of planes  := {θ : θ ∈ [0, 2π)} whose axis coincides with the t-axis.  plane θ ∈ , denote by h (θ ) the height of the orthogonal projection of fk [P ,P+1 ] onto θ . Let m :=  min{h (θ ): θ ∈ [0, 2π)} and M  := max{h (θ ): θ ∈ [0, 2π)}. Define N (ε) := ∈N0 m ε−1 and N (ε) := ∈N0 M ε−1 . Then K

K

N (ε) ≤ |C(ε)|c ≤ N (ε).

As in the proof of Theorem 104, functional inequalities for N (ε) and N (ε) are obtained, yielding the existence of two positive constants A and B such that ∗

∗

Aε−d ≤ |C(ε)|c ≤ Aε−d ,  d−1 where d∗ is the unique positivesolution of = 1. Note that the i∈NN si ai collinearity of k is equivalent to i∈NN si = 1. Remark 61. As dimb A(X) = d∗ < ∞, it might as well be assumed that X has (topological) dimension n := d∗  + 1. If Xk is a summand in a finite direct sum decomposition of X, then dim Xk =:m < n. If k is collinear (m = 1) or coplanar n (m > 1), then i∈NN sm i∈NN si ≤ 1. i = 1. Also, Theorem 108 can be generalized to vector-valued hidden-variable fractal functions [79]. This result is stated without our presenting its proof, with the reader being referred to Refs. [79, 106]. The arguments leading to functional inequalities in the vector-valued case are quite similar to those given in Theorem 104. The difficulty arises in generalizing Lemma 13; its proof becomes rather technical and involved (see Ref. [106]). Xk be a vector-valued hidden-variable fractal function. Theorem 109. Let fk : I →  Suppose dim Xk = m and that i∈NN sm i > 1. Then dimb graph fk = d∗ , where d∗ is the unique positive solution of N 

d−m sm = 1. i ai

(7.8)

i=1

Now the previously mentioned relationship between dimb graph fk and dim I × Xk can be proven. Theorem 110. Suppose  that fk ∈ HF (I, Xk ) and that dim Xk = m < n. Suppose d∗ = dimb graph fk . If i∈NN sm i > 1, then

Dimension of fractal functions

251

m ≤ d∗ ≤ (m + 1) −

m . n

(7.9)

Proof. The first inequality is clear. To prove the second, let q be such that m/n + 1/q = 1. The Cauchy-Schwarz inequality implies 1=



⎛ d sm i ai

∗ −m

≤⎝

i∈NN

But since

 i∈NN



⎞m/n ⎛ sni ⎠

⎝

i∈NN



⎛

⎞1/q q(d∗ −m)

ai

⎠

≤⎝

i∈NN



⎞1/q q(d−m) ⎠

ai

.

i∈NN

ai = 1, q(d∗ − m) ≤ 1. Therefore d∗ ≤ (m + 1) − (m/n).

4 Biaffine fractal functions In this section we derive a formula for the box dimension of the graphs of a class of biaffine fractal functions. The presentation closely follows Ref. [99]. Let I := [0, 1] ⊂ R be the unit interval and let I 2 := I × I. Assume that we are given a set of knots {0 =: x0 < x1 < · · · < xN := 1} on I. Furthermore, assume that {yj ∈ I: j ∈ N0N }

and

{yj ∈ I: j ∈ N0N }

are two finite sets of points with the property that 0 ≤ yj ≤ yj < 1 for all j ∈ N0N . Let Qn be the trapezoid with vertices An := (xn−1 , yn−1 ), Bn := (xn , yn ), Cn := (xn , yn ), and Dn := (sn−1 , yn−1 ), n ∈ NN . For each n ∈ NN , let ln : I → [xn−1 , xn ] be a collection of affine mappings and Bn : I × R → R, (x, y) → an x + bn y + cn xy + dn = (dn + an x) + (bn + cn x)y, an , bn , cn , dn ∈ R be a collection of biaffine mappings. Define mappings wn := (ln , Bn ): I 2 → Qn by requiring that wn

(0, 0) −→ An ,

wn

(1, 0) −→ Bn ,

wn

(1, 1) −→ Cn ,

wn

(0, 1) −→ Dn .

(7.10)

It follows readily from Eq. (7.10) that the affine mappings ln are given by Eq. (5.73) and the bilinear mappings Bn are given by Bn (x, y) = an x + [sn−1 + (sn − sn−1 )x] y + yi−1 ,

(7.11)

where we set an = yn − yn−1 , n ∈ NN , and sj := yj − yj , j ∈ N0N . Notice that 0 ≤ sj < 1 for all j = 0, 1, . . . , N. Set w := {w1 , . . . , wn }.

252

Fractal Functions, Fractal Surfaces, and Wavelets

Recall the definition of the metric dq given in Proposition 57. Here we set q ≡ 1 and denote the Lipschitz constant of the mapping ln by λn . Theorem 111. The biaffine IFS F = (I 2 , w) is contractive in the metric d1 with n α := 1 and 0 < β < 1−λ 2 . Proof. We show that each wn ∈ F is contractive with respect to the metric d1 . For this purpose, let (x, y), (x , y ) ∈ I 2 and let sn := sn − sn−1 , n ∈ NN . Then d1 (wn (x, y), wn (x , y )) = |ln (x) − ln (x )| + β|Bn (x, y) − Bn (x , y )| ≤ λn |x − x | + β(|an ||x − x | + |[sn−1 + sn x]y − [sn−1 + sn x ]y |) = λn |x − x | + β(|an ||x − x | + |[sn−1 + sn x]y − |[sn−1 + sn x ]y + |[sn−1 + sn x ]y − |[sn−1 + sn x ]y ) ≤ λn |x − x | + β(|an ||x − x | + |sn y||x − x | + |sn−1 + sn x ||y − y|) ≤ (λn + β(|an | + |sn |)) |x − x | + β|sn ||y − y | (since |x |, |y| ≤ 1) ≤ max {λn + 2β, sn } d1 ((x, y), (y , y )) (since |an |, |sn | ≤ 1). n If we choose 0 < β < 1−λ 2 , then max {λn + 2β, sn } can be made strictly smaller than 1. If we adapt Eqs. (5.73)–(5.75) to the current setting using {yj : j ∈ N0N } instead

of {yj : j ∈ N0N }, Theorem 93 shows that the associated operator : C∗ → C defined by

g := h + [sn−1 + sn ( · )] · (g − b) ◦ L

(7.12)

is contractive and its unique fixed point f is an element of C . Moreover, f satisfies the functional equations presented in Eq. (5.76). To derive a formula for the box dimension of the graphs of biaffine fractal functions arising from the above biaffine IFS (I 2 , w), we assume, without loss of generality, that y0 = yN = 0. (This can always be achieved by means of an affine transformation and leaves the box dimension invariant.) The proof of the following theorem is based on arguments first employed in Ref. [107]. Theorem 112. Let (I 2 , w) denote the biaffine IFS defined above and let G(f ) denote its attractor. Assume that the knots {xj := Nj : j ∈ N0N } are uniformly spaced on I, and  sn−1 +sn assume that s0 = sN . If γ := > 1 and G(f ) is not a straight line n∈NN 2 segment, then dimb G(f ) = 1 +

log γ ; log N

Dimension of fractal functions

253

otherwise dimb G(f ) = 1. Proof. In the computation of the box dimension of G(f ), it suffices to consider covers whose elements are squares of side N −r , r ∈ N0 . To this end, denote by C0 (r) a cover of G(f ) consisting of a finite number of squares of side N −r , r ∈ N0 . Now consider a specific cover C (r) of G(f ) of the form  C (r) :=

    k−1 k 1 , × a, a + r : r ∈ N0 ; k ∈ NN r ; a ∈ R . Nr Nr N

(7.13)

As G(f ) is compact, there exists a minimal cover C0∗ (r) of G(f ) and also a minimal cover C ∗ (r) of G(f ) of the form Eq. (7.13). Denote by N0 (r) and N (r) the cardinality of these minimal covers. As the covers from Eq. (7.13) are more restrictive, one has N0 (r) ≤ N (r). On the other hand, since each (N −r × N −r )-square in C0∗ (r) can be covered by at most two (N −r × N −r )-squares from a cover of the form Eq. (7.13), one obtains N (r) ≤ 2N0 (r). Therefore it suffices to consider covers from Eq. (7.13). Suppose r ∈ N0 is fixed. Let C (r) be a minimal cover of G(f ) of cardinality N (r) consisting of squares of side N −r whose interiors are disjoint. Let C (r, k) be k r the collection of all squares in C (r) that lie between x = k−1 N r and x = N r , k ∈ NN . Let N (r, k) denote the cardinality of C (r, k) and let 

R(r, k) :=

C.

C∈C (r,k)

Since C (r) is a cover of G(f ) of minimal cardinality, every square in C (r) must meet G(f ), and since f is continuous on I, the set R(r, k) must be a rectangle of width N −r  r and height N −r N (r, k). Moreover, N (r) = N k=1 N (r, k). Apply the mappings wn , n ∈ NN , defined in Eq. (7.10), to the rectangle R(r, k). Then wn (R(r, k)) is a trapezoid contained in the strip 

 l(k, n) − 1 l(k, n) , r+1 × R, N r+1 N

where l(k, n) := k + (n − 1)N r . Also, r

N (r + 1) =

N  N 

N (r + 1, l(k, n)).

n=1 k=1

The fixed-point equation for G(f ) implies that G(f ) ⊆

N  n=1

wn

 Nr  k=1

R(r, k) .

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Fractal Functions, Fractal Surfaces, and Wavelets

Depending on the sign of sn , there are 10 possible geometric shapes for the trapezoid wn (R(r, k)). In Fig. 7.1 we show one of these trapezoids and display the relevant geometric quantities. y2 C

an

wn

h

y1

D′

D C′

B′

an

1

1

B

Nr

Nr

Sn y1 Nr

A′ A

Sn y2 Nr

1

Fig. 7.1 An image of a rectangle under the map wn .

In the notation of Fig. 7.1, we write A < B if the y-coordinate of the point A is less than the y-coordinate of the point B . Similarly, we define A ≤ B . Case 1: sn ≥ 0. In this case, distance(A , C ) ≤ distance(B , D ). Hence there are five possible shapes given by the location of the vertices A , B , C , and D . These are B < D ≤ A < C , B < A ≤ D ≤ C , B ≤ A < C ≤ D , A ≤ B ≤ C < D , and A < C ≤ B < D . Each of these trapezoids is contained in a rectangle of width N −(r+1) and height at most

k sn−1 + sn · r N



(N −r N (r, k)) +

2(|an | + |sn |) , Nr

(7.14)

and meets a rectangle of width N −(r+1) and height at least

 k−1 2(|an | + |sn |) . (N −r N (r, k)) − sn−1 + sn · r N Nr

(7.15)

Therefore 

  k N (r, k) 2(|an | + |sn |) sn−1 + sn · r + N r+1 + 2 N Nr Nr 

k = N sn−1 + sn · r N (r, k) + 2N(|an | + |sn |) + 2, N

N (r + 1, l(k, n)) ≤

and

 k−1 N (r + 1, l(k, n)) ≥ N sn−1 + sn · N (r, k) − 2N(|an | + |sn |) − 2. Nr (7.16)

Dimension of fractal functions

255

Case 2: sn ≤ 0. Now distance(A , C ) ≥ distance(B , D ) and the five possible shapes are as in Case 2. Each of these trapezoids is contained in a rectangle of width N −(r+1) and height at most  k−1 2(|an | + |sn |) , (N −r N (r, k)) + sn−1 + sn · r N Nr



(7.17)

and meets a rectangle of width N −(r+1) and height at least

sn−1 + sn ·

k Nr



(N −r N (r, k)) −

2(|an | + |sn |) . Nr

(7.18)

Hence, similarly to Case 1, an upper bound and a lower bound for N (r + 1, l(k, n)) are obtained. They are given by

k−1 N (r + 1, l(k, n)) ≤ N sn−1 + sn · Nr

 N (r, k) + 2N(|an | + |sn |) + 2

(7.19) and

 k N (r + 1, l(k, n)) ≥ N sn−1 + sn · r N (r, k) − 2N(|an | + |sn |) − 2. N (7.20)

We denote by N± the set of all indices n ∈ NN for which sn ≥ 0 or sn ≤ 0. Then using Eqs. (7.14), (7.17) and summing over n, we obtain N  n=1

N (r + 1, l(k, n)) =



N (r + 1, l(k, n)) +

n∈N+



N (r + 1, l(k, n))

n∈N−



 k sn−1 + sn · r N (r, k) ≤N N n∈N+

  k−1 N (r, k) sn−1 + sn · +N Nr n∈N−

+

N 

[2N(|an | + |sn |) + 2] .

n=1

Note that k sn−1 + sn sn−1 + sn · r = + sn N 2



 1 k − , Nr 2

n ∈ N+ ,

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Fractal Functions, Fractal Surfaces, and Wavelets

and

k−1 sn−1 + sn k−1 = + s n Nr 2 Nr

sn−1 + sn k = − + sn 2 Nr

sn−1 + sn ·

Substituting these expressions into N 

N (r + 1, l(k, n)) ≤ N

n=1

 n∈NN



2

+N

N 



sn

n=1

+

N 

 n∈NN

N (r, k)

 1 k − N (r, k) Nr 2

N  −sn  + [2N(|an | + |sn |) + 2] . Nr

n∈N−

By assumption,

n ∈ N− .

N (r + 1, l(k, n)) yields

 N

 sn−1 + sn n=1

 1 − 2  1 −sn , + 2 Nr

n=1

sn = sN − s0 = 0, and thus we obtain

N (r + 1, l(k, n)) ≤ (Nγ ) N (r, k) +

N  −sn  + [2N(|an | + |sn |) + 2] Nr

n∈N−

n=1

n=1

≤ (Nγ ) N (r, k) + c1 ,

(7.21)

   |sn | where c1 := N n=1 2N(|an | + |sn |) + N + 2 . Summing Eq. (7.21) now over k, we obtain an upper bound for N (r + 1) in terms of N (r): N (r + 1) ≤ (Nγ ) N (r) + c1 N r .

Induction over r gives N (r) ≤ (Nγ ) N (0) + c1 N r

r−1

r−1 

γ .

=0

Two cases are possible, depending on the value of γ : Case A: γ ≤ 1. This implies that N (r) ≤ N r (N (0) + c1 r). Thus log N r (N (0) + c1 r) = 1. r→∞ log N r

dimb G(f ) ≤ lim

Dimension of fractal functions

257

Case B: γ > 1. As r−1 

γ ≤

=0

γr , γ −1

one obtains N (r) ≤ (γ N)r N (0) +

c1 (Nγ )r =: c2 (Nγ )r . γ −1

Hence log c2 (γ N)r log γ =1+ . r→∞ log N r log N

dimb G(f ) ≤ lim

As f is a continuous b G(f ) ≥ 1. If G(f ) is a line segment—that is, if  function, dim the set of data  := ( Nj , yj ): j ∈ N0N is collinear—then G(f ) = [0, 1], implying that dimb G(f ) = 1. To obtain a nontrivial lower bound for G(f ), the following lemma is required. Lemma 16. If γ := i∈NN si−12+si > 1, s0 = sN , and if G(f ) is not a line segment, then Nr = 0. r→∞ N (r) lim

Proof. As G(f ) is not a line segment, there exists at least one index n0 ∈ NN−1 such that δ := yn > 0. 0

By continuity of f on I, we have that N (r) ≥ δN r . Since I is mapped to the line segments (xn−1 , yn−1 ), (xn , yn ), we obtain for r ≥ 1 N (r) ≥

= =

N  

sn−1 + sn

n=1 N

 n=1 N 

n=1

sn−1 + sn 2

sn−1 + sn 2

n0  δ Nr N 



+

!  1 − sn δN r N 2

N

 n0 n=1

(δN r )

(as the sum over sn equals zero).

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Fractal Functions, Fractal Surfaces, and Wavelets

Inductively this yields N (r) ≥

N 

k  " sn

 + sn  (δ N r ), 2

 −1

n1 ,··· ,nk =1 =1

r ≥ k.

Hence N (r) ≥ [γ r δ − 1]N r ,

which, since γ > 1, implies the statement in the lemma. Assume then that γ > 1 and that G(f ) is not a line segment. As each C ∈ C (r, k) meets G(f ), wn (C), n ∈ NN , must also meet G(f ). Therefore, by Eqs. (7.16), (7.20), one obtains N 

N (r + 1, l(k, n)) =



N (r + 1, l(k, n)) +

n∈N+

n=1

≥N

sn−1 + sn ·

n∈N+

k−1 Nr



n∈N−

−

N 

N (r + 1, l(k, n))

n∈N−



+N



k sn−1 + sn · r N

 N (r, k)

 N (r, k)

[2N(|an | + |sn |) + 2] .

n=1

Computations similar to those for the upper bound yield N 

N (r + 1, l(k, n)) ≥ N

n=1

 N

 sn−1 + sn n=1

−

2

N (r, k)

N  sn  − [2N(|an | + |sn |) + 2] . Nr

n∈N+

Summing over k gives N (r + 1) ≥ (Nγ ) N (r) − c1 N r .

n=1

Dimension of fractal functions

259

Thus N (r) ≥ (Nγ )r−m N (m) − c1 N r−1

r−m−1 

γ

=0

  c1 N m−1 ≥ (Nγ )r−m N (m) − 1 − γ −1

for all m ∈ N, with 1 ≤ m ≤ r. By Lemma 16, one can choose r and m large enough so that N (m) −

c1 N m−1 > 0. 1 − γ −1

Hence N (r) ≥ c2 (γ N)r for a constant c2 > 0 and for large enough r. This gives, log γ dimb G(f ) ≥ 1 + log N.

Fractal functions and wavelets

8

Abstract Wavelet theory has its origin in several disciplines; the types of functions that are now called wavelets were studied in quantum field theory, signal analysis, and function space theory. In all these areas, wavelet-like algorithms replaced the classical Fourier-type expansion of a function. It was not until the mid-1980s that these, at first, seemingly different notions were described in a unified manner. The literature on wavelets has proliferated in recent decades, but the following very short list of references is sufficient for our purposes: [20, 21, 108–111, 114, 115, 119, 120, 132, 134, 136, 163, 221–228]. Because of the restricted scope of this monograph, we will not be able to address some ramifications of wavelet bases for so-called wavelet frames. The interested reader may find results and generalizations in Christensen [118]. The wavelet transform is an alternative to the classical windowed Fourier transform. The windowed Fourier transform serves as a means to describe or compare the fine structure of a function at different resolutions. Its basic building blocks are the integer dilates of the sine and cosine functions multiplied by a window function, usually a Gaussian. Although quite successful, Fourier analysis is not able to describe highly localized functions. For instance, the Fourier transform of the Dirac delta distribution δ supported at a single point x is 1, a function defined on all of R. (Recall that the Dirac delta distribution is that element of S  satisfying δ∗f =f

for all f ∈ S.

In other words, δ is the identity in the convolution algebra (S  , ∗).) This “spreading out” of a highly localized function is a direct consequence of the global support of sine and cosine. Using the windowed Fourier transform instead, and allowing the window to “move,” yields better localization properties. However, the window size is usually kept fixed, which for certain applications is a disadvantage. To overcome these problems, one replaces sine and cosine by a function that has compact support and is continuous, and whose dilates and translates form an orthonormal basis of L2 (R). (An example of a discontinuous orthonormal basis is the Haar system.) It can be shown that under rather mild conditions such a wavelet decomposition of L2 (R) exists. The famous Daubechies wavelets [109] are a class of such compactly supported, continuous, and orthonormal wavelet bases. The interested reader may consult [109, 110] or, if interested in spline wavelets, [108] and the references given therein, for a more detailed presentation of the construction of wavelets and their usefulness in applications. Wavelet decompositions are often obtained via a multiresolution analysis (see, eg, [21, 114]). In this chapter it is shown how a class of fractal functions can be used to define a multiresolution analysis yielding a (fractal) wavelet decomposition. Furthermore,

Fractal Functions, Fractal Surfaces, and Wavelets. http://dx.doi.org/10.1016/B978-0-12-804408-7.00008-4 Copyright © 2016 Elsevier Inc. All rights reserved.

262

Fractal Functions, Fractal Surfaces, and Wavelets

the existence of the free parameter s in the construction of fractal functions allows one to obtain a compactly supported, continuous, and orthogonal wavelet basis of L2 (R) that has a higher approximation order than the corresponding Daubechies wavelet 2ψ. In the first section a brief introduction to wavelet theory is given, developing the necessary notation and terminology needed later. It is assumed that the reader is familiar with the rudimentary concepts of Fourier analysis. A construction of wavelets based on fractal functions is given in the second section. There a method for constructing stable shift- and dilation-invariant function spaces using the fractal functions defined in Example 24 is given. This construction yields a multiresolution analysis on L2 (R). For some special classes of such translation- and dilation-invariant function spaces, exemplary wavelet decompositions of L2 (R) and also C0 (R) are presented. In the following section it is shown how a proper choice of s yields a compactly supported, continuous, and orthonormal wavelet basis of L2 (R). Finally, we state a result [128] that all compactly supported shift-invariant and refinable functions are piecewise fractal functions.

1 Basic wavelet theory Here the basic concepts of wavelet theory are introduced: refinable shift-invariant function spaces, scaling functions, multiresolution analyses, and reconstruction and decomposition algorithms. Because of the limited scope of this book the reader is referred to the extremely short and albeit incomplete list [21, 108–116] of wavelet publications and the references given therein for a more detailed introduction to this fascinating area. There are two approaches to obtain wavelet decompositions. The first uses the concept of multiresolution analysis. This multiresolution analysis yields, in general, a finite set of scaling functions which are then used to define the wavelets and the wavelet decompositions. The second method begins with a dilation equation for a given function or set of functions. The nontrivial solutions of this dilation equation then give scaling functions that may generate a multiresolution analysis. In Section 5 we will encounter a third method; it uses appropriately defined sets in the Fourier domain to construct wavelets.

1.1 Multiresolution analyses Let F = (F(Rn , K),  · F ) be a separable Banach space of functions f : Rn → K with Schauder basis {ei }i∈N . Here · denotes a Banach space norm on F and K is a subfield of C invariant under the involuntary automorphism z → z. Suppose that  is a lattice in Rn ; that is, a discrete subgroup of the topological group (R, +). A linear transformation δ is called a dilation for  iff δ satisfies the following properties: 1.  is δ-invariant (ie, δ ⊂ ). 2. ∀λ ∈ σ (δ): |λ| > 1.

Fractal functions and wavelets

263

These properties imply that 2 ≤ | det δ| ∈ N. An example of a dilation δ for any lattice  is given by δ := N idRn , where 2 ≤ N ∈ N. The dilation δ induces a dilation operator Uδ on F via the correspondence Uδ f := f ◦ δ −1 .

(8.1)

Definition 128 1. Assume that  is a lattice in Rn . The operator τγ : KR → KR defined by n

τγ f := f (· − γ ),

n

γ ∈ ,

(8.2)

is called a translation operator. 2. A linear subspace V of the function space F is called shift or translation invariant iff ∀f ∈ V ∀γ ∈ : f (· − γ ) ∈ V.

(8.3)

The function f (· − γ ) is called the shift or translate of f along . 3. A linear subspace V of F is said to be (δ-)refinable or (δ-)dilation invariant iff f ∈ V ⇒ Uδ f ∈ V.

(8.4)

The function Uδ f is called the δ-dilate of f .

For given functions φ 1 , . . . , φ A ∈ F, A ∈ N, denote by V[φ 1 , . . . , φ A ] the smallest norm closed shift-invariant subspace of F containing φ 1 , . . . , φ A ; that is, V[φ 1 , . . . , φ A ] = cl·F span {φ(· − γ ): γ ∈ } , where φ := (φ 1 , . . . , φ A )t ∈ F A . Then V[φ 1 , . . . φ A ] is called a finitely generated shiftinvariant space with generators φ 1 , . . . , φ A . If A := 1, the V[φ] is called a principal shift-invariant space. For a given lattice  and function space F, let ( (),  ·  ) be a Banach space of sequences c = (cγ ) ⊂ K such that the coordinate functionals {e∗i }i∈N are elements of (). (Recall that in a Banach space F with Schauder basis {e i }, the coordinate functionals are defined by e∗i : F → K, e∗i (f ) := f , ei  for F  f = i∈N f , ei ei .) The space of all finitely supported sequences c ⊂ K is denoted by 0 (). For a function f defined on Rn and a sequence c ⊂ (), the semiconvolution product  is defined by f  c :=



cγ f (· − γ )

γ ∈

whenever this sum makes sense. Note that if c ∈ 0 (), then f  c is well defined for all functions f : Rn → K. Suppose that φ has linearly independent shifts (ie, f  c = 0 iff c = 0). Then φ is called refinable iff V[φ] is. A refinable function is also called a scaling function.

264

Fractal Functions, Fractal Surfaces, and Wavelets

For k ∈ Z, denote by Vk the Uδ−k -dilate of a shift-invariant space V; that is Vk = {Uδ−k f : f ∈ V}. The shifts of a function f ∈ F are called stable iff there exist positive constants R1 and R2 such that ∀c ∈ (): R1 c ≤ f  cF ≤ R2 c .

(8.5)

One of the concepts that allows us to obtain wavelet decompositions of functions is introduced next. Definition 129. A sequence of function spaces {Vk }k∈Z ⊂ F forms a multiresolution analysis of F iff the following conditions hold: (M1) Nestedness: ∀k∈ Z: Vk ⊂ Vk+1 . Vk = F. (M2) Density: cl· k∈Z  Vk = {0}. (M3) Separation: k∈Z

(M4) Refinability: f ∈ Vk iff f (δ · ) ∈ Vk+1 , or equivalently Vk = Uδ−k V0 . (M5) Shift-invariance: V0 is τ -invariant; that is, f ∈ V0 ⇒ ∀γ ∈ : τγ f ∈ V0 . (M6) There exists a finite family of functions {φ a : a ∈ NA ∧ A ∈ N} ⊂ V0 such that for all a ∈ NA , φ a has stable shifts and V0 = V[φ 1 , . . . , φ A ].

At this point a few remarks are in order. Remark 62 1. The functions φ 1 , . . . , φ A in condition (M6) are not unique. 2. If Pk denotes the orthogonal projection of F := L2 (Rn ) onto Vk , then condition (M1) can be restated as Pk Pk = Pk Pk = Pk ,

k ≤ k.

Moreover, conditions (M2) and (M3) now read lim Pk f = f

k→∞

and

lim Pk f = 0

k→−∞

for all f ∈ L2 (Rn ). 3. Not all the conditions given in Definition 129 are independent. For instance, conditions (M1)– (M3) are really consequences of condition (M4). For a complete description of the relations among the items in Definition 129, see Refs. [113, 117]. 4. In view of conditions (M1) and (M4), the scaling functions φ 1 , . . . , φ A satisfy the following matrix dilation equation or refinement equation:  cγ φ(δx − γ ) (8.6) φ(x) = γ ∈

for some matrix-valued sequence (c)γ ∈ and with φ := (φ 1 , . . . , φ A )t ∈ V0A . The function φ will be called a vector scaling function. Note that if only a finite number of coefficients cγ are nonzero, then φ is compactly supported. The collection of matrices {cγ } is sometimes also called a matrix mask.

Fractal functions and wavelets

265

5. One mostly deals with the case F := L2 (Rn ). In this setting, is chosen to be 2 (), the 2 Banach space of all square-summable matrix-valued sequences c:  → RA with norm ⎛ c 2 := ⎝



⎞1/2 cγ 2 ⎠

,

γ ∈

where c = (cγ )γ ∈ and · denotes a matrix norm. Condition (M6) is then usually expressed in the following manner: (M6*) There exists a function φ ∈ V0 such that {τγ φ}γ ∈ is a Riesz basis for V0 .

The reader may recall that in a separable Hilbert space H, a Riesz basis is obtained from an orthonormal basis by means of a bounded bijective operator. Thus if H = L2 (Rn ), then condition (M6*) is equivalent to the following: There exist positive constants R1 and R2 , called the Riesz bounds, such that









cγ φ(· − γ )

≤ R2 c 2 (8.7) R1 c 2 ≤



γ ∈

2 n L (R )

for all c ⊂ 2 . For an introduction to Riesz bases and their generalizations (ie, frames), see Ref. [118]. Remark 63. If F = L2 (Rn ) one sometimes requires {τγ φ}γ ∈ to be an orthonormal basis for its closed linear span rather than a Riesz basis. This assumption is based on the fact that one can construct an orthonormal basis of L2 (Rn ) from a Riesz basis {τγ φ}γ ∈ . This is shown in great detail in Ref. [112] for n = 1 and A = 1. The case n ∈ N and A = 1 can be found in Refs. [111, 113], and the general case can be found in Ref. [82]. If {τγ φ}γ ∈ is an orthonormal basis of L2 (Rn ), then the constants R1 and R2 in Eq. (8.7) are equal to 1. By taking the Fourier transform of Eq. (8.7), one obtains the following equivalent condition for {τγ φ}γ ∈ to be orthonormal: 

†  φ(ξ + 2πγ ) φ (ξ + 2πγ ) = IA

(8.8)

γ ∈

for all γ ∈ . Here the dagger denotes the Hermitian conjugate and IA is the A × A identity matrix. An example of a multiresolution analysis is given by the Haar system. Example 49. Let  := Z, let δ := 2, and let Vk := {f ∈ L2 (R): f = constant on [ 2−k , ( + 1)2−k ) ∧ ∈ Z}. Then conditions (M1)–(M6) in Definition 129 are satisfied if φ := χ[0,1) . One can check that φ(· − ), ∈ Z, forms an orthonormal basis of V0 . The scalar dilation equation that φ has to satisfy is seen to be φ(x) = φ(2x) + φ(2x − 1).

266

Fractal Functions, Fractal Surfaces, and Wavelets

Furthermore, the projection Pk : L2 (R) → Vk is given by 2k ( +1) −k Pk f (x) = 2 f (t) dt, 2k

with x ∈ [ 2−k , ( + 1)2−k ). It is worthwhile remarking that because of the piecewise continuity of the functions in Vk , the projection Pk f converges very slowly to the function f . To obtain better convergence one has to impose a higher degree of regularity on the scaling function φ. This leads to the following definition. Definition 130. Let r ∈ N. A multiresolution analysis is called r-regular iff there exist positive constants Cm,α such that Dα φ(x) ≤ Cm,α (1 + x)−m

(8.9)

for all x ∈ Rn and m ∈ N0 and all multiindices α with |α| ≤ r. Remark 64. The derivatives Dα φ(x) are assumed to be only in L∞ (Rn ) and not necessarily in C0 (Rn ). Example 50. An example of an r-regular multiresolution analysis is given by the r (R): spline space Sr−1 V0 := Srr−1 (R) := {f ∈ L2 (R) ∩ Cr−1 (R): f |( , +1) ∈ Pr−1 , ∀ ∈ Z}, where Pr−1 denotes the R-linear space of all polynomials of order r − 1. It can be shown [108] that the space V0 generates a multiresolution analysis of L2 (R) r+1

if the scaling function φ is taken to be φ := ∗ χ[0,1] , the (r + 1)-fold convolution of i=1

the characteristic function on the unit interval. Clearly φ is r-regular. Assume that a multiresolution analysis on a function space F is given. Let φ be the associated vector scaling function. For k ∈ Z, let Wk be such that Vk ⊕ Wk = Vk−1 , with ⊕ denoting the direct sum. If F is a separable Hilbert space, then Wk is the direct orthogonal difference of the spaces Vk−1 and Vk . Assumption. In all but one case, F is assumed to be the separable Hilbert space L2 (Rn ), and this will therefore be the choice for F for the remainder of this section. The spaces Wk , k ∈ Z, are mutually disjoint and orthogonal: Wk ∩ Wk = ∅ and Wk ⊥ Wk for k = k . Moreover,

L2 (Rn ) = Wk . k∈Z

The spaces {Wk }k∈Z are called wavelet spaces. Let E be the set of all extreme points of the unit cube [0, 1]n in Rn ; that is, E = {(v1 , . . . , vn ) ∈ Rn : vi ∈ {0, 1}, ∀i ∈ Nn }. The proof of following theorem can be found in Refs. [21, 119] for A = 1 and in Ref. [117] for general A.

Fractal functions and wavelets

267

Theorem 113. Suppose {Vk }k∈Z is an r-regular multiresolution analysis on L2 (Rn ) with vector scaling function φ = (φ 1 , . . . , φ A )t . Then there exist B := (2n − 1)A functions ψ b , b ∈ (E \ {0})A , such that each ψ b is r-regular, and the collection ψ := {ψ 1 , . . . , ψ B } together with its -translates forms an orthonormal basis of W0 . As the wavelet spaces Wk are nested, it is a direct consequence of Theorem 113 that for a fixed k ∈ Z the collection of functions {(det δ)nk/2 ψ b (δ k · −γ ): γ ∈  ∧ b ∈ NB } is an orthonormal basis of Wk ; that is, Wk = clL2 (Rn ) span{(det δ)nk/2 ψ b (δ k · −γ ): γ ∈  ∧ b ∈ NB }. Hence {(det δ)nk/2 ψ b (δ k · −γ ): k ∈ Z ∧ γ ∈  ∧ b ∈ NB } is an orthonormal basis of L2 (Rn ). The functions ψ 1 , . . . , ψ B are called wavelets and ψ is called a multiwavelet. This allows the unique representation of a function f ∈ L2 (Rn ) in terms of a wavelet series or wavelet decomposition:  atkγ ψ kγ (x), (8.10) f (x) = k∈Z γ ∈

where the vector wavelet or multiwavelet ψ is defined by ψ := (ψ 1 , . . . , ψ B )t and ψ kγ (x) := (det δ)nk/2 ψ(δ k x − γ ) for all k ∈ Z and γ ∈ . Notation. In what follows, fkγ will always denote the δ k -dilate and γ -translate of a function f . The coefficients ak,γ are given by akγ = (f , ψ 1kγ , . . . , f , ψ Bkγ )t . Here, and in what follows, ·, · := ·, ·L2 (Rn ) denotes the L2 -inner product on Rn . The mapping 1 B t , . . . , f , ψkγ ) L2 (Rn )  f −→ (f , ψkγ

is called the discrete wavelet transform. It agrees with the wavelet transform Wψ 2 introduced in Eq. (1.81). Moreover, if f ∈ L2 (Rn ), then {ckγ } ∈ 2 ()B . Furthermore, since W0 ⊂ V1 , there exists a sequence of B × A matrices {d}γ ∈ such that

268

Fractal Functions, Fractal Surfaces, and Wavelets



ψ(x) =

dγ φ(δx − γ ).

(8.11)

γ ∈

Hence ψ also satisfies a matrix dilation equation. This dilation equation is used in the proof of Theorem 113 to obtain the wavelets. A family {ψ kγ : k ∈ Z ∧ γ ∈ } is called orthonormal iff ψ kγ ψ †kγ  dn x = δkk ,γ γ  IB (8.12) ψ kγ , ψ k γ   = Rn

for all k, k ∈ Z and γ , γ  ∈ . The “two-component” Kronecker delta δkk ,γ γ  is defined by δkk ,γ γ  := δkk δγ γ  , k, k ∈ Z, γ , γ  ∈ . Example 51. In this example it is shown how the wavelets can be constructed from a multiresolution analysis when n = 1 = A,  = Z, and φ := sinc(π·). Although this is a rather special setup, it nevertheless contains all the ingredients of the general construction and is thus of an exemplary nature. We define the Fourier-Plancherel transform F on L2 (R) by  f (x) e−ixξ dx. F f (ξ ) := f (ξ ) := R

Let f ⊂ [−π, π]}. V0 := V[φ] = {f ∈ L2 (R): supp It can be verified that {φ0, : φ = sinc π(·) ∧ ∈ Z} is an orthonormal basis for V0 . Since φ ∈ V0 ⊂ V1 , there exists an 2 (Z) sequence {c } such that  c φ1, , (8.13) φ=

∈Z

where c := φ, φ1 . The dilates of V0 are then explicitly given by Vk = {f ∈ f ∈ L2 (R) : supp f ⊂ [−2−k π, 2k π]},

k ∈ Z.

The orthonormality of φ implies that {c } 2 (Z) = 2. Applying the Fourier transform to Eq. (8.13) yields (ξ/2), (ξ ) = m0 (ξ/2)φ φ

(8.14)

Fractal functions and wavelets

269

where m0 (ξ/2) :=

1  −i ξ c e . 2

(8.15)

∈Z

The symbol m0 (ξ ) is 2π-periodic and of class C∞ . For φ = sinc π(·), one obtains  c0 = 1,

c =

c2 = 0,

c2 +1 =

0 = ∈ Z; 2(−1)

π(2 +1) ,

∈ Z.

In the present setup, Eq. (8.8) yields 

(ξ + 2π )|2 = 1 a.e., |φ

∈Z

and thus using Eq. (8.15), one obtains 

(ξ + 2π )|2 = 1, |m0 (ξ/2 + π )|2 |φ

a.e.

∈Z

Splitting the preceding sum into odd and even , using the periodicity of m0 (ξ ) and Eq. (8.14), one finally obtains |m0 (ξ/2)|2 + |m0 (ξ/2 + π)|2 = 1

a.e.

(8.16)

  The fact that 2−1/2 φ(·/2 − ) forms an orthonormal basis of V−1 leads to  (2ξ ): m is 2π-periodic and locally in L2 } V−1 = {mf (2ξ ) φ (ξ ): m is 2π-periodic and locally in L2 }. = {mf (2ξ ) m0 (ξ )φ In this characterization of V−1 , the fact was used that the Fourier transform of any (ξ ) for a 2π-periodic symbol mf (ξ ) = m(ξ ; f ) f = mf (ξ )φ f ∈ V0 can be expressed as  with the property that f L2 (R) = mf L2 ([0,2π] (ie, the mapping f → mf is unitary). To characterize W−1 , all 2π-periodic functions n(ξ ) that are orthogonal to all the functions mf (2ξ )m0 (ξ ) have to be found: f ∈ W−1 is equivalent to f ∈ V0 and f ⊥ V−1 . Hence  (2ξ ) dξ = 0 fφ ei (2ξ ) R

for all  ∈ Z, or equivalently



[0,2π]

ei ξ



∈Z

 (2ξ + 2π ) dξ = 0, f (2ξ + 2π )φ

270

Fractal Functions, Fractal Surfaces, and Wavelets

and thus 

 (2ξ + 2π ) = 0, f (2ξ + 2π )φ

(8.17)

∈Z

where the sum converges uniformly in L1 [− π2 , π2 ]. On substitution and regrouping into odd and even , one obtains the following condition on the symbol n(ξ ):

 [0,π]

 mf (2ξ )m0 (ξ )n(ξ ) + mf (2ξ )m0 (ξ + π)n(ξ + π) dξ = 0;

hence m0 (ξ )n(ξ ) + m0 (ξ + π)n(ξ + π) = 0,

a.e. on [0, π].

(8.18)

Since the vector with components m0 (ξ/2) and m0 (ξ/2+π) has length 1, the preceding orthogonality requirement is equivalent to 

   n(ξ ) −iξ m0 (ξ + π) = ν(ξ )e n(ξ + π) −m0 (ξ )

a.e. on [0, π]

for some function ν(ξ ). (For reasons that will become clear later, the factor e−iξ was added.) It follows directly from Eq. (8.16) that nL2 ([0,2π] = νL2 ([0,π] . It is not hard to see that the preceding matrix equation reduces to n(ξ ) = ν(ξ )e−iξ m0 (ξ + π)

a.e.

for some π-periodic function ν. Therefore the space W−1 consists of functions of the (ξ ), where n(ξ ) satisfies the preceding equation. Using the equality of the form n(ξ )φ −1 can be obtained from an orthonormal foregoing norms, an orthonormal basis of W 2 basis of L [0, π]. To this end, choose {2−1/2 e2i ξ : ∈ Z} as an orthonormal basis of L2 [0, π] and set ν(ξ ) := 2−1/2 e2i ξ . −1 of the form This leads to an orthonormal basis of W 2−1/2 e−iξ m0 (ξ + π) e2i ξ for ∈ Z. An orthonormal basis of W−1 is then given by {ψ−1,2 : ∈ Z}, where −1 = 2−1/2 e−iξ m0 (ξ + π). Dilation yields an orthonormal basis {ψ0, : ∈ Z} of ψ W0 with (ξ ) = e−iξ/2 m0 (ξ/2 + π)φ (ξ/2). ψ

(8.19)

Fractal functions and wavelets

271

This form of this  equation clearly shows that ψ satisfies a scalar dilation equation of  the form ψ(x) = ∈Z d φ(2x − ) for some 2 (Z) sequence {d }. The choice of ψ (ξ ) with ρ 2π-periodic and |ρ(ξ )| = 1 is not unique; any function of the form ρ(ξ )ψ a.e. will also do. Applying this last formula to the scaling function φ = sinc(π·) yields  −i(ξ/2+π) (ξ ) = e ψ 0

for π ≤ |ξ | ≤ 2π; otherwise.

 is easily computed and yields The inverse Fourier transform of ψ √ 2 2(cos(πx) − sin(2πx)) . ψ(x) = π(1 − x) The reader may wish to construct the wavelets for the multiresolution analysis considered in Example 49 and for {Vk }k∈Z , where V0 := S01 . In the former case the wavelet is given by ⎧ ⎪ for x ∈ [0, 12 ); ⎨1 ψ(x) = −1 for x ∈ [ 12 , 1); ⎪ ⎩ 0 otherwise. The latter case is presented in Ref. [110]. It should be noted that φ := χ[0,1) ∗ χ[0,1]) is not a scaling function; φ is not orthogonal to its integer translates. However, there is a way to obtain an orthonormal basis from a Riesz basis. This method will now be presented in the case n = 1 = A and  = Z. To this end, suppose that {φ0, } ∈Z is a Riesz basis of V0 . Taking the Fourier transform of Eq. (8.7) for n = 1 = A and  = Z yields 0 < R1 ≤



(ξ + 2π )|2 ≤ R2 < ∞. |φ

(8.20)

∈Z

(The proof uses Parseval’s identity.) Recall that Parseval’s identity states that for any two L2 (R) functions f and g, f , g =

1  f , g. 2π

Note that {φ0, } ∈Z is an orthonormal basis of V0 iff R1 = R2 = 1. Therefore our defining  (ξ ) :=  φ 

(ξ ) φ 

∈Z |φ (ξ

+ 2π )|2

(8.21)

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Fractal Functions, Fractal Surfaces, and Wavelets

leads to an orthonormal basis via the following arguments:   V0 = f ∈ L 2 : f =



 0, ∧ {  c φ c } ∈Z ∈ 2 (Z)

∈Z



 for some 2π-periodic  = f ∈ L2 :  f = m φ m ∈ L2 ([0, 2π])



   for some 2π-periodic m ∈ L2 ([0, 2π]) = f ∈ L2 :  f = mφ  = f ∈L :f = 2



 c φ0, ∧ {c } ∈Z ∈

2

= V0 .

∈Z

1.2 The dilation equation In the preceding presentation the multiresolution analysis and the associated (vector) scaling function provided the starting point for the construction of the (vector) wavelets. As seen earlier the symbol m0 (ξ ) played an important role in the construction of the wavelets. This importance is reemphasized in the second approach to the construction of wavelets: instead of beginning with a multiresolution analysis and associated (vector) scaling function, one considers the (matrix) dilation equation φ(x) =



cγ φ(δx − γ )

γ ∈

and finds conditions for the coefficients cγ that allow the construction of a (vector) scaling function for a multiresolution analysis. This method is briefly presented here for n = 1 and  = Z. Most of the results stated are taken from Refs. [109, 110]. Theorem 114. Suppose that the dilation equation φ(x) =



c φ(2x − )

∈Z

is given. Assume that the sequence {c } ∈Z satisfies:

 1. ∃ ε > 0:  ∈Z |c | | |ε < ∞; 2. ∀i,  j ∈ Z: ∈Z c −2i c −2j = δij ; 3.

∈Z c = 2.

Furthermore, assume that the symbol m0 (ξ ) :=  m0 (ξ ) =

1 + eiξ 2

M 

∈Z

 f e

i ξ

,

1 2



i ξ

∈Z c e

can be written as

Fractal functions and wavelets

273

with

 4. ∃ ε > 0: ∈Z |f || |ε < ∞; and   i ξ < 2M−1 : ξ ∈ R . 5. sup

∈Z f e

If d := (−1) c1− , (ξ ) := φ

∞ !

m0 (2−j ξ ),

j=1

ψ(x) :=



d φ(2x − ),

∈Z

then the family {φk : k, ∈ Z} defines a multiresolution analysis of L2 (R) with associated orthonormal wavelet basis {ψk : k, ∈ Z}. Proof. Only an outline of the proof will be presented. Condition 2 is essentially Eq. (8.16) necessary for orthogonality. Condition 1 guarantees that the infinite "and thus −j m (2 ξ ) converges pointwise for all ξ ∈ R, and uniformly on compact product ∞ j=1 0 subsets of R. The dilation equation together with the required orthogonality implies condition 3, which is also equivalent to m0 (0) = 1. Finally, conditions 4 and 5 imply that the iterates T m , m ∈ N, of the linear operator T: L2 (R) → L2 (R), defined by Tf (x) :=



ck f (2x − k),

(8.22)

k∈Z

and applied to the initial function χ[− 1 , 1 ) converge pointwise to a continuous function 2 2 " φ whose Fourier transform is given by j∈N m0 (2−j ξ ). The choice of coefficients dk for the wavelet is based on Eq. (8.19). For the detailed proof, see, for instance, Ref. [109]. The integer M in the preceding proof serves a twofold purpose: on the one hand, it is used in connection with the construction of compactly supported wavelets (all wavelets so far considered with the exception of the Haar system, were infinitely supported); on the other hand, it is a measure for the regularity of the scaling function and the wavelet. These remarks are now made more precise. As Eq. (8.16) has played a pivotal role throughout the previous presentation, it is of no surprise that the requirement of compact support is expressible in terms of the symbol m0 (ξ ): if the scaling function is to be compactly supported, then m0 (ξ ) must be a trigonometric polynomial; that is, the sum over must be finite. The converse is easily seen to be true also. Hence all solutions of Eq. (8.16) with only a finite number of nonzero terms c have to be characterized. The basic idea is to look for solutions of  M the form m0 (ξ ) = 12 [1 + eiξ ] Q(eiξ ), where Q is a polynomial. Since Eq. (8.16) resembles the well-known identity between the sine and cosine functions, one rewrites it in the form

274

Fractal Functions, Fractal Surfaces, and Wavelets

|m0 (ξ )|2 = | cos2 ξ/2|M |Q(eiξ )|2 . With this particular form it is not very difficult to show that there exist solutions |m0 (ξ )|2 of Eq. (8.16). To determine m0 (ξ ), the following lemma, which is due to Riesz, must be applied.  Lemma 17 (Riesz). Let A(ξ ) =

∈N0M a cos( ξ ), a ∈ R, be a positive trigonometric polynomial. Then there exists a trigonometric polynomial B(ξ ) of order M with real coefficients such that |B(ξ )|2 = |A(ξ )|. Proof. The simple and elegant proof can be found in Ref. [109]. This then leads to an explicit characterization of all trigonometric polynomials satisfying Eq. (8.16). Theorem 115. Any trigonometric polynomial solution of |m0 (ξ )|2 + |m0 (ξ + π)|2 = 1 is of the form  m0 (ξ ) =

1 + eiξ 2

M Q(eiξ ),

where M ∈ N, and where Q(eiξ ) is a polynomial satisfying |Q(e )| = iξ

2

M−1 

=0

   1 M−1+

2

2M cos ξ sin ξ/2 + (sin ξ/2)R 2

for some odd polynomial R. Proof. See, for instance, Ref. [109]. Next the regularity of scaling functions and wavelets is investigated [110, 120]. Smoothness in our context is most easily described in terms of the Fourier transform. Recall that if the Fourier transform of a given function f ∈ L2 (R) satisfies | f (ξ )| ≤ c(1 + |ξ |)−r−1−ε ,

ε > 0,

then f is of class Cr . If a scaling function satisfies this inequality, then the associated (ξ/2).) (ξ ) = e−iξ/2 m0 (ξ/2 + π)φ wavelet will also satisfy it. (This follows from ψ To derive a regularity result we need the next theorem, whose proof can be found in Ref. [109]. Theorem 116. Assume that f , g: R → K are two functions, not identically constant, satisfying fk, , gk ,   = δkk ,

 for all k, k , ,  ∈ Z. Let r ∈ N and suppose that |g(x)| ≤ c(1 + |x|)−α

Fractal functions and wavelets

275

for some c > 0 and for α > r + 1. Moreover, assume that f ∈ Cr (R, K) and that f (m) is bounded for all m ≤ r. Then xm g(x) dx = 0 for m ∈ N0r . R

Applying this theorem to f = g = ψ yields the following corollary. Corollary 11. Suppose that {ψk : k, ∈ Z} is an orthonormal set in L2 (R) such that for all k, ∈ Z, |ψk (x)| ≤ C(1 + |x|)−r−1−ε for C ∈ R, ψ ∈ Cm (R), and for all m ∈ N0r , |ψ (m) (x)| ≤ C1 for some C1 ∈ R. Then xm ψ(x) dx = 0 R

for m ∈ N0r . Note that this corollary implies#ψ (m) (0) = 0, for all m ∈ N0r . This, together with (0) = Eq. (8.19) and the fact that φ R φ dx = 0 (for otherwise this would entail φ ≡ 0), now implies dm m0 (0) = 0 dξ m for all m ∈ N0r . In other words, m0 (ξ ) has a zero of order r + 1 at ξ = π:  m0 (ξ ) =

1 + eiξ 2

r+1 M(ξ ),

r (R). Hence the following theorem holds. with M ∈ CK Theorem 117. Suppose that {ψk : k, ∈ Z} is an orthonormal wavelet basis of L2 (R) derived from a scaling function φ. If

|φ(x)|, |ψ(x)| ≤ C(1 + |x|)−r−1−ε for some C ∈ R, and ψ ∈ Cr (R) with ψ (m) bounded for all m ∈ N0r , then the symbol m0 , as defined in Eq. (8.15), factors as follows:  m0 (ξ ) =

1 + eiξ 2

r+1 M(ξ ),

where M ∈ Cr (R) is 2π-periodic.

(8.23)

276

Fractal Functions, Fractal Surfaces, and Wavelets

Now it is natural to ask what conditions must be imposed on the symbol m0 (ξ ) to guarantee that φ, and thus ψ, is of class Cα , 0 ≤ α < r + 1. Suppose that φ is a finitely supported scaling function satisfying the two-scale dilation equation φ(x) =



c φ(2x − ),

∈Z

  #with ∈Z c = 2. Without loss of generality it may also be assumed that φ (0) = R φ(x) dx = 1. Taking the Fourier transform of the dilation equation and using the definition of the symbol m0 gives (ξ ) = m0 (ξ/2)φ (ξ/2), φ

(8.24)

and thus (ξ ) = φ

∞ !

m0 (2−j ξ ).

j=1

Since φ is assumed to be finitely supported, m0 is a trigonometric polynomial that allows the factorization  m0 (ξ ) =

1 + e−iξ 2

r M(ξ )

for some trigonometric polynomial M(ξ ). Substituting this last equation into Eq. (8.24) yields (ξ ) = φ



1 − e−iξ iξ

r ! ∞

M(2−j ξ ).

(8.25)

j=1

" It is therefore necessary to estimate the growth of the infinite product j∈N M(2−j ξ ) to determine the regularity of φ. The next result states a condition of M(ξ ) that ensures that φ is of class Cα . Proposition 70. Suppose that sup{|M(ξ )|: ξ ∈ [0, 2π)} < 2r−α−1 . Then φ is of class Cα . Proof. In this proof C will denote a generic real-valued constant whose numerical value may differ from context to context. Since m0 , and thus M(ξ ), is a trigonometric polynomial with m0 (0) = M(0) = 1, there exists a constant C such that |M(ξ )| ≤ 1 + C|ξ |. Hence sup

∞ !

|ξ |≤1 j=1

−j

|M(2 ξ )| ≤ sup

∞ !

|ξ |≤1 j=1

exp(C2−j |ξ |) ≤ exp(C).

Fractal functions and wavelets

277

Now take ξ with |ξ | ≥ 1. Then one may find a j0 ∈ N such that 2j0 −1 ≤ |ξ | < 2j0 . Thus ∞ ! j=1

M(2−j ξ ) =

j0 !

M(2−j ξ )

j=1

∞ !

M(2−j−j0 ξ )

j=1

≤ (sup{|M(ξ )|: ξ ∈ [0, 2π)})j0 exp(C) ≤ C2j0 (r−α−1−ε) ≤ C(1 + |ξ |)r−α−1−ε .  gives |φ (ξ )| ≤ C(1 + |ξ |)−α−1−ε . Combining this with the equation for φ Remark 65 1. The higher the regularity of a scaling function (and wavelet), the larger the support (see Theorem 115 and notice that N is also the number of nonzero coefficients in the dilation equation): if φ and ψ are of class Cr , r ∈ N, then supp φ = [0, 2r−1] and supp ψ = [1−r, r]. 2. The family of functions {r φ, r ψ}, where r denotes the regularity, is referred to as the Daubechies scaling functions and wavelets. 3. It can be shown [109] that the Haar wavelet is the only symmetric wavelet in the Daubechies family of wavelets. 4. For r = 2 the scaling function 2 φ and the wavelet 2 ψ are in Cα (R) with 0 ≤ α < 1. In fact, one can show [110] that there exists a set of full measures on which 2 φ is differentiable. At the dyadic rationals in supp2 φ = [0, 3], one can prove that 2 φ is left differentiable but has Hölder exponent 0.55 when such a dyadic rational is approached from the right. Hence the graph of 2 φ exhibits fractal-like features. In fact, we show in Section 4 that 2 φ is an affine fractal function generated by two affine mappings λ1 and λ2 . 5. As seen earlier, a convenient way of constructing (vector) wavelets is via a multiresolution analysis. However, it should be emphasized that a (vector) wavelet ψ is really nothing but a special basis of L2 (Rn ); namely, a function ψ ∈ L2 (Rn ) such that the two-parameter family {(det δ)nk Uδ−k ◦ τγ (ψ): k ∈ Z ∧ γ ∈ } forms a Riesz basis for L2 (Rn ).

1.3 Dilation equation and the Read-Bajraktarevi´c operator The linear operator T given in Eq. (8.22) is reminiscent of the affine Read-Bajraktarevi´c operator (RB operator)  (Eq. 5.11). In this section we outline some of the commonalities of both operators, which give an interesting relationship between wavelets that are associated with linear operators of the form T and fractal functions that are constructed via affine operators of the form . We refer to Refs. [121, 122], where this connection is also discussed. Given an interval I ⊂ R and function space, say, C(I), one is interested in the convergence of iterates φn := Tφn−1 and fn := fn−1 starting with some initial function f0 ∈ C(I) to a fixed or limit point and the conditions under which this convergence will occur. To be specific, consider the linear operator T given by

278

Fractal Functions, Fractal Surfaces, and Wavelets

Tf (x) =



ck f (2x − k)

(8.26)

k∈Z

and compare it with the affine operator  given by g(x) = λ(x) +



sk g(2x − k),

(8.27)

k∈Z

where λ: [0, 1] → [0, 1] is given by ⎧ ⎪ for x ∈ [0, 12 ]; ⎨λ0 (2x) λ(x) := λ1 (2x − 1) for x ∈ [ 12 , 1]; ⎪ ⎩ 0 otherwise. Here λk are, say, two affine functions. The affine fractal function g generated by the RB operator (8.27) is supported on [0, 1] but has been extended to R by our setting it equal to zero off [0, 1]. Clearly the main difference between T and  is the existence of an inhomogeneous part g − 0 = λ in Eq. (8.27). Furthermore, whereas the shifts of f in general overlap on a set of positive measure, the shifts of g—by construction— overlap only on a set of measure zero. We will see in the following sections that the existence of the inhomogeneous part λ and the latter shift property of affine fractal functions can be used to generate scaling functions and wavelets satisfying equations of the type (8.26). In Section 10 we derived conditions for the fixed or limit point of the iterates fn to belong to C(I); namely, that {max{|si |: i ∈ NN } < 1 and all λi must belong to 0 (see, in particular, Theorem 91). In the following we present without proof a similar result for the linear operator T. To this end, choose I := [0, N]. The coefficients {ck } in the definition of the operator T must satisfy N 

ck = 2

(8.28)

k=0

to guarantee convergence in C(I). The fixed point φ ∈ C(I) of T satisfies the refinement equation φ=

N 

ck φ(2 · −k).

(8.29)

k=0

As the initial function φ0 ∈ C(I) one chooses the linear spline whose support points on the integer knots 1, . . . , N − 1 are given by {φ(1), . . . , φ(N − 1)}.

(8.30)

Fractal functions and wavelets

279

To determine the values, φ( ), ∈ NN−1 , the fixed-point equation is employed: φ( ) =

N 

ck φ(2 − k),

∈ NN−1 .

(8.31)

k=0

The N − 1 Eq. (8.31) constitute an linear algebraic system for the N − 1 unknowns φ(1), . . . , φ(N − 1). Using matrix notation, we rewrite this systems as follows: ⎛

⎞

⎛

c1 ⎜ ⎟ ⎜c3 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ .. ⎜ ⎟=⎜. ⎜ ⎟ ⎜ ⎝φ(N − 2)⎠ ⎝ 0 0 φ(N − 1) φ(1) φ(2) .. .

c0 c2 .. .

0 c1 .. .

c0 .. .

0 .. .

... ...

0

cN

cN−1

... ... .. . cN−2 cN

0 0 .. .

⎞⎛

φ(1) φ(2) .. .

⎞

⎟⎜ ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟. ⎟⎜ ⎟ ⎠ ⎝ cN−1 φ(N − 2)⎠ cN−1 φ(N − 1)

Setting φ := (φ(1), . . . , φ(N − 1))t and denoting the above matrix by C yields φ = Cφ. It can be seen that φ is an eigenvector with eigenvalue 1, and it can be shown that the matrix C is nonsingular. In particular, the following theorem [109, 123] holds. Theorem 118 (Daubechies Theorem; Daubechies-Lagarias Theorem). Suppose that N  k=0

N  ck = 2 and (−1)k km ck = 0

(m = 0, 1, . . . , M).

k=0

Then there exist nontrivial continuous L1 -solutions of φ = Tφ which are m times continuously differentiable. Moreover, φ − φn L∞ ≤ c2−αn with optimal Hölder exponent α. Proof. For the proof of this theorem and for values for the optimal Hölder exponent α, see Refs. [109, 123]. Remark 66 1. Choosing for the ck the Daubechies filter coefficients hk yields the family of Daubechies scaling functions N φ. Note that the (discontinuous) Haar function for which N = 1 is not included in Theorem 118. 2. Similar results were obtained by Micchelli and Prautzsch [124] in the context of subdivision schemes. See also the monograph by Cavaretta et al. [125]. 3. Micchelli and Xu [126, 127] also investigated these topics. 2φ

In Fig. 8.1 we depict the initial function φ0 , the first two iterates, and the fixed point for the operator T in the case N := 3.

280

Fractal Functions, Fractal Surfaces, and Wavelets

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 0.5

−0.2

1

1.5

2

2.5

3

−0.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 0.5

−0.2

1

1.5

2

2.5

3

−0.2

0.5

1

1.5

2

2.5

3

0.5

1

1.5

2

2.5

3

Fig. 8.1 Convergence of the iterates to the Daubechies scaling function 2 φ. The initial function φ0 (upper left), the first iterate φ1 (upper right), the second iterates φ2 (lower left) and the fixed point 2 φ (lower right)

The coefficients ck , k = 0, 1, 2, 3, are explicitly given by Daubechies [109] √ 1− 3 . 4 √ If we set φ(0) := 0 =: φ(3), φ(1) := 1, this then yields φ(2) = −2 + 3. In Section 4 we will show that 2 φ is an affine fractal function generated by an affine RB operator of the form Eq. (5.11) with two affine λi and with scaling factors √ √ s1 = 1+4 3 and s2 = 1−4 3 (see also Ref. [128]). c0 =

√ 1+ 3 , 4

c1 =

√ 3+ 3 , 4

c2 =

√ 3− 3 , 4

c3 =

1.4 The decomposition and reconstruction algorithms Next we introduce the decomposition and reconstruction algorithm for wavelets. This will be done in the more general setting of an arbitrary lattice  with dilation operator δ and V0 = V[φ 1 , . . . , φ A ] ⊂ L2 (Rn ). Let fk ∈ Vk , k ∈ Z. Since Vk = Vk−1 ⊕ Wk−1 , one may decompose fk into an averaged or “blurred” component fk−1 ∈ Vk−1 and a “fine structure” or error component gk−1 ∈ Wk−1 : fk =



atk,γ φ(δ k · −γ ) = fk−1 + gk−1

γ ∈

=



γ ∈

atk−1,γ φ(δ k−1 · −γ ) +

 γ ∈

btk−1,γ ψ(δ k−1 · −γ ),

Fractal functions and wavelets

281

with ak,γ , ak−1,γ , bk−1,γ ∈ 2 ()B . If the vector scaling functions φ kγ and the multiwavelets ψ kγ are fully orthogonal—that is, if for all k, k ∈ Z and for all γ , γ  ∈ , φ kγ , φ k γ   = δkk ,γ γ  IA , ψ kγ , ψ k γ   = δkk ,γ γ  IB , φ kγ , ψ k γ   = OA×B , where OA×B denotes the A × B zero matrix—then the coefficient matrices ak−1,γ and bk−1,γ can be obtained easily from the coefficient matrices akγ via the Mallat transform 2

2

2

M: 2 ()B → 2 ()A × 2 ()B , M({ak }) := ({ak−1 }, {bk−1 }),

(8.32)

with ak−1,γ =



cγ  −δγ akγ 

and

bk−1 =

γ  ∈



dγ  −δγ akγ  .

(8.33)

γ  ∈

Proceeding with the decomposition of fk−1 into fk−2 ∈ Vk−2 and gk−2 ∈ Wk−2 , one obtains after j steps the following decomposition: fk = fk−j +

j 

gk−i ∈ Vk−j ⊕

i=1

j

Wk−i .

(8.34)

i=1

This equation describes the decomposition algorithm for fk . In terms of the sequences of coefficient matrices, this decomposition can also be expressed as follows: {ak }

→ {ak−1 } →   {bk−1 }

{ak−2 } {bk−2 }

→ ···  ···

→ 

{ak−j } {bk−j }.

To reverse the preceding process, one may reconstruct the original function fk by taking—at decomposition level j—the “blurred” approximation fk−j and its fine structure correction gk−j to obtain fk−j+1 = fk−j + gk−j ∈ Vk−j+1 , etc., until one arrives at fk . The coefficient matrices for this reconstruction algorithm are given by the inverse Mallat transform M −1 : 2 ()A × 2 ()B → 2 ()A , 2

2

2

M −1 ({ak−1 }, {bk−1 }) = {ak },

(8.35)

282

Fractal Functions, Fractal Surfaces, and Wavelets

with ak,γ =



atk−1,γ  cγ −δγ  + dtk−1,γ  dγ −δγ  .

(8.36)

γ  ∈

This reconstruction can also be graphically represented by a diagram that is dual to the decomposition diagram: {ak }

← {ak−1 }  {bk−1 }

← {ak−2 }  {bk−2 }

← ···  ···

← 

{ak−j } {bk−j }.

The decomposition and reconstruction algorithms can be used to approximate a given function f ∈ L2 (Rn ). Without loss of generality one may assume that f ∈ V0 . Then one decomposes f into a finite series f = fk−j + gk−1 + gk−2 + · · · + gk−j consisting of a “blurred” or coarse-scale approximation fk−j and a finite number of fine-scale corrections gk−1 , . . . , gk−j . These summand functions are known from the expansion of f in terms of the vector scaling functions φ, and the coefficient matrices are obtained by application of the Mallat transform successively. For the reconstruction, however, one does not use all the coefficient matrices b, but uses only those that are above a predetermined threshold. In this way one decompresses the graph of f to obtain a satisfactory approximation  f . This procedure has been successfully applied to image compression denoising and signal analysis, and the interested reader is referred to the vast literature on this subject; see, for example, Refs. [115, 129–132]. Finally, a few approximation properties of the Daubechies scaling functions and wavelets are presented. These properties play an important role in the application of wavelets to the theory of numerical solutions of differential equations. See, for example, Ref. [133] and the references therein. For the remainder of this section, I denotes either an open interval (a, b) of R or the real line R itself. Let D(I) denote the linear space of all infinitely differentiable functions with compact support contained in I. Let p

H0 (I) := clW 2,p (I) D(I)

(1 ≤ p ≤ ∞),

(8.37)

where W 2,p (I) is the Sobolev space of functions defined on I. The restriction of functions in Vk and Wk to I is denoted by Vk (I) and Wk (I), respectively. To state the next result, the concept of best approximation must be introduced. To this end, let f : I → R be any function and let P m denote the set of all real polynomials of degree less than or equal to m. Define Em (f ; I) := min max |f (x) − p(x)|. p∈P m x∈I

(8.38)

Fractal functions and wavelets

283

The following result is well known in approximation theory and is usually referred to as an estimate of Whitney-Jackson type. These theorems estimate the approximation entirely in terms of the approximant and not the approximating elements. Proposition 71. Let n ∈ N and f ∈ D(R). If C := f (n) ∞ , then for any compact interval I, En−1 (f ; I) ≤

2C|I|n . 4n n!

(8.39)

Proposition 71, Corollary 11, and Schwarz’s inequality imply the next result (the details are left to the reader). Proposition 72. Let f ∈ D(R), let C := f (n) ∞ , and let r ψ be an r-regular Daubechies wavelet. Then ∀k, ∈ Z: |f , ψk | ≤ C1 2−k(r+1/2) ,

(8.40)

where C1 := 2C(2r − 1)r+1/2 /(4r r!). p Now consider the Sobolev space H0 (R). Since the Daubechies scaling functions and wavelets form an r-regular family, one can find an r ≥ s such that r φ and r ψ ∈ p H0 . Let J be the smallest interval such that supp f + 2r ⊆ J, and let C1 be as in Proposition 71. p Proposition 73. Let f ∈ D(R) and let r ∈ N0 be such that r φ, r ψ ∈ H0 . Let k ∈ N. Then f − Pk f W 2,p (R) ≤



|f , ψκ |ψκ W 2,p (R) ≤ C2 2−k(r−p) ,

(8.41)

κ≥k ∈Z

where Pk : L2 (R)  Vk denotes the √ orthogonal projection of f onto Vk and where the constant C2 is given by C2 := C1 |J|(2r − 1)(1 − 2−(r−p) )−1 ψW 2,p (R) . Proof. Use Proposition 72 and that ψk W 2,p (R) ≤ 2kp ψW 2,p (R) . The details are left to the diligent reader. p Using the classical result that {f |I : f ∈ D(R)} is dense in H0 (see, eg, Ref. [19]) and the just-stated proposition, one arrives at the following approximation theorem for Daubechies scaling functions and wavelets. This result lays the foundation for the application of wavelets to Galerkin methods [133, 134]. p p Theorem 119. Let p ∈ N0 , let r ∈ N0 be such that r φ, r ψ ∈ H0 , and let f ∈ H0 . Then ∀ε > 0 ∃ k ∈ Z: f − ϕW 2,p (R) < ε, where ϕ ∈ Vk (I).

(8.42)

284

Fractal Functions, Fractal Surfaces, and Wavelets

2 Fractal function wavelets In this section it is shown how fractal functions can be used to generate a multiresolution analysis on L2 (R) and C0 (R), respectively. The type of fractal functions that allows this construction was introduced in Example 24. The space V0 will consist of piecewise fractal functions parameterized by free parameters s1 , . . . , sN and can thus be thought of as a parameterized spline space. It will be seen that for particular values of the parameters s1 , . . . , sN , the space V0 reduces to a known spline space. The spaces V0 result from a linear isomorphism between the space of real-valued bounded functions on compact subsets of R and a certain function space  whose elements are sequences λ := {λ ,i : i ∈ N0N ∧ ∈ Z} of functions bounded on [0, 1]. The lift of the dilation operator Uδ to the function space  gives then rise to dilationand shift-invariant linear spaces of sequences λ. They are then used to obtain the scaling functions and wavelets. However, because of the construction, the spaces V0 will be generated by more than one (fractal) scaling function. Necessary and sufficient conditions on the sequences λ are given to ensure that the resulting vector scaling functions and multiwavelets are of class Cr , r ∈ N0 . The multiwavelets derived from these vector scaling functions are, however, not fully orthogonal; they are so-called prewavelets: only for k = k one has ψ kγ , ψ k γ   = δkk ,γ γ  IB . The free parameters s1 , . . . , sN allow the construction of continuous, compactly supported, and orthogonal wavelets. These fractal vector scaling functions and wavelets and their properties are compared with Daubechies scaling functions and wavelets. The presentation follows closely the original articles [82, 135].

2.1 The general construction Let I := [0, 1) and let B(I) := B(I, R) denote the Banach space of all"bounded Rvalued functions on I endowed with the supremum norm. Let BN := j∈NN−1 B(I) denote the N-fold direct product of B(I). Elements of BN will be denoted by λ := (λ0 , . . . , λN−1 ), where each λj ∈ B(I), j ∈ N0N−1 . For such a λ, let b: [0, 1] → [0, 1] be defined by b(x) =

N−1 

u−1 i (x)χIi (x)

i=0

and v(x, ·): R → R be defined by v(x, y) =

N−1 

(λi (x) + si y)χIi (x),

i=0

where Ii := ui (I), with ui : I → I given by ui (x) :=

1 N (x + i),

i ∈ N0N−1 .

Fractal functions and wavelets

285

The unique fixed point fλ := fλ of the associated RB operator λ is an element of B(I). In the event that each component λi of λ is continuous and vi+1 (0, fλ (0)) = vi (1− , fλ (1− )),

i ∈ N0N−1 ,

fλ ∈ C(I); see Section 1. The graph Gλ of fλ satisfies Gλ =

N−1 &

wi (Gλ ),

j=0

where wi : I × R → I × R is given by wi (x, y) := (ui (x), vi (x, y)),

i ∈ N0N−1 .

Note that each image wi (Gλ ) satisfies a similar equation; namely, wi (Gλ ) =

N−1 &

(wi ◦ wj ◦ w−1 i )(wi (Gλ )).

(8.43)

j=0

Now define the rescaling functions i : Ii × R → I × R by i := u−1 i × idR ,

i ∈ N0N−1 .

The application of i to Eq. (8.43) yields i ◦ wi (Gλ ) =

N−1 &

wi,j (i ◦ wi (Gλ )),

(8.44)

j=0

with wi,j := (i ◦ wi ) ◦ wj ◦ (i ◦ wi )−1 . It is not difficult to show that the maps wi,j : I × R → I × R are equal to wi,j = (uj , vi,j ), where vi,j = λi ◦ uj + si λj − sj λi + sj idR =: λi,j + sj idR .

(8.45)

If λ(i) := (λi,0 , λi,1 , . . . , λi,N−1 ) ∈ BN , then the preceding calculations and the fixed-point property of fλ imply graph fλ(i) = i ◦ wi (Gλ ) = graph fλ ◦ ui .

(8.46)

286

Fractal Functions, Fractal Surfaces, and Wavelets

In other words, the horizontally scaled image of fλ , fλ ◦ ui , is a fractal function in its own right generated by maps λ(i) ∈ BN . The immediate goal is to find a representation of the dilation operator Uδ , where δ := NidR :  →  in terms of the functions λi and the associated sequences λ. Assumption. Here and in what follows we assume without loss of generality that  = Z. " To this end, let : BN → j∈N0 BN be given by N−1

(λ) := (λ(0), λ(1), . . . , λ(N − 1)). Now let

"

BN :=

"

N n∈Z B ,

and let λ ∈

"

(8.47) BN . Define :

(λ)Nn+j := (λn )j

"

BN →

"

BN by (8.48)

for all j ∈ N0N−1 and n ∈ Z. With Eqs. (8.45), (8.47), Eq. (8.48) can also be expressed in the following form: (λ)Nn+j,i = λn,j ◦ ui + sj λn,i − si λn,j .

(8.49)

Shortly it will be proven that the operator  is the representation of the dilation operator Uδ in terms of the biinfinite sequences λ. However, before this can be done, the following result, which gives the basic correspondence between the elements in BN and functions in BR (I), is needed. It is a special case of Theorem 76, and the proof will therefore not be repeated. θ

a linear isomorphism from BN to B(I). Theorem 120. The mapping λ −→ fλ is" Next one has to identify elements in BN with R-valued functions on R. To this end, let (Bc (R),  · ∞ ). denote the Banach space of R-valued functions that are bounded on compact subsets of R. An element f := {fn }n∈Z ∈

!

BN

is identified with an f ∈ Bc (R) via the linear isomorphism τ

f −→



fn (· − n)χ[n,n+1) .

(8.50)

n∈Z

Note that the linear isomorphism θ defined in Theorem 120 canonically induces a linear isomorphism !

θ:

!

BN →

!

BN .

In summary, a biinfinite sequence of functions {λ(n)}n∈Z where each λn is itself a finite sequence of functions {λn,0 , λn,1 , . . . , λn,N−1 } gives rise to a piecewise fractal function

Fractal functions and wavelets

287

fλ : R → R with the property that the restriction of fλ to any interval [n, n + 1), n ∈ Z, is itself a fractal function generated by the RB operator λ(n) . The basic idea is to obtain properties for the fractal function fλ by imposing conditions on the biinfinite sequence functions λ. Fig. 8.2 shows an example of such a piecewise fractal function consisting of four affine fractal functions. 2.0

1.5

1.0

0.5

−1

1

2

3

−0.5 −1.0

Fig. 8.2 The graph of a piecewise fractal function.

" N The following theorem representation of the dilation B . δ on " Noperator " U " gives the " B → BN be defined Theorem 121. Let BR (I) := n∈Z BR (I), and let : as in Eq. (8.47). Then the following diagram commutes: " N  B BN −−−−→ ⏐ ⏐ " " ⏐ ⏐ θ( ( θ " " BR (I) BR (I) ⏐ ⏐ ⏐ ⏐τ τ( ( "

Uδ

Bc (R) −−−−→ Bc (R) Proof. Let f = {fn }n∈Z ∈ (Uδ ◦ τ )(f )(x) =



" 

fn

n∈Z

BN . Then x − nN N

 χ[n,n+1)

x N

  x − iN   N−1 fi = χ[iN+j,iN+j+1) (x). N i∈Z j=0

288

Fractal Functions, Fractal Surfaces, and Wavelets

Hence (τ −1 ◦ Uδ ◦ τ (f ))iN+j = fi



·+j N

 = fi ◦ uj

for all x ∈ I. Eqs. (8.46), (8.48) then give  !   ! −1 ◦ Uδ ◦ τ ◦ θ . = τ◦ θ

The shift or translation operator τγ : Bc (R) → Bc (R) can also be lifted to is not hard to see that τγ corresponds to the right-shift operator !

τγ :

!

BN →

!

"

BN . It

BN

given by {λn }n∈Z −→ {λn−1 }n∈Z ; that is, !

 ! −1  !  τγ = τ ◦ θ ◦ τγ ◦ τ ◦ θ .

It natural to ask what happens to certain subspaces of Bc (R) under the lift " *−1 . This question is answered for a particular subspace; namely, Cr (I). (This τ◦ θ subspace is the most natural one to consider, and its characterization is also needed for later developments.) ) " *−1 f . As f |I is the For this purpose, let f ∈ Cr (I), r ∈ N, and let λ = τ ◦ θ unique fixed point of the RB operator (λ)0 , one has

)

f ◦ ui = λ0,i + si f

(8.51)

for x ∈ I and i ∈ N0N−1 . Now let  Cr (I) := {g: g = f |I , where f ∈ Cr (I)}. Here I := cl I. As f ∈  Cr (I), it follows that (λ)0 ∈ 0 derivative, m ∈ Nr , of Eq. (8.51) with respect to x gives (m)

f (m) ◦ ui = N m λ0,i + si N m f (m) ,

i ∈ N0N−1 .

"N−1 j=0

 Cr (I). Taking the mth

Fractal functions and wavelets

289

Using the continuity of f (m) at 0 and that ui−1 (1− ) = ui (0) = L0m λ :=

i N

yields

1 − sN−1 N m (m) (m) λ (0) − λ−1,N−1 (1− ) = 0 1 − s0 N m 0,0

(8.52)

and (m) Lim λ := λ(m) 0,i (0) − λ0,i−1 (1) +

si N m si−1 N m (m) λ (0) − λ(m) (1) = 0. 1 − s0 N m 0,0 1 − sN−1 N m 0,N−1 (8.53)

Here the linear operators Lim :

!

BN → R,

i ∈ N0N−1 m ∈ N0r ,

were introduced. As for all n ∈ Z, τγn f ∈ Cr (R), it follows that r N−1 ! n + + ker Lim . τ λ∈

(8.54)

m=0 i=0

The reader is asked to compare these results with those obtained in Chapter 5. Now let !

⎧ r N−1 ⎨ ! n ! + + ! N−1 r  Cr := λ ∈ CR (I): τ λ∈ ker Lim , ⎩ Z j=0

m=0 i=0

⎫ ⎬

∀n ∈ Z . ⎭

It then follows from the foregoing discussion that 

τ◦

! −1 ! θ Cr (R) ⊆ Cr .

The reverse set containment is implied by the next result. Theorem 122. Suppose N r max{|si |: i ∈ N0N−1 } < 1. Then 

τ◦

! ! −1 Cr (R) = Cr . θ

Proof. " Theorem 121 and the following arguments. Let " 1The proof follows from λ ∈ C and let fλ := (τ ◦ θ )−1 λ. Since λn , n ∈ Z, satisfies the hypothesis of Theorem 121, fλ |[n,n+1) ∈ C1 ([n, n + 1)). Observe that fλ (n+ ) =

Nλn,0 (0) 1 − s0 N

290

Fractal Functions, Fractal Surfaces, and Wavelets

and fλ (n− ) =

Nλn−1,N−1 (1− ) 1 − sN−1 N

.

1 (R). This, together with the arguments given in the proof Hence, by Eq. (8.52), fλ ∈ CR of Theorem 121, yields" the result for r = 1. The general case now follows " by induction ∈ Cm ," with si replaced by Nsi , whenever λ ∈ Cm+1 .) on r. (Note that Nλ " Corollary 12.  Cr ⊆ Cr . Example 52. Let P d (I) denote the linear space of all real polynomials of degree d or less supported on I, and let ! ! ! P d := P d (I). n∈Z j∈N0

N−1

Note that if λi , λj ∈ P d (I) then λi,j = λi ◦ uj + si λj − sj λi ∈ P d (I). Hence Eq. (8.48) implies that ! ! Pd ⊆ Pd.  If we set !

Prd :=

that is, λ∈

!

!

Prd

Pd ∩

⇐⇒

!

Cr ,

λ∈

!

P d and satisfies Eq. (8.53),

Theorem 122 immediately gives 

!

Prd ⊆

!

Prd .

For most of the remainder of this chapter the case d = 1 is of particular interest. In this situation the functions λi are affine, and the continuous fractal functions generated of affine fractal functions supported on I. Since by these λi belong to the "class Aff 1 fλ ∈ C(I), the space θ ( N−1 j=0 P ) ∩ C(I) =: C is seen to be (N + 1)-dimensional. Hence the elements g of C are completely determined by their values g( Ni ), i ∈ N0N−1 . This allows the following geometric interpretation: Given a y := (y0 , y1 , . . . , yN−1 )t ∈ RN+1 ,

Fractal functions and wavelets

291

there exists a unique continuous fractal function fy having the interpolation property with respect to {( Ni , yi ): i ∈ N0N−1 } (see Example 26). The affine functions λi are then determined by Eqs. (5.19)–(5.22). The linear form of these equations implies the following result, whose proof is left to the reader. Proposition 74. The mapping RN+1  y −→ λ ∈ C is a linear isomorphism. Later it will be seen that this proposition adds a geometric component to the multiresolution analysis generated by fractal functions. " d The following example illustrates how λ ∈ N−1 j=0 P ∩ C(I) can be obtained. Example 53. Let N := 2, d := 2, and r := 1. Furthermore, let si := s, i = 0, 1 with |s| < 1/2. The linear space P 2 (I) is identified with R3 via the correspondence a + bx + cx2 → (a, b, c). Therefore λ = (λ0 , λ1 ) ∈

1 !

P 2 (I),

j=0

and the latter space can be identified with R6 via λ(x) = (λ0 (x), λ1 (x)) = (a0 + b0 x + c0 x2 , a1 + b1 x + c1 x2 ) → v := (a0 , b0 , c0 , a1 , b1 , c1 ), and λ ∈

"

P 2 with

"

6 n∈Z R :

λ → vn := {(an,0 , bn,0 , cn,0 , an,1 , bn,1 , cn,1 )}n∈Z . "1 2 Note that a function f ∈ C(I) iff λ ∈ j=0 P satisfies Eq. (8.52) for m = 0 and m = 1. These equations, however, are equivalent to the “more geometric” conditions nm , vE = 0,

m = 0, 1,

where n0 := (−1, −1, −1, 1, 0, 0)t + s(2, 1, 1, −2, −1, −1)t , n1 := (0, −1, −2, 0, 1, 0)t + s(0, 4, 4, 0, −4, −4)t , and ·, ·E denotes the Euclidean inner product in R6 . Hence the space

(8.55)

292

Fractal Functions, Fractal Surfaces, and Wavelets

S1 :=

1 !

P d ∩ C(I)

j=0

is a four-dimensional vector space, and each element f ∈ S1 is uniquely determined by f (0), f  (0), f (1), and f  (1). Suppose the base elements {f1 , f2 , f3 , f4 } of S1 are chosen according to f1 (0) = f2 (0) = f3 (0) = f4 (0) = 1 and f1 (0) = f1 (1) = f1 (1) = f2 (0) = · · · = f4 (1) = 0. Then the corresponding functions λ(k), k = 1, 2, 3, 4, are given by   1 1 1 λ(1) = θ −1 f1 = 1 − s, 0, s − , − s, 2s − 1, − s , 2 2 2   1 3 1 1 1 λ(2) = θ −1 f2 = 0, − s, s − , , − , , 2 8 8 4 8   1 1 1 −1 λ(3) = θ f3 = 0, 0, − s, , 1 − 2s, s − , 2 2 2   3 1 1 −1 λ(4) = θ f4 = 0, 0, s − , s − , −s, , 8 4 4 Next we use the preceding results to define a nested sequence of function spaces necessary for a multiresolution analysis. For this purpose, let  ! ! θ Prd , V0 := τ ◦

(8.56)

or, explicitly,   V0 := f : R → R: ∀j ∈ Z ∃ g ∈ F r [j, j + 1] so that f |(j,j+1) = g|(j,j+1) , (8.57) and define Vk , k ∈ Z, by requiring f ∈ Vk ⇐⇒ Uδ−k f ∈ V0 ,

k ∈ Z.

(8.58)

Proposition 75. The family {Vk }k∈Z forms a nested sequence of linear spaces.

Fractal functions and wavelets

293

Proof. It is clear that the Vk , k ∈ Z, are linear spaces. The nestedness follows from the observation that ! ! Prd ⊆ Prd , Uδ Vk ⊆ Vk+1 ⇐⇒  and the validity of the second statement was shown in Example 52. Example 54. This is a continuation of Example 53 with the notation and terminology used there. Set, as above,  ! ! θ ( P 2 )1 . V0 := τ ◦ Then f ∈ V0 is uniquely determined by our specifying the values of f and f  at the integers. Note the restriction of elements in V0 to any interval of the form [n, n + 1), n ∈ Z, defines a two-dimensional linear space. Thus if φ 1 , φ 2 ∈ V0 are such that supp φ i ⊆ [−1, 1], i = 1, 2, φ 1 (0) = φ 2 (0) = 1 and φ 1 (0) = φ 2 (0) = 0, then any f ∈ V0 can be expressed in the form f =



f ( )φ 1 (· − ) + f  ( )φ 2 (· − ).

∈Z

The functions φ 1 and φ 2 can be constructed in the following way: Let λ1 , λ2 ∈ be defined by

"

C1

⎧ ⎪ ⎨λ(3) if i = −1; λ1i = λ(1) if i = 0; ⎪ ⎩ 0 otherwise and

⎧ ⎪ ⎨λ(4) if i = −1; 2 λi = λ(2) if i = 0; ⎪ ⎩ 0 otherwise.

The graphs of φ 1 and φ 2 for s = 3/10 are depicted in Fig. 8.3. 1.0

0.10

0.8

0.05

0.6 −1.0

0.4

−0.5 −0.05

0.2

−0.10 −1.0

−0.5

0.5

1.0

Fig. 8.3 The graphs of φ 1 (left) and φ 2 (right) for s = 3/10.

0.5

1.0

294

Fractal Functions, Fractal Surfaces, and Wavelets

This example will be continued later; in particular, it will be shown that φ 1 and φ 2 generate a multiresolution analysis on L2 (R). To establish that the nested linear function spaces Vk as defined in Eqs. (8.56), (8.57), respectively, form a multiresolution analysis of L2 (R), a few general results concerning the conditions for a multiresolution analysis have to be proven. To this end, suppose that φ := (φ 1 , . . . , φ A )t is a finite vector function whose components are bounded and compactly supported functions in L2 (R). Let V0 := V0 [φ 1 , . . . , φ A ] := clL2 (R) span {φ a (· − ): a ∈ NA ∧ ∈ Z}; in other words, V0 is a finitely generated shift-invariant space with generators φ 1 , . . . , φ A , and Vk := Uδ−k V0 ,

k ∈ Z.

The next proposition gives sufficient conditions on φ guaranteeing that the density condition (M2) and the separation condition (M3) hold. Proposition 76. Let {φ a }a∈A be a finite set of bounded and compactly supported functions in L2 (R). 1. Separation:



Vk = {0}.

k∈Z

2. Density: Assume that there exists an a = {aa }a∈A such that A  

aa φ a (x − ) = 1,

a.e. x ∈ R.

a=1 ∈Z

Then



Vk is dense in L2 (R).

k∈Z

Proof. 1. Let In := [n, n + 1], n ∈ Z, and let U := {f ◦ χI0 : f ∈ V0 }. The boundedness and compact support of the functions φ a implies that U is a finite-dimensional linear space over R. Hence the norms  · 2 and  · ∞ are equivalent on U. Therefore there exists a positive constant c such that f ∞ ≤ cf 2 , ∀f ∈ U. Using the translation invariance of the space V0 , one obtains f ◦ χIn ∞ ≤ cf ◦ χIn 2 for all f ∈ V0 . Hence f ∞ = sup {f ◦ χIn ∞ } ≤ c n∈Z



f ◦ χIn 2 = cf 2

n∈Z

for all f ∈ Vk . Following Ref. [136], note that f ∞ ≤ cN −k/2 f 2 for all f ∈ Vk . Hence if f ∈



k∈Z Vk , then f ∞ = 0.

Fractal functions and wavelets

295

 2. Because of the translation and dilation invariance of Vk , it suffices to show that χ[0,1] ∈ k∈Z  Vk . Without loss of generality, suppose that supp φ a ∈ [0, M], a ∈ A. Choose a clL2 (R) k∈Z

j ∈ N such that N j > M, and let j

SN j :=

A N  

aa φ a (· − ).

=0 a=1

Then supp SN j ⊆ [0, N j + M], SN j ≡ 1 on [M, N j ], and SN j ∞ ≤ MA max{aa φ a ∞ }. a∈A

This then implies that SN j (N j ·) − χ[0,1] ∞ → 0. As SN j (N j ·) ∈



Vk , the result follows.

k∈Z

Next necessary and sufficient conditions are given for the set of translates of {φ a } to be a Riesz basis of V0 . To this end, let  †  φ(ξ + 2π ) φ (ξ + 2π ). (8.59) Eφ (ξ ) :=

∈Z

Note that Eφ is an A × A matrix. Using the Poisson summation formula, one verifies that    † φ(y − )φ (y)dy eiξ

Eφ (ξ ) =

∈Z

=

R

M−1 

=−M+1

 R

 φ(y − )φ † (y)dy eiξ

for some M > 0.

(8.60)

Observe that in the last equality the fact was used that φ has  compact support. Recall that if a function f ∈ L1 (Rn ) is such that

∈Z f (x + 2π ) converges  f ( )eix converges everywhere, everywhere (in L1 ) to a continuous function and ∈Z then    f ( )eix . f (x + 2π ) = (8.61)

∈Z

∈Z

Eq. (8.61) is called the Poisson summation formula. Theorem 123. The collection Bφ := {φ a (· − ): a ∈ NA ∧ ∈ Z}

forms a Riesz basis for V0 iff Eφ is nonsingular for ξ ∈ [0, 2π].

296

Fractal Functions, Fractal Surfaces, and Wavelets

Proof. The proof consists in finding the Riesz bounds R1 and R2 ; see Eq. (8.7). Note that by Parseval’s inequality and the shift property of the Fourier transform, we have that 2 2   1 φ(ξ ) dξ c† φ(x − ) dx = eiξ c†  2π R R ∈Z

∈Z / 02 1  c† (ξ ),  φ(ξ ) dξ , = 2π R  iξ † where  c(ξ ) :=

∈Z e c . As the Fourier transform of the symbol c(ξ ) is 2πperiodic, the last of the preceding equations can be rewritten as 1  2π



2

∈Z [0,2π]

 c† (ξ ) φ(ξ + 2π ) dξ   1  †  c† (ξ ) φ(ξ + 2π ) φ (ξ + 2π ) c(ξ ) dξ 2π

∈Z [0,2π] 1  c† Eφ (ξ ) c(ξ )dξ . = 2π [0,2π] =

Using Parseval’s identity again, we obtain 1 1 2  c† (ξ ) c(ξ )dξ =: ||| c|||2L2 (R) . {c } ∈Z L2 (R) = 2π [0,2π] 2π

(8.62)

Now let L2 ([0, 2π], CA ) := {f: [0, 2π] → CA : |||f|||L2 (R) < ∞}. Then Eq. (8.7) is equivalent to R1 ||| c|||2L2 (R)

≤

[0,2π]

 c† Eφ (ξ ) c(ξ )dξ ≤ R2 ||| c|||2L2 (R)

(8.63)

for all  c ∈ L2 ([0, 2π]; CA ). From Eq. (8.59) it follows that Eφ (ξ ) is self-adjoint and positive, and thus has real nonnegative eigenvalues λa , a ∈ NA . Moreover, Eφ is continuous in ξ , implying that the eigenvalues λa are also continuous functions in ξ . Let m(ξ ) := min{λa : a ∈ NA } and let M(ξ ) := max{λa : a ∈ NA }. Then m(ξ ) c† (ξ ) c(ξ ) ≤  c† (ξ )Eφ (ξ ) c(ξ ) ≤ M(ξ ) c† (ξ ) c(ξ ), and so Eq. (8.63) holds with R1 := min{m(ξ ): ξ ∈ [0, 2π]} and

R2 := max{m(ξ ): ξ ∈ [0, 2π]}.

Fractal functions and wavelets

297

Observe that since Eq. (8.63) holds for all  c ∈ L2 ([0, 2π]; CA ), these are the best possible bounds. As M(ξ ) is always finite, Bφ is a Riesz basis provided min{m(ξ ): ξ ∈ [0, 2π]} > 0, which is true iff the symbol Eφ (ξ ) is nonsingular for ξ ∈ [0, 2π]. With use of a counting argument, it is not hard to see that there are B = A(N − 1) wavelets that span W0 = V1 " V0 . Corollary 13. The collection Bψ := {ψ b (· − ): b ∈ NB ∧ ∈ Z}

forms a Riesz basis for W0 iff: 1. W0 ⊆ clL2 (R) span Bψ ;   + 2π )ψ † (ξ + 2π ) is nonsingular for ξ ∈ [0, 2π]. 2. Eψ (ξ ) := ∈Z ψ(ξ

As ψ is a linear combination of φ, there must exist a relation between the symbols Eφ and Eψ . The next theorem derives this relation.  Theorem 124. Let ζ := e−iξ/N and let D(ζ ) := N1 ∈Z d ζ . Then

 †    (−2πijζ )/N E ((ξ + 2πj)/N) D e(−2πijζ )/N ; 1. Eψ (ξ ) = N−1 D e φ j=0 2. supposing that Eφ (ξ ) is nonsingular, the symbol Eψ (ξ ) is nonsingular for all ζ if †  N−1  ker D e(−2πijζ )/N = {0}. j=0

Proof. 1. Eq. (8.11) and the definition of Eψ (ξ ) yield ⎛ ⎞    1   /N ξ + 2π

−i(ξ +2π )

 ⎝ ⎠ Eψ (ξ ) = d  e φ N  N

∈Z

∈Z ⎛ ⎞ †    1 ξ + 2π

⎠ ×⎝ d  e−i(ξ +2π ) /N  φ N  N

∈Z

=

  ξ + 2πj  + 2π  φ N j=0

 ∈Z   † ξ + 2πj + 2π  D(e(−2πijζ )/N )† , × φ N N−1 

D e(−2πijζ )/N



where = N  + j. 2. For simplicity, set E(ξ , ζ ; j, N) := D(e(−2πijζ )/N )Eφ ((ξ + 2πj)/N) D(e(−2πijζ )/N )† . As D(ζ )Eφ (ξ )D(ζ )† is a positive matrix for all ξ , it follows from proof 1 that Eψ is nonsingular N−1  if ker E(ξ , ζ ; j, N) is trivial. By the hypotheses, however, j=0

†  ker E(ξ , ζ ; j, N) = ker D e(−2πijζ )/N .

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Fractal Functions, Fractal Surfaces, and Wavelets

Example 55. This is the continuation of the two preceding examples. Here it is shown that the scaling functions φ 1 and φ 2 defined in Example 54 generate a multiresolution analysis of L2 (R). As before, one defines the finitely generated shiftinvariant space   V0 := V0 [φ 1 , φ 2 ] := clL2 (R) span φ a (· − ): a = 1, 2 ∧ ∈ Z . Since λ(1) + λ(3) = (1 − s, 0, 0, 1 − s, 0, 0), one has θ (λ(1) + λ(3)) ≡ 1, implying that φ 1 forms a partition of unity:  φ 1 (x − ) ≡ 1.

∈Z

As both scaling functions are bounded, compactly supported, and satisfy condition 2 in Proposition 76 with a = (1, 0), the density and separation properties hold. It remains to be shown that the integer translates of the vector scaling function φ form a Riesz basis of V0 . This can be done via Theorem 123 by our proving that the symbol Eφ is nonsingular for all ξ ∈ [0, 2π]. Note that for this example the number M in Eq. (8.60) is equal to 2 since φ 1 and φ 2 are both supported on [−1, 1]. Setting e :=

R

φ(y − )φ † (y)dy,

= −1, 0, 1,

one can show—after some considerable algebra and by using Proposition 56—that these 2 × 2 matrices are given by ⎛

92 − 69s − 78s2 + 56s3 ⎜ 60(2 − s)(1 − s2 ) e0 = ⎜ ⎝ 0 ⎛ 28+9s−42s2 +4s3

et−1

⎜ 120(2−s)(1−s2 ) = e1 = ⎜ ⎝ −13 − 9s + 27s2 − 4s3 240(2 − s)(1 − s2 )

⎞ 0

5 − 4s 240(1 − s2 ) ⎞ 13 + 9s − 27s2 + 4s3 240(2 − s)(1 − s2 ) ⎟ ⎟. ⎠ −3 − 4s + 8s2 480(1 − s2 )

Hence Eφ (ξ ) = e−1 e−iξ + e0 + e1 eiξ ,

ξ ∈ [0, 2π],

and therefore det Eφ (ξ ) = where ζ = eiξ and

⎟ ⎟, ⎠

r 0 + r1 ζ + r 2 ζ 2 + r1 ζ 3 + r 0 ζ 4 , 57,000(s − 2)2 (s2 − 1)ζ 2

Fractal functions and wavelets

299

r0 := − 1 − 40s − 147s2 + 200s3 − 16s4 , r1 := 544 + 640s − 3912s2 + 3520s3 − 896s4 , r2 := − 3006 + 8400s − 7722s2 + 2160s3 − 96s4 . Note that the denominator in the preceding expression for det Eφ does not vanish for |s| 1 < 1, and the roots of the quartic equation in the numerator are of the form ρ ± ρ 2 − 1, with ρ = −(r1 /4r0 ) ±

 1 2 + (r1 /2r2 )2 − (r2 /r0 ), 2

assuming r0 = 0. It follows directly from the form of the zeros of det Eφ that they lie on the unit circle in C iff ρ ∈ R and ρ 2 ≤ 1. This, however, is equivalent to R(s) := (2r0 + r2 )2 − 4r12 ≤ 0. The polynomial R(s) allows the factorization R(s) = 30,720(2 − s)3 (2s − 1)2 (32 − 39s − 18s2 + 26s3 ). Methods from elementary calculus can be used to establish that the cubic polynomial in the preceding factorization is strictly positive for |s| < 1. Hence R(s) > 0 for s = 1/2. Therefore φ generates a multiresolution analysis of L2 (R) for any s ∈ (−1, 1) \ { 12 }. If, in addition, |s| < 12 , then φ 1 and φ 2 are also of class C1 (R). In this case the coefficient matrices c can be calculated by use of Eq. (8.51) and the characterization of a function f ∈ V0 given in Example 54. A straightforward computation gives  c−1 =

1 2

− 18

 1 − 2s , s − 14

 c0 =

1 0

0 1 2

 ,

c1 =

1 2 1 8

 2s − 1 . 1 4 −s

Next the multiwavelet associated with the vector scaling function in the preceding example is constructed. It will be seen that it is possible to construct wavelets supported / {−1, 0, 1, 2, 3}. Let on [−1, 2]. This corresponds to our choosing d = 0 for ∈ ψ ∈ V1 be one of the two components of ψ (since N = A = 2, one also has B = 2). Then there exist coefficients d a ∈ R such that ψ=

3  1 

d a φ a (2x − ).

=−1 a=0

A necessary condition for ψ ∈ W0 ⊆ V−1 is φ a (· − ), ψ = 0

(8.64)

300

Fractal Functions, Fractal Surfaces, and Wavelets

for a = 0, 1 and = −1, 0, 1, 2. (That only these values for have to be considered follows from the lengths of the supports of φ a and ψ.) Using the two-scale dilation equation for φ, we can rewrite this last condition as 3  3 

1 

−1  =−1 a ,a =0









a  a a ca,a

 φ (2 · −2 − ), φ (2 · − )d = 0

(8.65)



for a = 0, 1 and k = −1, 0, 1, 2, 3 (here c = (ca,a

)). Observe that the preceding equations form a linear system of 8 equations in the 10 unknowns d a . A tedious 



computation shows that there exist two solutions d0 = (d 0,a ) and d1 = (d 1,a ) that form a basis of the null space of this linear system.  Now let d = (d a,a )a,a =0,1 , and define the multiwavelet ψ = (ψ0 , ψ1 )t by ψ

x N

:=

3 

d φ(x − ).

(8.66)

=−1

Clearly ψ 0 , ψ 1 ∈ W0 . Moreover, it can be shown [82] that Bψ forms a Riesz basis for W0 . But the multiwavelet ψ is not fully orthogonal; ψ 0 and ψ 1 are orthogonal to their dilates; that is, 0 /  a , ψka  = 0, k = k , ψk

where a, a = 0, 1, k, k , , ∈ Z. However, it will be shown shortly that there exists a way of obtaining fully orthogonal multiwavelets. This construction is given in Section 3.

2.2 A multiresolution analysis of C0 (R) Here a multiresolution analysis of C0 (R) is presented. To this end, let si = s, i ∈ NN , |s| < 1, and let V0 := {f : R → R: ∀j ∈ Z ∃ g ∈ F [j, j + 1]: f |(j,j+1) = g|(j,j+1) }.

(8.67)

Define V0 := V0 ∩ C0 (R)

(8.68)

and f ∈ Vk ⇐⇒ f (N k ·) ∈ V0 ,

k ∈ Z.

(8.69)

Let en ∈ ZNN , n ∈ NN , be such that en (j) = δnj for j ∈ N0N . Define an interpolating set n by

Fractal functions and wavelets

2 n :=

301

 3 j , en (j) : j ∈ N0N , N

n ∈ N0N ,

and let f n ∈ F |n [0, 1] be the affine fractal function interpolating n . We will use the set {f 0 , f 1 , . . . , f N } to define a basis for V0 . It is not difficult to see that the functions N φ 0 := f 0 + f0,−1 ,

φ i := f i ,

i ∈ NN−1 ,

(8.70)

form a basis for V0 (note that C0 (R) is separable). Thus a Vk = span {φk

: n ∈ N0N ∧ k, ∈ Z} ∩ C0 (R).

Theorem 125. The set of functions {φ 0 , φ 1 , . . . , φ N } generates a multiresolution analysis of C0 (R). Remark 67. Condition (M6*) is equivalent to ∀k ∈ Z ∃ R1 , R2 ∈ R+ : R1 c ∞ (Z) ≤ φ  c ≤ R2 c ∞ (Z) , where φ := (φ 0 , φ 1 , . . . , φ N )t and Z := N0N−1 × Z. Proof. That the Vk , k ∈ Z, are linear spaces follows directly from Proposition 74, and the nestedness follows from Eqs. (8.43), (8.45). We can generate a bounded piecewise linear function on R by choosing collinear interpolation points on each interval of the form [ N k , ( + 1)N k ], k, ∈ Z. Thus the space Vk contains all boundedfunctions that are linear on [ N k , ( + 1)N k ], k, ∈ Z. Vk is dense in C0 (R). This gives the density property. Since N k → 0 as k → −∞, k∈Z

The separation property is implied by the following argument: all constant functions Vk . For k ∈ N0 , let k be the RB are elements of Vk , k ∈ Z. Let f ∈ k∈Z

operator generating that fractal function on Ik := [0, N k ] which interpolates the points (iN k−1 , f (iN k−1 )), i ∈ N0N . Let Lk denote the linear interpolant through (0, f (0)) and (N k , f (N k )). As f ∈ Vk−1 ∩ Vk , it follows that k (Lk )|[0,N k−1 ] = Lk−1 |[0,N k−1 ] , and thus for any 0 ≤ m ≤ k, m ◦ · · · ◦ k (Lk )|[0,N k−1 ] = Lm |[0,N m−1 ] . Note that f |[0,N j ] is a fixed point of j for all j ∈ N0 . Hence (f − Lm )|[0,N m−1 ] ∞ = m ◦ · · · ◦ k (f )|[0,N m−1 ] − m ◦ · · · ◦ k (Lm )|[0,N m−1 ] ∞ ≤ sk−m+1 (f − Lk )|[0,N k ] ∞ ≤ 2sk−m+1 f ∞ . Letting k → ∞ gives f |[0,N m−1 ] = Lm |[0,N m−1 ] , and thus f |[0,∞) = f (0). Similarly, one shows f |(−∞,0] = f (0). This implies the result.

302

Fractal Functions, Fractal Surfaces, and Wavelets

a : a ∈ N0 ∧ ∈ Z} is a Riesz basis for V , k ∈ Z, Finally, to show that the set {φk

k N one only needs to set

 N 





n

cn f : max |cn | = 1 > 0 R1 := min



0≤n≤N n=0

∞

and

 N 





n

R2 := max

cn f : max |cn | = 1 .



0≤n≤N n=0

∞

The wavelet spaces Wk are defined as the orthogonal complement of Vk in Vk+1 . A possible choice for W0 is given by  a  W0 := span φ1,

: a ∈ NN−1 ∧ ∈ Z .

(8.71)

a that are supported on [0, 1] and their integer Hence one can choose the functions φ1,

translates to generate the wavelet space W0 . Therefore a ψ b := φ1,

,

(8.72)

where b := (N − 1) + a and ∈ N0N−1 . Hence   b Wk = span ψk

: b ∈ NN(N−1) ∧ ∈ Z ,

k ∈ Z.

(8.73)

It is worthwhile mentioning that any function f ∈ V0 has a very simple expansion in terms of the scaling functions φ a ; namely, f =

+∞ N   a=0 =−∞

 a a f + φ . N 0,

(8.74)

3 Orthogonal fractal function wavelets In this section it is shown that a proper choice of the scaling factors si yields fully orthogonal fractal function wavelets. This construction is presented for N := 2 and d := 1, and the underlying fractal functions are constructed via the procedure outlined in Example 26. The presentation follows closely the original "articles [82, 137]. In the case N := 2 and d := 1, the space C = θ ( 1j=0 P 1 ) ∩ C(I) is threedimensional. The first objective is to find an orthonormal basis for C. To this end, let y1 , y2 , and y3 be vectors in R3 , and let ei := eyi , i = 1, 2, 3, denote the fractal function

Fractal functions and wavelets

303

interpolating the set {( 2j , yi,j ): j = 0, 1, 2}, with yi = (yi,0 , yi,1 , yi,2 )t . Furthermore, it is assumed that e1 (0) = e1 (1) = 0 = e2 (0) = e3 (1). These functions may, without loss of generality, also be rescaled so that   1 e1 = e2 (1) = e3 (0) = 1. 2 Thus one may take y1 = (0, 1, 0)t , y2 = (0, p, 1)t , and y3 = (1, q, 0)t for some p, q ∈ R. The idea is to determine p, q, and the scaling factors si so that the functions e1 , e2 , and e3 are mutually orthogonal. Using, for instance, Eqs. (5.19)–(5.22), one obtains the following formulae for the functions λi , i = 0, 1: e1 : λ0 = idR , e2 : λ0 = (p − s0 )idR , e3 : λ0 = (q + s0 − 1)idR + (1 − s0 ),

λ1 = −idR + 1; λ1 = (1 − s1 − q)idR + (1 − s0 ); λ1 = (s1 − q)idR + (q − s1 ).

In other words, B2 λ(1) = (idR , −idR + 1), λ(2) = ((p − s0 )idR , (1 − s1 − q)idR + (1 − s0 )), λ(3) = ((q + s0 − 1)idR + (1 − s0 ), (s1 − q)idR + (q − s1 )). Our requiring that ei ej dx = 0 [0,1]

for i = j,

yields—using Proposition 56 and some rather tedious algebra—the following equations for the unknowns p and q, p=

−(4 − 6s0 − 2s0 s1 − 4s20 − 4s21 + 3s30 + 3s20 s1 )

(8.75)

16 + 4s0 s1 − 4s20 − 4s21

and q=

−(4 − 6s1 − 2s0 s1 − 4s20 − 4s21 + 3s31 + 3s0 s21 ) 16 + 4s0 s1 − 4s20 − 4s21

,

(8.76)

as well as the following algebraic equation to be satisfied by s0 and s1 : p(s0 , s1 ) := 2s41 +6s31 −7s0 s31 + 18s0 s21 − 29s21 − 7s30 s1 + 18s20 s1 − 14s0 s1 + 12s1 + 2s20 + 6s30 − 28s20 + 12s0 + 8 = 0.

304

Fractal Functions, Fractal Surfaces, and Wavelets

The preceding arguments proved the next result. Proposition 77. The affine fractal functions e1 , e2 , and e3 constitute an orthogonal basis for C only for those scaling factors si , i = 0, 1, which satisfy |si | < 1 and p(s0 , s1 ) = 0. Corollary 14. The only scaling factors s0 , s1 such that the basis { ey1 , ey2 , ey3 } with y2 := (0, b, 1)t , and  y3 := (1, c, 0)t , a, b, c ∈ R, constitutes an  y1 := (0, 1, a)t ,  orthogonal basis of C are those which satisfy |si | < 1 and p(s0 , s1 ) = 0. The same ey2 , ey3 } with  y1 = (a, 1, 0)t ,  y2 = (0, b, 1)t , and is true for bases of the form { ey1 , t  y3 = (1, c, 0) . Proof. Let s0 and s1 be scaling factors that satisfy the hypotheses of the corollary, ey2 , ey3 } with  y1 := (0, 1, a)t ,  y2 := (0, b, 1)t , and  y3 := (1, c, 0)t , a, b, c ∈ and let { ey1 , R, be the corresponding orthogonal basis. Suppose that a = 0, for otherwise the result follows from Proposition 77. e∗y2 , e∗y3 } be the orthonormal basis constructed from { ey1 , ey2 , ey3 }. If Let { e∗y1 , ∗   ey1 (1) = a and  e∗y2 (1) = b , set e∗y1 − a e∗y2 w1 := b

and

w2 := a e∗y1 + b e∗y2 .

Note that a2 + b2 = 1. Then {w1 , w2 , e∗y3 } is an orthonormal basis of C with w1 (0) = w1 (1) = 0 and w2 (0) = 0. The result now follows from Proposition 77. A similar argument can be applied to obtain the second statement. Definition 131. A triple (ey1 , ey2 , ey3 ) of mutually orthogonal affine fractal functions, each of which possesses the interpolation property with respect to {( 2j , yi,j ): j = 0, 1, 2}, for i = 1, 2, 3, will be called a fundamental basis for C. Now suppose that the scaling factors si , i = 0, 1, are such that |si | < 1 and p(s0 , s1 ) = 0. Set 

ey1 (x) for x ∈ [0, 1]; 0 otherwise, ⎧ ⎪ for x ∈ [0, 1]; ⎨ey2 (x) 1 φ (x) := ey3 (x − 1) for x ∈ [1, 2]; ⎪ ⎩ 0 otherwise, 0

φ (x) :=

(8.77a)

(8.77b)

and φ i∗ :=

φi φ i L2 (R)

,

i = 0, 1.

Theorem 126. Suppose that the scaling factors si , i = 0, 1, admit the existence of an orthogonal basis of affine fractal functions. Then the functions φ i∗ generate a multiresolution analysis of L2 (R). i∗ := 2−k/2 φ i∗ (2k · − ), i = 0, 1, k, ∈ Z, and define V as usual: Proof. Let φk

k

Fractal functions and wavelets

305

  i∗ Vk = clL2 (R) span φk

: i = 0, 1 ∧ k, ∈ Z ∩ L2 (R). It is a direct consequence of {ey1 , ey2 , ey3 } being a fundamental basis for V0 [0, 1) that {ey1 (2·), ey2 (2·), ey3 (2·)} and {ey1 (2·−1), ey2 (2·−1), ey3 (2·−1)} are fundamental bases for V−1 [0, 12 ) and V−1 [ 12 , 1), respectively. Therefore 

  3 φ 0∗ (x) c φ(2x − ). = φ 1∗ (x)

φ(x) =

(8.78)

=0

Consequently, V0 ⊆ V1 , and thus Vk ⊆ Vk+1 for all k ∈ Z. As χ[0,1] = (1 − p − q)ey1 + ey2 = ey3 , it follows that 

(1 − p − q)φ 0∗ (x − ) + φ 1∗ (x − ) = 1,

x ∈ R.

∈Z

Proposition 76 now gives the result. The coefficient matrices c , = 0, 1, 2, 3, may be computed directly from Eq. (8.78) either by use of the inner product 

φ, φ =

φφ dx = †

R

R

φ 0∗ , φ 0∗  φ 1∗ , φ 0∗ 



φ 0∗ , φ 1∗  φ 1∗ , φ 1∗ 

dx

or by evaluation of the equation at x = 4j , j = 1, . . . , 8. Either method yields  c0 = 

 s0 − p + 12 1 , 1 2 (p − s0 ) + p(s0 − p)p

 s1 − q + 12 0 , 1 2 (1 − p − s1 ) + p(s1 − q) 1   0 0 , c2 = 1 (1 − p − s0 ) + q(s0 − p) q  2 0 0 c3 = 1 . (q − s ) + q(s − q) 0 1 1 2

c1 =

Next it will be shown that, even if longer supports are considered, compactly supported, continuous, and orthogonal scaling functions can be constructed only from those scaling factors which allow the existence of a fundamental basis. Proposition 78. Suppose that for a given finite-dimensional subspace V of Cr [0, 1] there is no orthonormal basis with more than one function vanishing either at x = 0 or at x = 1. Then any pair of functions φ 1 , φ 2 ∈ Cr [0, 1] with compact support that is a linear combination of the basis elements of V and their integer translates and is constructed so that the set

306

Fractal Functions, Fractal Surfaces, and Wavelets

F := {φ a (x − ): a = 1, 2 ∧ ∈ Z} spans V and 

φ a , φ a (· − ) = δaa , 0 ,

a, a ∈ {1, 2}, ∈ Z,

must have the property that the leftmost nonzero components of φ 1 and φ 2 are linearly dependent, as are their rightmost nonzero components. Proof. Suppose that supp φ a = [0, Ma ], a = 1, 2. By continuity we have that  a φ (0) = 0 and φ a (x+Ma −1)|x=1 = 0. Furthermore, φ a |[0,1] , φ a (·+Ma −1)|[0,1]  = 0 for a, a = 1, 2. Since F spans V, the result follows. It should be noted that—by a rotation—we may always assume that M1 = M2 . Proposition 79. Suppose that the hypotheses of Proposition 78 are satisfied. Then there exists no such pair of functions φ 1 and φ 2 . Proof. Suppose that dim V = k, that M1 > M2 , and that φ 1 and φ 2 exist. Let y := φ 1 /φ 1 L2 (R) and z := φ 2 /φ 2 L2 (R) . Then y and z are orthonormal and there exists a basis in which y and z have the following representation: y = (y1 , y2 , . . . , yM1 , 0, . . . , 0)t ∈ V = ((y1,1 , 0∗ ), (y2,1 , y∗2 ), . . . , (0∗ , yM1 ,k ), (0, 0∗ ), . . . , (0, 0∗ ))t and z = (z1 , z2 , . . . , zM2 , 0, . . . , 0)t ∈ V = ((z1,1 , 0∗ ), (z2,1 , y∗2 ), . . . , (0∗ , zM2 ,k ))t , where 2 ≤ M1 < M2 , and y∗i (i = 2, 3, . . . , M1 ), z∗j , (j = 2, 3, . . . , M2 ), and 0∗ each have k − 1 components, and z1,1 > 0. Now consider the following rotation R: V → V defined on pairs of vectors of the preceding form: ) *t 1 z1,1 y − y1,1 z, y1,1 y + z1,1 z . R(y, z) := ( y, z)t =  y21,1 + z21,1 Note that  y and z can be written as  y = ((0, 0∗ ), ( y2,1 , y∗2 ), . . . , ( yM1 ,1 , y∗M1 ))t , t   * ) zM2 ,1 , y21,1 + z21,1 , 0∗ ,  z∗2 , . . . , ( z∗M2 ) .  z= z2,1 , Define a right-shift map σ+ : range R → V by σ+ ( y, z) := (( y2 , y3 , . . . , yM1 , 0), ( z1 , z2 , . . . , zM2 ))t = ( y, z)t .

Fractal functions and wavelets

307

Both R and σ+ are continuous and  y =  z = 1. (Here  ·  denotes a norm on the finite-dimensional vector space V.) Moreover,  y and z are orthogonal. The functions corresponding to the vectors  y and  z are in Cr (R). Proposition 78 z1 are linearly dependent, and thus  y∗1 = 0. Hence σ+ maps into implies that  y1 and  range R. This allows the iteration of the map σ+ ◦ R on the pairs (y, z). In this way a sequence {(y(j) , z(j) )}j∈N of pairs of vectors is produced and, since this sequence is contained in a compact set, it has a limit point (¯y, z¯) in range R. The continuity of the inner product now implies that ¯y = ¯z = 1, that y¯ and z¯ are orthogonal, and that the functions corresponding to y¯ and z¯ are elements of Cr (R). (j) (j) Observe that the sequence {z1,1 } is monotone increasing in j and, z1,1 being a (j)

component of a unit vector, bounded above by 1. Let z¯1,1 := limj→0 z1,1 . Hence (j)

(j)

the sequence {(z1,1 )2 }j∈N converges to (¯z1,1 )2 . Therefore the sequence {(y1,1 )2 }j∈N must converge to zero. Using the following induction argument, one can show that (j) (j) {(yi,1 )2 }j∈N converges to zero for all i = 2, 3, . . . , M1 . So suppose that {(yi ,1 )2 }  converges to zero for some i . Then for all j ∈ N, (j) (j)

(j+1)

yi ,1

=

(j) (j)

z1,1 yi +1,1 − y1,1 zi +1,1  . (j) (j) (y1,1 )2 + (z1,1 )2

Note that the denominator in the preceding equation is bounded above by Thus (j+1)

yi ,1

≥

√

2 < 2.

 1  (j) (j) (j) (j) z1,1 yi +1,1 | − |y1,1 zi +1,1 , 2

or (j) (j)

(j+1)

z1,1 yi +1,1 ≤ 2 yi ,1 (j)

(j) (j)

(j+1)

+ y1,1 zi +1,1 ≤ 2 yi ,1

(j)

+ y1,1 .

Since z1,1 ≥ z1,1 > 0 and since the left-hand side of the last expression converges to zero, the claim is proven. Because of this last result,  yi,1 = 0 for i ∈ NM1 . By Proposition 78 the leftmost z1 = ( z1,1 , 0∗ ). Hence  yp has to nonzero tuple in  y, say,  yp , is linearly dependent on  ∗ yp,1 , 0 ), implying that  yp = 0. This contradiction proves the be of the form  yp = ( proposition. Combining the results in Propositions 77, 78, and 79 yields the next theorem. Theorem 127. If s0 and s1 do not satisfy the algebraic equation p(s0 , s1 ) = 0, then there exist no continuous, compactly supported scaling functions φ 1 and φ 2 formed from affine fractal functions generated with these scaling factors. Proof. By Proposition 79, only scaling factors s0 and s1 that allow an orthogonal basis of C in which at least two basis elements vanish at either 0 or 1 have to be considered. But in Propositions 77 and 78 it was shown that this can occur only when p(s0 , s1 ) = 0.

308

Fractal Functions, Fractal Surfaces, and Wavelets

The preceding theorem shows that, to construct continuous and compactly supported scaling functions, the zero-set of the polynomial p(s0 , s1 ) has to be investigated. The following proposition gives a characterization of this zero-set. Proposition 80. The zero-set of the polynomial p(s0 , s1 ) = 2s41 + 6s31 − 7s0 s31 + 18s0 s21 − 29s21 − 7s30 s1 + 18s20 s1 − 14s0 s1 + 12s1 + 2s20 + 6s30 − 28s20 + 12s0 + 8 = 0 has two connected components C1 and C2 . Component C1 is a closed convex curve and component C2 consists of a pair of asymptotically linear curves that intersect transversely at exactly one point. Proof. A change of variables, T: R2 → R2 , (s0 , s1 ) → (x − y, x + y), transforms the polynomial p(s0 , s1 ) into a new polynomial q(x, y): q(x, y) = 18y4 + (24x2 − 42)y2 − 10x4 + 48x3 − 70x2 + 24x + 8. The zero-set of p(s0 , s1 ) is the preimage of the√zero-set of q(x, y) under T. Since T is just a rotation by π4 followed by a dilation by 2, the essential properties of the zeroset of p(s0 , s1 ) are preserved under T. Hence it suffices to find the zero-set of q(x, y). Also, note that the polynomial q(x, y) is symmetric with respect to y for all x. To find the zero-set of q, the equation q(x, y) = 0 is solved for y in terms of x. This gives 4 y=±

7 2 2 1√ 1 4 3 12x − 32x3 + 28x2 − 16x + 11 − x + 6 3 6

or 4 y=±

7 2 2 1√ 1 4 3 12x − 32x3 + 28x2 − 16x + 11. − x − 6 3 6

The first solution yields a pair of asymptotically linear curves that intersect transversely at (x, y) = (2, 0) or (s0 , s1 ) = (2, 0). This component C1 consists of pairs (s0 , s1 ) that are outside the range of si -values necessary to construct fractal functions. The second solution, which is real valued only for x ∈ [− 15 , 1], gives two halves of a symmetric closed curve γ . Denote by γ + the positive branch of the curve γ . If it can be established that γ + is the graph of a convex function, then—by symmetry—γ is a convex curve. To show that γ + is indeed the graph of a convex function f , it suffices to prove that f · f is a convex function. This can be done by our showing that f  |[− 1 ,1] is nonpositive; that 5 is, that 3 + 4

√ 3(48x3 − 96x2 + 56x − 16)2 3/2

24p1

√ −

3(144x2 − 192x + 56) 1/2

12p1

≤ 0,

Fractal functions and wavelets

309

where p1 (x) := 12x4 − 32x3 + 28x2 − 16x + 11. Since p1 > 0 for all x ∈ R, the preceding inequality is equivalent to 3/2

−

−4p1

√ 3(−144x6 + 576x5 − 888x4 + 832x3 − 780x2 + 528x − 122) ≤ 0.

To prove that this inequality holds for x ∈ [− 15 , 1], the expression √ approximated by the linear polynomial p2 := 3( 74 − 34 x). Note that p1 − p22 =

√

p1 is first

1 (x − 1)(192x3 − 320x2 + 101x − 29) ≥ 0 16

√ 3/2 for x ∈ [− 15 , 1], and thus p2 ≤ p1 on [− 15 , 1]. Replacing p1 in the preceding inequality by p1 p2 and expanding the resulting expression gives 3/2

− 4p1

√ 3(−144x6 + 576x5 − 888x4 + 832x3 − 780x2 + 528x − 122) √ −4 3 ≤ (5x − 4)2 (4500x4 − 11,925x3 + 11,415x2 − 9729x + 9128) 3125 ≤ 0.

−

The last inequality follows since the second and third factors are nonnegative for all x ∈ R. Now let (¯s0 , s¯1 ) be that pair which solves p(s0 , s1 ) = 0 and also dp/ds0 = 0. Then a careful examination of the polynomial p(s0 , s1 ) with use of Proposition 80 shows that there is at least one but at most two values of s1 ∈ (−1, 1) with p(s0 , s1 ) = 0. If s1 = s0 , then x = 0 in the polynomial q(x, y), and thus 0 = q(0, y) = −2(5y + 1)(−2 + y)2 . 3 Hence y = s0 = s1 = − 15 , implying p = q = 10 . Consequently, for s0 = s1 = 1 − 5 there exists a fundamental basis (ey1 , ey2 , ey3 ) of C with y1 = (0, 1, 0)t , y2 = 3 3 (0, − 10 , 1)t , and y3 = (1, − 10 , 0)t . The associated scaling functions φ 1 and φ 2 are depicted in Fig. 8.4. Note that the graph of φ 1 is symmetric about the line x = 12 , whereas the graph of φ 2 is symmetric about the line x = 1. Normalizing the vector scaling function φ = (φ 1 , φ 2 )t gives the following matrix coefficients:

 c0 =

√ 3 2 10 1 − 20

 c2 =

0 9 20

4 5√ − 3202

0√

− 3202



 ,

c1 =



 ,

c3 =

√ 3 2 10 9 20

0 1 − 20

 0

√ 2 2

 0 . 0

,

310

Fractal Functions, Fractal Surfaces, and Wavelets

1.0

1.0

0.8

0.8 0.6

0.6

0.4

0.4

0.2

0.2 −0.2 0.2

0.4

0.6

0.8

0.5

1.0

1.5

2.0

1.0

Fig. 8.4 The scaling functions φ 1 (left) and φ 2 (right) for s0 = s1 = − 15 .

The vector scaling function φ constructed above has become known as the GHM scaling vector (after the first letter in the surnames of the three authors of Ref. [82]: Geronimo, Hardin, and Massopust). When referring to this particular scaling vector, we will add the subscript GHM. Before the wavelets are constructed, the regularity and approximation order of the scaling functions φ 1 and φ 2 as defined in Eqs. (8.77a), (8.77b) is briefly discussed. But first a definition is needed. Definition 132. Let I ⊆ R be an interval and let f ∈ C(I). The Hölder exponent αx of f at x ∈ I is defined as 3 log |f (x) − f (y)| αx := lim inf : y ∈ Bε (x) , ε→0 log |x − y| 2

(8.79)

where Bε (x) denotes a ball of radius ε > 0 centered at x ∈ I. The Hölder exponent α of f on I is then defined by α := inf{αx : x ∈ I}.

(8.80)

Definition 133. Let S be a closed subspace of L2 (R) and let E(f ; S) := min{f − sL2 (R) }. s∈S

Let h > 0 and denote by Uh : L2 (R) → L2 (R) the dilation operator defined by Uh f := f (·/h). The subspace S is said to provide approximation order k iff for all f ∈ W 2,k (R) E(f ; Sh ) ≤ Chk f W 2,k (R) ,

(8.81)

where Sh := {Uh s: s ∈ S} and C > 0. Proposition 81. Let si , |si | < 1, i = 0, 1, be a given pair of scaling factors, and let f ∈ C. If |si | < 12 , i = 0, 1, then f is of class Lip1 (R). If, on the other, hand s := max{|si |: i = 0, 1} > 12 and if graph f is not a straight line, then f has Hölder exponent α = − log s/ log 2.

Fractal functions and wavelets

311

Remark 68. The second statement in Proposition 81 was proven in Ref. [138]. Since this proof is not very involved, it is repeated here. Proof. Let i(n) = (i1 i2 . . . in ) ∈ {0, 1}N be a finite code of length n ∈ N, and let a(i(n)) :=

in i2 i1 + 2 + ··· + n. 2 2 2

Let f ∈ C. Since f is the unique fixed point of an RB operator , we have the following identity: f (x) =

n k−1  !

 k−1

s ij 2

k=1 j=1

k  im x + bik − m−1 2

 +

m=1

n !

 s ij f

j=1

2 x− n

n 

 n−m

2

im

m=1

(8.82) for all x ∈ Ia(n) := [a(i(n)), a(i(n)) + 2−n ]. Here bi denotes λi − lin λi , i = 0, 1, with lin representing the linear part of the affine function λi . Now suppose that x, y ∈ Ia(n) with 2−(n+1) ≤ |x − y| ≤ 2−n . Then it follows directly from the preceding formula that |f (x) − f (y)| ≤

n  (2s)k−1 |x − y| + sn C, k=1

where C := 2f ∞ . First assume that s < 12 . Then |f (x) − f (y)| ≤

n  (2s)k−1 |x − y| + (2s)n (2C)|x − y|, k=1

where the fact that 1 ≤ 2n+1 |x − y| was used to obtain the last term in the preceding inequality. Hence |f (x) − f (y)| ≤

1 + 2C |x − y|. 1 − 2s

Now suppose that x and y satisfy 2−(n+1) ≤ |x − y| ≤ 2−n , but x ∈ Ix and y ∈ Iy are not inside the same dyadic interval. Let x0 := Ix ∩ Iy . Then |f (x) − f (y)| ≤ |f (x) − f (x0 )| + |f (x0 ) − f (y)|, and thus |f (x) − f (y)| ≤

2(1 + 2C) |x − y|. 1 − 2s

312

Fractal Functions, Fractal Surfaces, and Wavelets

Since this latter inequality holds for all x, y ∈ [0, 1], f is of class Lip1 (R). If s > 12 , assume without loss of generality that f (0) = f (1) = 0. (This can always f := f + L1 + be achieved by the addition of two linear functions L1 and L2 to f so that  f (0) =  f (1) = 0.) L2 ∈ C and  If x, y ∈ Ia(n) with 2−(n+1) ≤ |x − y| ≤ 2−n , then—as in the previous case—it is easy to see that  |f (x) − f (y)| ≤ s

n  (2s)k−n−1 C+

n



 =s

n

k=1

C+

2rs 1−

 .

1 2s

If one chooses α := log s/ log 12 and notes that 2−(n+1) ≤ |x − y| ≤ 2−n , one finds that |x − y|α ≥ 2−α(n+1) ≥ 2−α sn . Thus  |f (x) − f (y)| ≤ 2

α

C+



2s 1−

1 2s

|x − y|α ,

provided x and y lie in the same dyadic interval of length 2−n . If this is not the case, then as before one finds   2¯s α+1 C+ |x − y|α |f (x) − f (y)| ≤ 2 1 1 − 2s for all x, y ∈ [0, 1]. It remains to be shown that α is the largest possible Hölder exponent. Suppose then without loss of generality that s = |s0 |. Since f vanishes at x = 0 and x = 1, and f ( 12 ) = 0, there exist x0 , y0 ∈ [0, 1] such that f (x0 ) = f (y0 ) and m := (x0 − y0 )/(f (x0 ) − f (y0 )) > 0. (Here use was made of the fact that graph f is not a straight line.) Setting xn := x0 2−n and yn := y0 2−n , n ∈ N, and using Eq. (8.77a), we obtain f (xn ) − f (yn ) =

n−1 

 (2s0 )

k=1

k−1

x0 − y0 2n

 + sn0 (f (x0 ) − f (y0 )).

Therefore  n−2 2 1 − 1 2s0 1 |f (xn ) − f (yn )| = sn0 |f (x0 ) − f (y0 )| 1 + m 2s0 1 − 2s1 

0

≥

sn0 C

≥ C1 |x − y| . α

The next theorem states one of the main results in this section.

Fractal functions and wavelets

313

Theorem 128. Suppose that the scaling factors si , i = 0, 1, give rise to a fundamental basis. Then the space V0 allows approximation order 2. Moreover, if s := max{|si |: i = 0, 1}
12 , then φ a∗ , a = 1, 2, are of class Lipα (R), where α = − log s/ log 2. Proof. Note that to show that V0 has approximation order 2, it suffices to prove that the hat function h: [0, 2] → [0, 1], h = χ[0,1] ∗ χ[0,1] , is in V0 [139]. First, note that for y := (0, 12 , 1)t and any scaling factors si with |si | < 1, i = 0, 1, the affine fractal function fey interpolating {( 2j , yj ): j = 0, 1, 2} is the identity on [0, 1]. Consequently,  h=

   1 1 0∗ 1∗ −p φ +φ + − q φ 0∗ (· − 1). 2 2

If |si | < 12 , i = 0, 1, then Proposition 81 gives the first statement. The second conclusion holds if it can be shown that the graphs of φ a∗ , a = 1, 2, are not straight lines. This is clearly true for φ 1∗ . To show that it is also true for φ 2∗ it suffices to prove that p = 12 = q since only in this case does the graph of φ 2∗ degenerate to a line. However, if p is equal to 12 , then [0,1]

e1 e2 dx =

s0 − 2 = 0. (s0 + s1 − 2)(s0 + s1 − 4)

Hence graph e2 is not a straight line. A similar argument shows that graph e3 is not a straight line. Before commencing with the construction of the wavelets, we compute the Fourier transforms of the previously constructed continuous, compactly supported, and orthogonal scaling functions. To this end, set N := 2 in Eq. (5.58). Using the expressions for the functions λi , i = 0, 1, one shows that 1  ey (ξ ) = 2

[0,1]

−iξ x

λ0 e

1 dx + 2

[0,1]

λ1 e−iξ x dx +

 1 s0 + s1 e−iξ/2  ey (ξ/2). 2

Hence 0∗ (ξ ) = 8 e−iξ/2 φ

 sin2 ξ/2 1  −iξ/2 0∗ + + s e s φ (ξ/2). 0 1 2 ξ2

(8.83)

Setting C(ξ ) := 12 (s0 + s1 e−iξ/2 ) and h1 (ξ ) := 8ξ −2 e−iξ/2 sin2 ξ/2, and iterating the equation yields

314

Fractal Functions, Fractal Surfaces, and Wavelets



0∗ (ξ ) = φ

⎞ ⎛ n ! ⎝ C(2−j+1 ξ )⎠ h1 (2−n ξ ).

(8.84)

j=1

n∈N0

Since |C(2−j ξ )| ≤ 12 (|s0 | + |s1 |) < 1 for all j ∈ N, and since | sin x/x| ≤ 1 for all x ∈ R, the series converges uniformly for all ξ ∈ R. To compute the Fourier transform of φ 1∗ , one writes 1∗ (ξ ) = φ



[0,2]



[0,1]

= =

[0,1]

e−iξ x φ 1∗ (x) dx e−iξ x φ 1∗ (x) dx +

[0,1]

e−iξ x e2 (x) dx + e−iξ

e−iξ(x+1) φ 1∗ (x + 1) dx



[0,1]

e−iξ x e3 (x) dx.

Again using Eq. (5.58), one computes the Fourier transforms of e2 and e3 as 1  e2 (ξ ) = 2



1 e (p − s0 )x dx + 2 [0,1]  1 + s0 + s1 e−iξ/2  e2 (ξ/2) 2 −iξ x/2

[0,1]

e−iξ(x+1)/2 ((1 − s1 − p)x + p) dx

and 1 2

 e3 (ξ ) =

[0,1]

e−iξ x/2 ((q + s0 − 1)x + 1 − s0 ) dx

1 e−iξ(x+1)/2 ((s1 − q)x + q − s1 ) dx 2 [0,1]  1 + s0 + s1 e−iξ/2  e3 (ξ/2). 2 +

Integration gives  e2 (ξ ) = h2 (ξ ) + C(ξ ) e2 (ξ/2) and  e3 (ξ ) = h3 (ξ ) + C(ξ ) e3 (ξ/2), where  h2 (ξ ) :=

(1 − s1 )e−iξ/2 − s0 −iξ



−iξ/2

e

− 4e

−iξ/4 sin ξ/4

ξ



Fractal functions and wavelets

315

and  h3 (ξ ) :=

(s0 − 1 − s1 )e−iξ/2 −iξ

  −iξ/4 sin ξ/4 1 − 4e . ξ

Consequently, 1∗ (ξ ) = φ

 n∈N0

⎛ ⎝

n !

⎞

  C(2−j ξ )⎠ h2 (2−n ξ ) + e−iξ h3 (2−n ξ ) ,

(8.85)

j=1

and this infinite series converges uniformly for all ξ ∈ R. Now the construction of the multiwavelet ψ is presented. This is done in a slightly more general setting than before. It is assumed that φ = (φ 1 , . . . , φ A )t and that ψ = (ψ 1 , . . . , ψ A )t . Furthermore, since supp φ = [0, 2]A , it is also assumed that supp ψ = [0, 2]A . The reason for this approach is that it provides another way of constructing vector scaling functions and multiwavelets; namely, by solving matrix dilation equations. Note that under the preceding assumptions the coefficient matrices {c } ∈Z and {d } ∈Z for φ and ψ are A × A matrices. The orthogonality conditions φ, φ 0,  = δ0, IA ,

∈ Z,

ψ, ψ 0,  = δ0, IA ,

∈ Z,

and

can be reexpressed in terms of these coefficient matrices as 3 

c c† −2  = δ0,  IA

(8.86)

d d† −2  = δ0,  IA .

(8.87)

=0

and 3 

=0

As ψ ∈ W0 ⊆ V−1 , the component functions ψ a |[ j , j+1 ] are affine fractal functions 2 2 interpolating the elements of the set Z/4: The condition that W0 is the orthogonal complement of V0 in V1 means that 3 

=0

c d† −2  = 0,

 ∈ Z,

(8.88)

316

Fractal Functions, Fractal Surfaces, and Wavelets

and the fact that V0 ⊕ W0 = V1 means 3 

c†m−2 cn−2  + d†m−2 dn−2  = δmn IA ,

m, n ∈ Z.

(8.89)

=0

Setting h1 := (c0 , c1 ), h2 := (c2 , c3 ), g1 := (d0 , d1 ), and g2 := (d2 , d3 ), we can rewrite these equations c as h1 h†2 = 0,

(8.90)

h1 h†1 + h2 h†2 = I2A ,

(8.91)

and likewise g1 g†2 = 0,

(8.92)

g1 g†1 + g2 g†2 = I2A , h2 g†1 = 0, h1 g†2 = 0, h1 g†1 + h2 g†2 = 0,

(8.93) (8.94) (8.95) (8.96)

and finally, h1 h†1 + h2 h†2 + g1 g†1 + g2 g†2 = I2A .

(8.97)

The general solution of Eq. (8.90) is given by h†2 = P1 η, where P1 : R2A → R2A denotes an orthogonal projection onto the null space of h1 , and η is an arbitrary 2A × A matrix. Likewise, it follows from Eq. (8.95) that g†2 = P1 ξ is the general solution, with ξ representing an arbitrary 2A × A matrix. Defining h†1 := (I2A − P1 )η, one verifies that Eqs. (8.90), (8.91) are satisfied, provided the matrix η is chosen in such a way that η† η = I2A .

Fractal functions and wavelets

317

If η is chosen as η = h†1 + h†2 , then the above requirement is fulfilled. Eqs. (8.92)–(8.94) now suggest the following choice for g†1 : g†1 = (I2A − P1 )ξ , with ξ † ξ = I2A . Moreover, Eq. (8.96) implies that η† ξ = 0. Suppose that {e1 , . . . , e2A } is an orthonormal basis for R2A . Let η = (η1 η2 . . . ηA ), where ηj , j ∈ NA , denotes the jth column in the matrix η. Define ξ := (η1 η2 . . . ηA eA+1 . . . e2A ). Then one verifies that with this choice of η and ξ , Eq. (8.97) is also satisfied. These arguments have now proven the following theorem. Theorem 129. Let c0 , c1 , c2 , and c3 be given A × A matrices satisfying 3 

c c† −2  = δ0,  IA

=0

for all  ∈ Z. Then there exist B × A matrices d0 , d1 , d2 , and d3 such that Eqs. (8.87)– (8.89) are satisfied. Theorem 129 allows the explicit construction of the wavelet vector ψ for A = 2. To this end, let ⎧ eya1 , ⎪ ⎪ ⎪ ⎨e a , y2 ψ a := ⎪ e ya , ⎪ ⎪ ⎩ 3 eya4 ,

x ∈ [0, 12 ]; x ∈ [ 12 , 1]; x ∈ [1, 32 ]; x ∈ [ 32 , 2];

a = 0, 1.

Here R3  yaj = ψ a ((2(j − 1) + k)/4),

j = 1, 2, 3, 4, k = 0, 1, 2, a = 0, 1.

(8.98)

318

Fractal Functions, Fractal Surfaces, and Wavelets

To ensure continuity, the vectors yaj have to be of the following form: ya1 = (0, ba1 , ba2 )t , ya3 = (ba4 , ba5 , ba6 )t ,

ya2 = (ba2 , ba3 , ba4 )t , ya4 = (ba6 , ba7 , 0)t .

The unknowns ban , n = 1, . . . , 7, are determined so that ψ is orthogonal to φ, its nonzero integer translates, and so that ψ 0 , ψ 1  = 0. In the case when s0 = s1 = − 15 , the inner product formula for affine fractal functions gives

5 a 41 a 5 3 b1 + b2 + ba3 + ba4 , 32 96 32 64 [0,2] 3 5 41 5 ψ a (x)φ 0 (x + 1) dx = ba4 + ba5 + ba6 + ba7 , 64 32 96 32 [0,2] 1 1 3 107 3 ψ a (x)φ 1 (x) dx = − ba1 − ba2 + ba3 + ba4 , + ba5 48 20 16 240 16 [0,2] 1 a 1 a − b6 − b4 , 20 48 3 a 1 1 1 a a 1 ψ (x)φ (x + 1) dx = b1 − ba2 − ba6 − b , 16 20 48 1602 4 [0,2] 1 a 1 1 3 ψ a (x)φ 1 (x − 1) dx = − b4 − ba5 − ba6 + ba7 , 160 48 20 16 [0,2] ψ a (x)φ 0 (x) dx =

(8.99) (8.100)

(8.101) (8.102) (8.103)

and [0,2]

ψ 0 (x)ψ 1 (x) dx =

25 0 1 73 0 1 (b b + b03 b13 + b05 b15 + b07 b17 ) + (b b + b04 b14 + 2b06 b16 ) 96 1 1 192 2 2 5 3 (b1 b0 + b04 b12 + b06 b14 + b16 b04 ) + (b02 b11 + b01 b12 + b03 b12 + 128 4 2 64 + b04 b13 + b03 b14 + b05 b14 + b04 b15 + b06 b15 + b05 b16 + b07 b16 + b06 b17 ) (8.104)

for a = 0, 1. Setting Eqs. (8.99)–(8.104) equal to 0 determines all but 3 of the 14 unknowns baj , a = 0, 1, j = 1, . . . , 7. Two of these three are then used to normalize the integrals of ψ 0 and ψ 1 . One degree of freedom remains because of the existence of a one-parameter family of rotations taking the multiwavelet ψ into other multiwavelets. It is worthwhile remarking that once Eqs. (8.100), (8.102), (8.103) are satisfied, then ψ, ψ 0,  = 0 for all ∈ Z \ {0}. A solution to the preceding equations yielding a normalized vector wavelet ψ with smallest support is given by b01

√ 3 2 =− , 200

b02

√ 9 2 = , 20

b03

√ 273 2 =− , 200

√ b04

=

2 , 2

Fractal functions and wavelets

b05

√ 5 2 = , 200

b06

319

√ 3 2 =− , 20

b07

√ 2 = , 200

and b11 = 0, b15 =

b12 = 0,

48 , 25

b13 =

3 b16 = − , 5

3 , 10

b17 =

b14 = −1, 1 . 50

This then yields the following wavelet matrix coefficients: √

2 20

d0 =  d2 =

0

√ 3 3 20 √ 3 6 10

√  3 6 20 ,

0

√  6 −√ 20 , − 53



√ √  6 − 9203 6√ d1 = , 0 − 33  √  3 −√ 0 60 d3 = . − 306 0

Next it is shown that neither component of the multiwavelet ψ constructed from the vector scaling function φ defined by Eqs. (8.77a), (8.77b) can have support on [0, 1]. More precisely, the minimum length of the support of any of its components is at least 3 2 , and at least one component must have support larger than or equal to 2. Proposition 82. Suppose that the scaling factors s0 and s1 admit a fundamental basis of C. Then there does not exist a multiwavelet whose components are supported on [0, 1]. Proof. Let U1 := V1 ([0, 1]) and let U0 := {f ∈ U1 : f (0) = f (1) = 0}. Note that dim U0 = 5 and dim U1 = 3. To simplify the notation, the following convention is adopted: ϕ 1 := φ 1 |[0,1] ,

ϕ 2 := φ 1 (· + 1)|[0,1] ,

ϕ 3 := ψ 1 |[0,1] ,

ϕ 4 := ψ 1 (· + 1)|[0,1] .

Now suppose ψ is a wavelet supported on [0, 1]. Since the scaling functions and the wavelets are mutually orthogonal, span {ϕ 1 , ϕ 3 }, span {ϕ 2 , ϕ 4 }, span {φ 0 }, and span {ψ} are mutually orthogonal spaces. If, in addition to {ϕ 2 , ϕ 4 } being linearly independent, {ϕ 1 , ϕ 3 } are also linearly independent, then span {φ 0 , ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 } is five-dimensional and hence equals U1 . But then ψ ∈ span {φ 0 , ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 }⊥ = {0}, which is not permissible. Therefore either ϕ 1 and ϕ 3 or ϕ 2 and ϕ 4 are linearly dependent. Assume that the former is true. Then two cases need to be considered: ϕ 3 = 0 and ϕ 3 = ϕ 1 (the latter can always be achieved by our scaling ψ 1 ). If ϕ 3 = 0, then ϕ 4 ∈ U0 , and thus {φ 0 , ϕ 4 , ψ} forms an orthogonal basis of U0 . Since ϕ 1 is orthogonal to these three vectors, ϕ 1 ∈ U0⊥ .

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Fractal Functions, Fractal Surfaces, and Wavelets

In the other case, ϕ 5 := ϕ 2 − ϕ 4 ∈ U0 . As φ 1 = ψ, ϕ 5 = 0. Hence {φ 0 , ϕ 5 , ψ} forms an orthogonal basis of U0 and, again, ϕ 1 ∈ U0⊥ . The conclusion of the theorem follows if we can show that it is not possible for ϕ 1 to be in U0⊥ . To this end, let ϕ 6 := ϕ 1 (2 · −1) = φ 1 (2 · −1)|[0,1] . Note that ϕ 6 = ϕ 1 but 6 ϕ (0) = ϕ 1 (0) and ϕ 6 (1) = ϕ 1 (1). Since span {φ 0 (2·), φ 0 (2 · −1), φ 1 (2·)} = U1 , it follows that ϕ 6 ∈ U0⊥ . / U0⊥ . This contradiction shows Now, ϕ 1 − ϕ 6 ∈ U0 and thus ϕ 1 = ϕ 6 + (ϕ 1 − ϕ 6 ) ∈ 1 3 that ϕ and ϕ cannot be linearly dependent. A similar argument shows that ϕ 2 and ϕ 4 cannot be linearly dependent. The preceding Proposition 82 implies the next theorem. Theorem 130. Let s0 and s1 be scaling factors that admit the existence of a fundamental basis of C, and let ψ = (ψ 0 , ψ 1 )t be a vector wavelet. Then the support of one of the components of ψ must have length greater than or equal to 32 , whereas the support of the other one must be of length greater than or equal to 2. Proof. The preceding Proposition 82 shows that the support of each component ψ a must have length at least 32 . Note that ψ 0 and ψ 1 cannot both be supported on [0, 32 ]. In this case ψ 0 |[0, 1 ] and ψ 1 |[0, 1 ] , or ψ 0 |[1, 3 ] and ψ 1 |[1, 3 ] would have to be linearly 2 2 2 2 dependent. But then a wavelet supported on [0, 1] could be found by our simply rotating the components. This contradiction proves the assertion. It remains to establish that there is a multiwavelet whose components are supported on [0, 32 ] and [0, 2], respectively. So suppose that supp ψ 0 = [0, 32 ] and supp ψ 1 = [0, 2]. In this case the wavelet matrix coefficients d , = 0, 1, 2, 3, take the form 

d d0 = 1 0  d d2 = 7 d8

 d2 , 0  0 , d8



 d3 d4 d1 = , d5 d6   0 0 d3 = . d10 d11

Defining a := c0 d†2 + c1 d†3 ,

b := c2 d†0 + c3 d†1 ,

c := c0 d†0 + c1 d†1 + c2 d†2 + c3 d†3 ,

and using the explicit form of the scaling function matrix coefficients c , = 0, 1, 2, 3, we obtain for the components (Aij ) of a, (Bij ) of b, and (Cij ) of c A11 =

1 (2s0 + 1 − 2p)d7 , 2

B22 = (1 − q)(q − s1 )d5 ,

1 (1 − 2p)(p − s0 )d7 , 2 1 = (2s1 + 1 − 2q)d5 . 2

A21 = C12

Note that if one sets d7 = 0, then the preceding equations yield p = 12 and s0 = 0. These values for p and s0 now imply that 12 = (1 + s21 )/(4 − s21 ). This last equation has, however, no solution for |s1 | < 1. Likewise, one can show that if d5 were equal

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to zero, then s0 would be outside the admissible domain. Therefore d5 = d7 = 0. For ψ 0 , ψ 1  = 0, d4 d6 = 0. Hence one of them must have a support of length 1, which is impossible by Proposition 82. Next the question of the symmetry of wavelets is briefly addressed. It was shown in Ref. [109] that the Haar wavelet is the only symmetric wavelet in the Daubechies family of continuous, compactly supported, and orthogonal wavelets. One of the advantages of the fractal-geometric construction of wavelets over other constructions is that symmetric wavelets exist. The next theorem makes this statement more precise. Theorem 131. Let φ be a vector scaling function whose components are affine fractal functions with s0 = s1 . Then there exists a set of interpolation points such that the components of the associated multiwavelet are symmetric or anti-symmetric affine fractal functions. Proof. Since s0 = s1 , both scaling factors must be equal to − 15 . Furthermore, if ψ := (ψs , ψa )t , ψs is symmetric about the line x = 1 and ψa is antisymmetric with respect to it. Bearing this in mind, one sets ys1 = (0, bs1 , bs2 )t , ya1 = (0, ba1 , ba2 )t ,

ys2 = (bs2 , bs3 , bs4 )t , ya2 = (ba2 , ba3 , 0)t ,

ys3 = (bs4 , bs3 , bs2 )t ,

ys4 = (bs2 , bs1 , 0)t ,

ya3 = (0, −ba3 , −ba2 )t , ya4 = (−ba2 , −ba1 , 0)t.

Then Eq. (8.102) is automatically satisfied and Eqs. (8.99)–(8.101) yield bs1 = 1,

bs2 = −30,

bs3 = 111,

ba1 = 1,

ba2 = −30,

ba3 = 81.

bs4 = −100

and

If the resulting wavelets ψ s and ψ a are normalized, then the wavelet matrix coefficients {d } are given by 

1 −√ 20 d0 = − 2  20

d2 =

9 20√ − 9202

√  − 3202 , 3 − 10 √  − 3202 3 10

 d1 = ,

d3 =

9 20 √ 9 2  20 1 −√20 2 20

√  − 22 , 0  0 . 0

The graphs of ψs and ψa are shown in Fig. 8.5. The multiwavelet ψ = (ψs , ψa ) has been called in the literature the DGHM multiwavelet (after the first letter in the surnames of the four authors of Ref. [137]: Donovan, Geronimo, Hardin, and Massopust). We will add the subscript DGHM when referring to this particular multiwavelet. For certain applications such as sampling on the interval there arises the need to construct scaling functions and wavelets on a compact interval. For the Daubechies family of scaling functions and wavelets

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100 50 50

0.5

1.0

1.5

0.5

2.0

−50

1.0

1.5

2.0

−50

−100

Fig. 8.5 The symmetric and antisymmetric wavelets ψs (left) and ψa (right).

this cannot be done without the introduction of boundary functions [130]. For the scaling vectors φ = (φ 1 , φ 2 ) and multiwavelets ψ = (ψ 1 , ψ 2 ) constructed above, the restriction to an interval poses no problems and can be done without our having to add boundary functions to preserve orthogonality at and near the boundary points. The reason why it is possible for φ and ψ lies in the fact that the four functions are piecewise fractal interpolation functions defined on a compact interval. To this end, choose any level of approximation, say, k = 0, as your lowest level and define a V k := Vk ∩ L2 [0, 1], k ∈ Z+ 0 , and φ k := φk |[0,1] , a

a = 0, 1.

With these definitions one shows that {V k }k∈Z+ is a multiresolution analysis on L2 [0, 1] 0

and that {φ k : a = 0, 1 ∧ ∈ Z} is an orthonormal basis for V k , k ∈ Z+ 0 . Now define a

φka := φ a |[k−1,k] and ψka := ψ a |[k−1,k] ,

a = 0, 1; k = 1, 2.

 Let R :=

 a b , |a|2 + |b|2 = 1, be a rotation in R2 so that −b a

ψ + := R ◦ ψ satisfies ψk+,0 , φk1  = 0,

k = 1, 2.

(8.105)

To see that such a rotation exists, note that the preceding equation is equivalent to our requiring that aψ10 , φ11  + bψ11 , φ11  = 0 and aψ20 , φ21  + bψ21 , φ21  = 0.

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These last two equations are not linearly independent since ψ 0 , φ 1  = ψ10 , φ11  + ψ20 , φ21  = 0 and ψ 1 , φ 1  = ψ11 , ψ11  + ψ21 , φ11  = 0. Thus there exist numbers a and b such that Eq. (8.105) is satisfied. For k ∈ Zk0 , define  0 ψ k

+,0 ∩ [0, 1]c = 0; for ψk

otherwise

0 +,0 ψk

:=

and

1

1 ψ k := ψk

|[0,1] .

(8.106)

If W k := V k+1 " V k , then it follows that the collection of functions 

ψ k := (ψ k , ψ k )t : k ∈ Z+ 0 ∧ ∈Z 0

1



forms an orthonormal basis of W k . These arguments have now established the following theorem. Theorem 132. The collection of functions   a φ k : a = 0, 1 ∧ k ∈ N0 ∧ ∈ {−a, . . . , 2k − 1} is an orthonormal basis for V k = Vk ∩ L2 [0, 1], and the family of functions   a ψ k : a = 0, 1 ∧ k ∈ N0 ∧ ∈ {a − 1, . . . , 2k − a − 1} W k = V k+1 " V k . Furthermore, we have the forms an orthonormal basis for5 W k. decomposition L2 [0, 1] = V 0 ⊕ k∈N0

Remark 69 0

1. In the case s0 = s1 = − 15 , we see that ψ = ψs |[0,1] and  1

ψ =

ψa |[0,1] 0

for supp ψa ⊆ [0, 1]; otherwise.

2. The GHM scaling vector φ GHM is interpolatory in the following sense: given a set of interpolation points Y := {Yk }k∈K for some at most countable index set K supported on 1 Z, there exists a set of vector coefficients {α } ⊂ R2 such that k 2 

α tk φ GHM (x − k)

k∈K

interpolates Y.

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4 Wavelets are piecewise fractal functions In this section we will present an idea and procedure to generate orthogonal generators for a refinable shift-invariant space. This idea and procedure were originally introduced and explored in Ref. [140] and then investigated further in, for instance, Refs. [141–144]. To this end, reconsider the hat function explicitly defined by h: [−1, 1] → [0, 1] h(x) := (1 − |x|)+ := max{1 − |x|, 0}. The hat function is—by definition—a linear cardinal spline with knot sequence {(−1, 0), (0, 1), (1, 0)}. Define a principal shift-invariant space by V0 [h] := clL2 (R) span{h(· − ): ∈ Z}. As h(x) =

1 1 h(2x − 1) + h(2x) + h(2x + 1) 2 2

and has linearly independent shifts, the space V0 [h] is refinable and h generates an multiresolution analysis for L2 (R). However, this multiresolution analysis is not orthogonal in the sense that the shifts of h are not orthogonal. Indeed,

1 −1

h(x)h(x ± 1) dx =

1 = 0. 6

The idea is now to try to find another function, say, w, supported on [0, 1] which will be used to modify h so that this modified h together with w generates a shift-invariant space where all the shifts are orthogonal. More precisely, let w ∈ L2 (R) have support [0, 1] and consider the finitely generated shift-invariant space V[h, w]. Try to find a function u ∈ L2 (R) with the properties: 1. 2. 3. 4.

supp u = [−1, 1]; u is a linear combination of h, w, and w(· + 1); u is orthogonal to its translates u(· ± 1) and to w; and V[h, w] = V[u, w].

Define u(x) := (I − projV[w] )h = h −

h, w h, w(· + 1) w− w(· + 1). w, w w, w

Here I denotes the identity operator on L2 (R), projV[w] denotes the projection onto the shift-invariant space V[w], and ·, · denotes the L2 (R) inner product.

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The unknown function w is already orthogonal to its shifts and so we need only 0 = u, u(· − 1) = h, h(· − 1) −

h, ww, h(· − 1) . w, w

Hence the function w has to satisfy the following condition: h, h(· − 1) =

h, ww, h(· − 1) . w, w

(8.107)

Imposing that V0 := V[h, w] is to be refinable implies that w(·/2) ∈ V0 ; that is, w(·/2) must be a linear combination of h(· − 1), w, and w(· − 1): w(·/2) = h(x − 1) + s0 w(x) + s1 w(· − 1).

(8.108)

Note, however, that Eq. (8.108) is the fixed-point equation for an affine fractal function provided |s0 |, |s1 | < 1. (In our previous terminology, λ0 (x) = x and λ1 (x) = 2 − x.) Our choosing s0 = s1 =: s causes w to be symmetric about the line x = 12 . Employing Proposition 56 to compute the inner products of w itself, h and h(· − 1), as well as the integral over w yields 1 , 2(1 − s) 2+s w, w = , 6(1 − s)2 (1 + s) w, 1 =

h, w = h(· − 1), w =

1 . 4(1 − s)

Substitution into Eq. (8.107) together with h, h(· − 1) = Normalizing u and w yields φ 1 := √

w w, w

and

φ 2 := √

1 6

gives for s the value − 15 .

u , u, u

reproducing the GHM scaling vector φ GHM . Similar arguments can be used to construct, for example, piecewise quadratic orthonormal scaling vectors [144]. A construction to obtain so-called biorthogonal scaling vectors and multiwavelets is presented in, for example, Ref. [145]. In the same spirit one can show that the Daubechies scaling function 2 φ is a piecewise fractal function consisting of the sum of the linear spline L: R → R defined by ⎧ x if 0 ≤ x ≤ 1; ⎪ ⎪ √ ⎪√ ⎨ ( 3 − 3)x + (4 − 3) if 1 ≤ x ≤ 2; √ √ L(x) := ⎪ (2 − 3)x + 3( 3 − 2) if 2 ≤ x ≤ 3; ⎪ ⎪ ⎩ 0 otherwise

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and the piecewise fractal function depicted in Fig. 8.6 [128]. 0.2 0.1 0.5

1.0

1.5

2.0

2.5

3.0

−0.1 −0.2 −0.3 −0.4

Fig. 8.6 The piecewise fractal function needed to obtain a refinable generator.

Note that √ L is the linear spline interpolating the sequence of knots {(0, 0), (1, 1), (2, −2 + 3), (3, 0)} (see p. 280) and can also be expressed in the form 7 6 √ L(x) = h(x − 1) + (−2 + 3)h(x − 2) χ[0,3] (x). It is shown in Ref. [128] that the scaling vector φ GHM and the Daubechies scaling function 2 φ are the only refinable generators of a shift-invariant space with approximation order 2 and local dimension 3. The former is a finitely generated shiftinvariant refinable space, whereas the latter is a principal shift-invariant refinable space. Definition 134. A finitely generated shift-invariant space V has accuracy k iff V ⊃ Pk . Suppose that V = V[φ] has accuracy k. If the generators φ := {φ1 , . . . , φn } have linearly independent shifts and are compactly supported functions in Lp (R), 1 ≤ p ≤ ∞, then V provides Lp -approximation order k [146]. Definition 135. Suppose that V ⊂ L2 (R) is a refinable shift-invariant space and Vj := 2j V = {f (2j ·): f ∈ V}, j ∈ N. Then V provides Lp -approximation order k if for every sufficiently smooth function g ∈ Lp (R), inf {f − gLp } ≤ c2−jk ,

g∈Vj

where c is a positive constant depending only on f . The careful reader may have noticed that the above construction of φ GHM can be done by our shifting the generator h so that it generates a basis on the interval [0, 1]. This basis is given by the three functions {h|[0,1] , h(· + 1)|[0,1] , w}. Note that h|[0,1] and h(· + 1)|[0,1] are affine fractal functions with collinear interpolation points. Hence the collection {h|[0,1] , h(·+1)|[0,1] , w} is a fractal basis for S2 [0, 1] (cf. Proposition 46 with d := 2 =: N). To this end, we define the following spaces.

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Definition 136 (Ref. [128]). Let d ∈ N0 and let p: R → R, x → xd χ[0,1] , the restriction of the polynomial x → xd to the unit interval [0, 1]. Define Pd := span{pd : d ∈ N0d }. We remark that, on the basis of the above observation, a fractal basis for S2 [0, 1] is given by {p0 , p1 , w} with w ∈ F |[0, 1], where  := {(0, 0), ( 12 , 1), (1, 0)} and the scaling factors are |s0 |, |s1 | < 1. Definition 137 (Ref. [128]). Let V be a refinable shift-invariant space. The restriction of V to [0, 1] is defined by V := Vχ[0,1] = {f χ[0,1] : f ∈ V}. The dimension of V is called the local dimension of V. The following two results were established in Jia [147, 148] and Hardin and Hogan [149], respectively: 1. A refinable space V has accuracy k; that is, Pk ⊂ V iff Pk ⊂ V. 2. V = V[φ], φ compactly supported, has approximation order k iff Pk ⊂ V.

Notice that both φ GHM and 2 φ have accuracy/approximation order 2 and local dimension 3. The next result, which is proven in Ref. [128], shows that all refinable, compactly supported, and continuous scaling vectors φ are piecewise fractal interpolation functions. The setting is more general than the one we have considered so far. The results in Ref. [128] are valid for hidden-variable fractal interpolation functions of the type mentioned in (2) of Remark 50 with Xk := [0, 1] and Y := Rn . We state the theorem under these assumptions. The current setting is obtained by our choosing n := 1. Theorem 133. Suppose that φ is a refinable, compactly supported, and continuous generator of a refinable shift-invariant space V. Assume that V has accuracy k ≥ 2 and local dimension m. Then V = Sk [0, 1] for some pair of matrices s0 , s1 ∈ R(m−2)×(m−2) with joint spectral radius less than 1. Proof. See Ref. [128] for the setting and proof.

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9

Abstract This chapter deals with Rm -valued multivariate continuous fractal functions f : X ⊆ Rn → Rm and some of their properties. The existence and construction of these multivariate functions is already implicitly contained in Theorems 71–73; simply choose X to be a nonempty compact subset of Rn and Y := Rm . Here, however, the issues have to be readdressed. There are two reasons for this: firstly, the more complex geometry is hidden within the construction and needs to be investigated more closely; secondly, the graphs of these Rm -valued multivariate continuous fractal functions, the so-called fractal surfaces, can be used to construct wavelet bases in Rn . This construction is based on results from the theory of Coxeter and affine reflection groups, and certain issues involving the geometry of fractal surfaces need to be clarified. Chapter 10 will deal exclusively with this last question. Fractal surfaces constructed via iterated function systems were first systematically introduced in Massopust [150]. The fractal surfaces introduced there required planar boundary conditions but used different vertical scaling factors. A different construction of fractal surfaces was later presented in Geronimo and Hardin [151]. This construction used arbitrary boundary values but only one vertical scaling factor. Rm -valued multivariate fractal functions f : X ⊆ Rn → Rm were investigated in Hardin and Massopust [106]. The latter two constructions use recurrent iterated function systems. Of course, it is always possible to construct fractal surfaces as tensor products of univariate continuous fractal functions. However, these tensor product fractal surfaces lack most of the exciting features of the aforementioned fractal surfaces. Dubuc and his coworkers have also constructed fractal surfaces using a multidimensional iterative interpolation process. Other constructions of fractal surfaces can be found in the literature; for instance, in [152– 154, 156, 157]. The first section in this chapter introduces tensor product fractal surfaces. The construction of nontensor product fractal surfaces is presented next. This construction is based on recurrent iterated function systems and emphasizes the interpolatory nature of fractal surfaces in Rn+m . The special case m = 1 is then considered as an illustrative example. A few comments are made about Dubuc’s fractal surfaces. In the third section, properties of fractal surfaces such as Hölder continuity, oscillation, box dimension, and regularity are discussed. The final section deals with fractal surfaces of higher smoothness; that is, fractal surfaces of class Ck , k ∈ N.

1 Tensor product fractal surfaces Here it is shown how the tensor product can be used to obtain fractal-like surfaces from univariate R-valued continuous fractal functions. Fractal Functions, Fractal Surfaces, and Wavelets. http://dx.doi.org/10.1016/B978-0-12-804408-7.00009-6 Copyright © 2016 Elsevier Inc. All rights reserved.

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Fractal Functions, Fractal Surfaces, and Wavelets

Before we commence with the construction, the definition of a tensor product is needed. The definition that will be given here is not the most general one but it is more than adequate for our purposes. Definition 138. Let X and Y be K-vector spaces of dimension n and m, respectively. With the spaces X and Y is associated an nm-dimensional vector space X ⊗ Y in the following way: Let x ∈ X and y ∈ Y, and let θ: X × Y → X ⊗ Y, θ

(x, y) −→ x ⊗ y be a mapping that is required to have the following three properties: 1. Distributive law: If x, x1 , x2 ∈ X and y, y1 , y2 ∈ Y, then x ⊗ (y1 + y2 ) = x ⊗ y1 + x ⊗ y2 , (x1 + x2 ) ⊗ y = x1 ⊗ y + x2 ⊗ y. 2. Associative law: If x ∈ X, y ∈ Y, and α ∈ K, then αx ⊗ y = x ⊗ αy = α(x ⊗ y). 3. If {x1 , . . . , xn } and {y1 , . . . , ym } are bases of X and Y, respectively, then the nm elements xi ⊗ yj ,

i = 1, . . . , n, j = 1, . . . , m,

are to form a basis of X ⊗ Y.

If such a mapping exists, then the vector space X ⊗ Y is called the tensor product of X with Y. Example 56. An example of a tensor product is provided by the linear spaces P d (R). Suppose that d = 1. Then {e1 := 1, e2 := idR } is a basis for the vector space P 1 (R). The tensor product P 1 (R) ⊗ P 1 (R) is then the vector space of all bivariate real polynomials p of the form p(x, y) =

1 1  

aij xi yj ,

i=0 j=0

where xi yj := (ei ⊗ ej )(x, y) and the aij , i, j = 0, 1, are real numbers. In what follows the notation and terminology of Section 2, are used. Let X := [x0 , xN ] be a nonempty compact interval of R, and let 1  and 2  be finitedimensional dilation-invariant subspaces of BN . Define spaces 1 r and 2 s by     1 r := θ 1  ∩ Cr (X) and 2 s := θ 2  ∩ Cs (X), r, s ∈ N0 .

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Furthermore, let {f 1 , . . . , f n } and {g1 , . . . , gm } be bases for these spaces, and let f and g be fractal functions generated by the maps f

f

f

λf = (λ0 , λ1 , . . . , λN−1 ) ∈ 1 

and

g

g

g

λg = (λ0 , λ1 , . . . , λN−1 ) ∈ 2 ;

that is, f =

n 

αi f i

and

i=1

g=

m 

βj gj .

j=1

Then the tensor product of f with g, f ⊗ g, is the bivariate fractal function f ⊗g=

n  m 

αi βj f i ⊗ gj .

(9.1)

i=1 j=1

The graph of the fractal function f ⊗ g is called a tensor product surface. 1 fractal 1 2 Example 57. Let N := 2, let X := [0, 1], let 1  := k=0 P =: , and let r := 0 =: s. Then the fractal functions f and g are affine. The space   1  1 P ∩ C(X) θ k=0

is thus three-dimensional, and a basis is, for instance, given by {f 1 , f 2 , f 3 }, where f i is the affine fractal function that interpolates the set of points 

1 , δ2i , (1, δ3i ) , i = 1, 2, 3. (0, δ1i ) , 2 If the scaling factors are denoted by s0 and s1 , then the following expressions for the functions λ ∈ B2 are obtained: f 1 :λ1 = ((s0 − 1)(idR − 1), s1 (idR − 1)) , f 2 :λ2 = (idR − s0 , −idR + 1 − s1 ) , f 3 :λ3 = (−s0 (idR + 1), (1 − s1 )idR − s1 ) . The tensor products f i ⊗ f j satisfy the functional equations fi ⊗ fj =

1  

−1 i λik ◦ u−1 k + sk f ◦ uk



 j −1 j + s f ◦ u λ ◦ u−1 χuk X χu X ,  k 

k,=0

with uk = 12 idR + (k − 1). Note that if (x, y, z) is a coordinate system in R3 and if the graph of f ⊗ f is analytically represented by z = f ⊗ f (x, y), then the intersection of graph f ⊗ f with the hyperplane y = c, c ∈ [0, 1], is a rescaled version of graph f ⊗ f .

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2 Affine fractal surfaces in Rn +m In this section nontensor product fractal surfaces defined on certain polyhedra in Rn are introduced. The construction of such surfaces is given in a general setting; that is, recurrent iterated function systems (IFSs) are used. Although some of the first results stated in this section follow from earlier theorems in Chapter 5, their proofs are repeated here. The reason for this repetition lies in the fact that these proofs explicitly use the underlying geometry of the construction, an issue that will be taken up again in more detail and more generality in Chapter 10. But first a few definitions are needed. Definition 139. Let {q0 , q1 , . . . , qn } be a set of geometrically independent points in Rk , k ≥ n. The n-dimensional geometric simplex, or n-simplex, denoted by σ n , is defined as ⎧ ⎫ n n ⎨ ⎬   αj qj ∧ αj = 1 . σ n = x ∈ Rn : ∃ α0 , α1 , . . . , αn ∈ R+ so that x = ⎩ ⎭ j=0

j=0

(9.2) The points q0 , q1 , . . ., qn are called the vertices of the n-simplex. A simplex σ m is called a face of the simplex σ n , m ≤ n, if each vertex of σ m is also a vertex of σ n . Definition 140. Two simplices σ n and σ m are called properly joined if either σ n ∩ m σ = ∅ or σ n ∩ σ m is a face of both σ n and σ m . Definition 141. A simplicial complex in Rn is a finite family τ n of properly joined p-geometric simplices σ p , p ≤ n, such that each face of σ p is also a member of τ n . The n-simplex σ n endowed with the Euclidean subspace topology of Rk is denoted by |σ n | and is called the geometric carrier of σ n .

2.1 The construction Let X be a polyhedron made up of finitely many n-simplices σin ⊆ Rn , i ∈ NN . Denote by Q the set of all vertices qj ∈ X, j ∈ NM of X. Let := {(qj , zj ) ∈ X × Rm : j ∈ NM } be an interpolating set. Let τkn be an n-simplicial complex in X that is a union of some of the σkn , k ∈ NK . After relabeling—if necessary—we denote the vertices of τkn by q1 , . . . , qL . A function : NM → NL is called a labeling map if whenever qj1 , . . . , qjn+1 are the vertices of some σin , then q(j1 ) , . . . , q(jn+1 ) are the vertices of some τkn . Now let ui : Rn → Rn be the unique affine map such that ui (q(j) ) = qj

for all qj ∈ σin ,

(9.3)

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333

i ∈ NN . The maps ui can be represented as ui (x) = Ai x + Di ,

(9.4)

where Ai ∈ Mn (R), the unitary noncommutative associative algebra of all invertible n × n matrices over R, and Di ∈ Rn . Let B ∈ Mm (R) and suppose that the spectral radius s of B is less than 1. Note that there exists a norm  · B on Rm such that the induced matrix norm of B equals s. Let vi : Rn × Rm → Rm be the unique affine map of the form vi (x, y) = Ci x + By + Ei ,

(9.5)

where Ci ∈ Mmn (R), Ei ∈ Rm , are such that vi (q(j) , z(j) ) = zj

(9.6)

for all j such that qj ∈ σin , and for all i ∈ NN . Let C∗ (X, Rm ) := {f ∈ C(D, Rm ): f (qj ) = zj ∧ j ∈ NM }. Define a norm  · ∞ on C∗ (X, Rm ) by f ∞ := sup{f (x)B : x ∈ X} and let : C∗ (X, Rm ) → C(D, Rm ) be the Read-Bajraktarevi´c operator (RB operator) defined by ( f )(x) =

N 

−1 n vi (u−1 i (x), f ◦ ui (x))χσi .

(9.7)

i=1

Theorem 134. The mapping in Eq. (9.7) is well defined, maps C∗ (X, Rm ) into itself, and is contractive in  · ∞ .   Proof. Clearly (f ) is continuous on each σin . Let φ i := (f ) σin , i ∈ NN . Suppose σin and σin intersect along a face; that is, σin ∩ σin = F, where F is a p-simplex with p < n. To prove that is well defined, it suffices to show that φ i (F) = φ i (F). Note that φ i (qj ) = zj = φ i (qj ) for each vertex qj ∈ F. But Eq. (9.4) and the fact that each x ∈ F is a linear combination of the vertices of F imply that φ i (x) = φ i (x) for all x ∈ F. By Eqs. (9.4), (9.6) we have that (f )(qj ) = zj , j ∈ NM . Therefore maps C∗ (X, Rm ) into itself. Now let f , g ∈ C∗ (X, Rm ). Then

334

Fractal Functions, Fractal Surfaces, and Wavelets

 (f ) − (g)∞ −1 −1 −1 = sup {vi (uu−1 i (x), f (ui (x))) − vi (ui (x), g(ui (x)))B : x ∈ X} 1≤i≤N

−1 = sup {B(f (u−1 i (x)) − g(ui (x)))B : x ∈ X} ≤ sf − g∞ . 1≤i≤N

The unique fixed point f ∈ C∗ (X, Rm ) is called an Rm -valued multivariate (continuous) affine fractal function and its graph is called an affine fractal surface in Rn+m . Note that f ∈ RF (X, Rm ). Furthermore, graph f is the attractor of the recurrent IFS (X × Rm , w), where 



x ui (x) wi (i ∈ NN ). = vi (x, y) y

(9.8)

The fractal surfaces constructed previously can be used to define hidden-variable fractal surfaces. To this end, assume that ui = Ai x + Di , where Ai is a similitude of norm Ai  = ai , and vi (x, y) = Ci x + By + Ei , where B = s for an isometry , i ∈ NN . Consider the components of the fixed point f = (f ,1 , . . . , f ,m )t . The graph of f ,j is the orthogonal projection of f onto Rn ×0×· · ·×R×· · ·×0, where the factor R is in the jth position. Since fj, still depends continuously on all the variables, it is referred to as a hidden-variable multivariate fractal function, and its graph is called a hidden-variable fractal surface.

2.2 Fractal hypersurfaces in Rn+1 Here we construct a class of special attractors of IFSs; namely, attractors that are the graphs of bounded functions f :  ⊂ Rn → R, where  ∈ H(Rn ), n ∈ N. To this end, we assume that {ui :  → }: i ∈ NN is a family of bounded bijective mappings with the following property: (PS) {ui (): i ∈ NN } forms a set-theoretic partition of ; that is,  = uj () = ∅ for all i = j ∈ NN .

We introduce the set B() := B(, R) := {f :  → R: f is bounded on } and endow it with the metric d(f , g) := sup|f (x) − g(x)|. x∈

Then (B(), d) is a complete metric linear space.



i∈NN

ui () and ui () ∩

Fractal surfaces

335

For i ∈ NN , let vi : i × R → R be a mapping that is uniformly contractive in the second variable; that is, there exists an L ∈ [0, 1) so that for all y1 , y2 ∈ R |vi (x, y1 ) − vi (x, y2 )| ≤ L|y1 − y2 |,

∀x ∈ .

(9.9)

−1 vi (u−1 i (x), f ◦ ui (x))χui () (x).

(9.10)

Define an RB operator : B() → R by

f (x) :=

N  i=1

Note that is well defined, and since f is bounded and each vi is contractive in the second variable, f is again an element of B(). Moreover, by Eq. (9.9) we obtain for all f , g ∈ B() the following inequality: d( f , g) = sup | f (x) − g(x)| x∈ −1 −1 −1 = sup |v(u−1 i (x), fi (ui (x))) − v(ui (x), gi (ui (x)))| x∈

−1 ≤ L sup |fi ◦ u−1 i (x) − gi ◦ ui (x)| ≤ Ld(f , g).

(9.11)

x∈

 To simplify the notation, we set v(x, y) := i∈NN vi (x, y)χui () (x) in Eq. (9.11). In other words, is a contraction on the complete metric linear space B() and, by the Banach fixed-point theorem, therefore, has a unique fixed point f in B(). This unique fixed point will be called a multivariate real-valued fractal function (for short, fractal function) and its graph will be called a fractal hypersurface of Rn+1 . Next we consider a special choice for the mappings vi . Define vi :  × R → R by vi (x, y) := λi (x) + Si (x)y,

i ∈ NN ,

(9.12)

where λi ∈ B() and Si :  → R is a function. Then vi given by Eq. (9.12) satisfies condition (9.9) provided that the functions Si are bounded on  with bounds in [0, 1), for then |vi (x, y1 ) − vi (x, y2 )| = |Si (x)y1 − Si (x)y2 | = |Si (x)| · |y1 − y2 | ≤ Si ∞, |y1 − y2 | ≤ s|y1 − y2 |. Here we denoted by  · ∞, the supremum norm on  and defined s := max{Si ∞, : i ∈ NN }. Thus for a fixed set of functions {λ1 , . . . , λN } and {S1 , . . . , SN }, the associated RB operator (9.10) now has the form

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Fractal Functions, Fractal Surfaces, and Wavelets

f =

N 

λi ◦ u−1 i χui () +

i=1

N 

−1 (Si ◦ u−1 i ) · (f ◦ ui )χui () ,

(9.13)

i=1

or equivalently

f ◦ ui = λi + Si · f on  and for all i ∈ NN . Thus we have arrived at the following result. Theorem 135. Let  be an element of H(Rn ) and suppose that {ui :  → }i∈NN is a family of bounded bijective mappings satisfying property (PS). Further suppose that the vectors of functions λ := (λ1 , . . . , λN ) and S := (S1 , . . . , SN ) are elements of N

BN () := X B(). i=1

Define a mapping : BN () × BN () × B() → B() by

(λ)(S)f =

N 

λi ◦ u−1 i χui () +

i=1

N  −1 (Si ◦ u−1 i ) · (f ◦ ui )χui () .

(9.14)

i=1

If the quantity s = max{Si ∞, : i ∈ NN } < 1, then the operator (λ)(S) is contractive on the complete metric linear space B() and its unique fixed point f satisfies the self-referential equation f =

N 

λi ◦ u−1 i χui () +

i=1

N  −1 (Si ◦ u−1 i ) · (f ◦ ui )χui () ,

(9.15)

i=1

or equivalently f ◦ ui = λi + Si · f

(9.16)

on  and for all i ∈ NN . Remark 69. The fractal function f :  → R generated by the RB operator defined by Eq. (9.14) depends on the two N-tuples of bounded functions λ, S ∈ BN (). The fixed point f should therefore be written more precisely as f (λ)(S). However, for notational simplicity we usually suppress this dependence for both f and . Remark 70. The above construction can also be extended to local IFSs yielding local fractal surfaces. One considers subsets i of , defines a local IFS (, (i , ui )), and requires condition (PS) to hold for {ui (i )}. The details are left to the reader.

Fractal surfaces

337

Now assume that the vector of functions S is fixed. Then the following result found in Ref. [82] and in more general form in Ref. [83] describes the relationship between the vector of functions λ and the fixed point f generated by it. It is a special case of Theorem 76 but we state it again for our special case. Theorem 136. The mapping λ → f (λ) defines a linear isomorphism from BN () to B(). We will see later that this theorem allows us to define bases for multivariate fractal functions (see also Refs. [78, 104, 135]).

2.3 Affinely generated fractal surfaces In this section we specialize our construction of fractal functions even further. It is our goal to obtain continuous fractal hypersurfaces that are generated by affine mappings λi :  → R on specially chosen domains  ⊂ Rn (ie, n-simplices) and by constant functions Si := si ∈ (−1, 1). This type of fractal surface was first systematically introduced in Ref. [150] and was generalized in Ref. [151]. All these constructions are based on use of certain types of simplicial regions and affine mappings for . Later, different types of fractal surface constructions not necessarily based on simplicial regions and affine mappings were published; see, for example, Refs. [152–157].

2.3.1 The construction To this end, consider the set C(σ n ) := C(σ n , R) of continuous functions over the n-simplex σ n . Then (C(σ n ),  · ∞,σ n ) is a complete metric linear space. Now suppose that {σin : i ∈ NN } is a family of nonempty compact subsets of σ n with the properties that: (P1) {σin : i ∈ NN } is a set-theoretic partition of σ n . (P2) σin is similar to σ n for all i ∈ NN . (P3) σin is congruent to σjn for all i, j ∈ NN .

Note that conditions (P1), (P2), and (P3) imply the existence of N unique contractive similitudes ui : σ n → σin given by ui = a Oi + τi ,

i ∈ NN ,

(9.17)

where a < 1 is the similarity constant, or the similarity ratio, for σin with respect to σ n , Oi is an orthogonal transformation on Rn , and τi a translation in Rn . the subsimLet V be the set of vertices of σ n . Denote the set of all distinct vertices of Vi → V plices {σin : i ∈ NN } by Vi . Suppose there exists a labeling map : ∪ Vi := i∈NN

so that ∀i ∈ NN ∀v ∈ Vi : ui ((v)) = v.

(9.18)

338

Fractal Functions, Fractal Surfaces, and Wavelets

Now suppose that {λi : σ n → R}i∈NN is a collection of N affine functions and {si }i∈NN is a set of N real numbers. As in the previous section, we set λ := (λ1 , . . . , λN ) and s := (s1 , . . . , sN ). Let us denote by An := Aff(Rn ) the R-vector space of all affine mappings λ: Rn → R. We define an RB operator 

(λ)(s)f :=

n (λi ◦ u−1 i )χσi +

i∈NN



n (si f ◦ u−1 i )χσi

(9.19)

i∈NN

on a subspace C0 (σ n ) of C(σ n ) so that

(λ)(s):

N



× An × RN × C0 (σ n ) → C0 (σ n ).

i=1

The affine mappings λi are usually determined by interpolation conditions. Thus consider the interpolation set   Z := (v, zv ) ∈ σ n × R: v ∈ ∪Vi .

(9.20)

Then for all i ∈ NN , the affine mapping λi is uniquely determined by the interpolation conditions λi ((v)) + si z(v) = zv ,

∀v ∈ Vi .

(9.21)

For f to be well defined on and continuous across adjacent triangles σin and σjn , one needs to impose the following join-up conditions: −1 −1 −1 λi ◦ u−1 i (x, y) + si f ◦ ui (x, y) = λj ◦ uj (x, y) + sj f ◦ uj (x, y)

(9.22)

for all (x, y) ∈ eij := σin ∩ σjn , i, j ∈ NN with i = j. The quantity eij is called a common edge of σin and σjn . The next result, which is essentially Theorem 6 in Ref. [82] and Theorem 134, gives conditions for Eq. (9.22) to be satisfied. Theorem 137. Let σ n be an n-simplex and {σin : i ∈ NN } a family of nonempty compact subsets of σ n satisfying conditions (P1), (P2), and (P3). Let Z be an interpolation set of the form Eq. (9.20) and let   C0 (σ n ) := f ∈ C(σ n ): f (v) = zv ∧ v ∈ ∪Vi . Suppose that there exists a labeling map  as defined in Eq. (9.18). Further suppose that s := (s, . . . , s), with |s| < 1. Then the RB operator defined by Eq. (9.19) maps C0 (σ n ) into itself, is well defined, and is contractive on the complete metric subspace (C0 (σ n ),  · ∞,σ n ) of (C(σ n ),  · ∞,σ n ).

Fractal surfaces

339

The unique fixed point of the RB operator in Theorem 137 is called a multivariate real-valued affine fractal function and its graph is called an affinely generated fractal hypersurface or an affine fractal hypersurface.

0.25

0

0.5

0.75

1

0.4 0.2 0 0

0.25

0.5

0.75

1

Fig. 9.1 A 2-simplex and its associated partition.

Example 58. Let n := 2 and suppose we are given the 2-simplex and its associated partition as depicted in Fig. 9.1. A simple computation yields for the four mappings ui the following expressions:

 1 x = 2 y 0

 1 x u3 = 2 y 0

u1

  1 x + 2 , 1 y 0 2    0 x 0 + 1 , 1 y −2 2 0

 1   1 x −2 0 x = + 2 1 y y 0 0 2

 1    0 x x 0 u4 = 2 1 + 1 . y y 0 2 2

u2

The affine functions λi are then given by λ1 (x, y) = λ2 (x, y) = −z1 x + (z2 − z1 )y + z1 , λ3 (x, y) = λ4 (x, y) = (z2 − z3 )x − z3 y + z3 , where z1 , z2 , and z3 are the nonzero z values (see Fig. 9.1). The sequence of graphs in Fig. 9.2 shows the generation of the fractal surface and its view from above.

340

Fractal Functions, Fractal Surfaces, and Wavelets

2.3.2 Affine fractal basis Using Theorem 137, we can construct a basis for fractal surfaces. For this purpose, we denote by S(σ n ; An ) the set of all fractal functions generated by the RB operator (9.19) subject to the conditions stated in Theorem 137. The set S(σ n ; An ) becomes for

0 1

0.5 0.25

0.75

10

0.25 0.5

0.75

1

0.5

0.4 1

0

0.2 z

0.75

–0.5

y

–1

0.5 0.25 0 0

0.25 1

0.5

0.75

1

0

x

0.75 0.5

y 0.25 0 0.6

0.4 1 0.75

0.4 z

0.2 z

0.5 y 0.25 0 0

0.5 x 0.25

0.75

0.2

0 1

0 1

0.75

y

0.5

0

0.25

0.25

0.25

x 0.5

0.5

0.75

0 1

x

0.75 1.0 0.8 0.6 0.4 0.2 0.0 z

0 0.8

0.75

0.6 0.4 z

0.5 y

0.2 10

0.25

0.75 0.25

0.5 x

0

0

Fig. 9.2 The generation of an affine fractal surface and its view from above (bottom right).

Fractal surfaces

341

fixed s a complete metric linear space inheriting its metric from C(σ n ). Note that dim S(σ n ; An ) = 3N − card (∪Vi ); three free parameters for each of the N affine functions λi and card (∪Vi ) many interpolation conditions at the vertices. This observation suggests the construction of an n-dimensional system of Lagrange interpolants for each affine fractal function f of the form   B := bv : σ n → R: v ∈ ∪Vi ,

(9.23)

where each bv is the unique affine fractal hypersurface interpolating the set   Zv := (v , δvv ): v ∈ ∪Vi ,

v ∈ ∪Vi .

We also refer to the set of these affine fractal functions as an affine fractal basis. Thus we have the following result. Theorem 138. Let f be an affine fractal function with associated interpolation set Z as defined in Eq. (9.20). Then there exists an affine fractal basis of Lagrange-type (Eq. 9.23) of cardinality | ∪ Vi |c . such that S(σ n ; An )  f =



z v bv .

v∈∪Vi

Example 59. Three of the six graphs of the affine fractal basis functions for the affine fractal surface constructed in Example 58 are displayed in Fig. 9.3. As S(σ n ; An ) is finite dimensional, we can apply the Gram-Schmidt orthogonalization process to obtain an orthonormal basis for S(σ n ; An ) consisting of affine fractal basis functions (see also Refs. [78, 104, 158, 159]). This orthonormal basis will play an important role in the connection with wavelet sets and affine Weyl groups.

2.3.3 Coplanar boundary data The preceding construction shows that for the RB operator in Eq. (9.19) to be well defined, there can be only one free parameter |s| < 1. Next it is shown that there is another construction that allows more free parameters but, unfortunately, requires the boundary data of the interpolating set to be coplanar (see also Ref. [150]). To this end, let again m = 1, X = σ 2 , and let (X × R, w) be an IFS. Further, let  be a nonvertical hyperplane in R3 . Let C (σ 2 , R) =: {f ∈ C(σ 2 , R): (x, f (x)) ∈ , ∀x ∈ ∂σ 2 }. Furthermore, suppose there exist affine contractions ui : σ 2  ui σ 2 =: σi2 such that {σi2 : i ∈ NN } is a partition of σ 2 . Let {qj : j ∈ NM } be the collection of all distinct  2 σi , with q1 , q2 , and q3 being the vertices of σ 2 . Define a labeling map vertices of i∈NN

: NN × {1, 2, 3} → NM by requiring that {q(i,j) : j = 1, 2, 3} are the vertices of σi2 given that {qj : j = 1, 2, 3} are the vertices of σ 2 .

342

Fractal Functions, Fractal Surfaces, and Wavelets

y 0

0.25

0.5

0.75

10

x 0.5

0.25

0.75

0.25

0.5

0.25

x 0.5

0.75

1

0

1.5

z

y

1

1 0

0.75

1.5

1

1 z 0.5

0.5 0

0

1.5

1 z 0.5

0 0.25

0.5 x 0.75

0

1 0

0.25

0.5 y

0.75

1

Fig. 9.3 Some affine fractal basis functions.

Let := {(qj , zj ) ∈ X × R: j ∈ NM } be an interpolating set. Note that ui is the unique affine contraction satisfying ui (qj ) = q(i,j) ,

j = 1, 2, 3; i ∈ NN .

(9.24)

As before, define unique affine mappings vi : σ 2 × R → R, vi (x, y, z) := bi x + ci y + si z + ei , where bi , ci , and ei are uniquely determined by vi (qj , zj ) = z(i,j) , and |si | < 1, i ∈ NN .

j = 1, 2, 3; i ∈ NN ,

(9.25)

Fractal surfaces

343

Proposition 83. Suppose that := {(qj , zj ) ∈ X × R: j ∈ NM } is an interpolating set such that {(qj , zj ): qj ∈ ∂σ 2 } ⊂ . Then the RB operator , as defined in Eq. (9.19), but with vi as before, is well defined and contractive in the norm ·∞ with contractivity factor s := max{|si |: i ∈ NN }. Moreover, (f )(qj ) = zj for all j ∈ NM , and the unique fixed point f of is an element of C (σ 2 , R). Proof. Let f ∈ C (σ 2 , R). Then graph f contains the three line segments (qj , zj )(qj , zj ),

j, j = 1, 2, 3 j = j .

Therefore the line segments (q(i,j) , z(i,j) )(q(i ,j ) , z(i ,j ) ) are a subset of graph f . Hence −1 −1 −1 vi (u−1 i (x), f ◦ ui (x)) = vi (ui (x), f ◦ ui (x))

(9.26)

for all x ∈ qj qj whenever qj qj = σi2 ∩ σi2 . The well definiteness and continuity of f now follows from the fact that f |qj qj = (qj , zj )(qj , zj ). Furthermore, since

f |∂σ 2 consists of line segments that join points in {(qj , zj ): qj ∈ ∂σ 2 } ⊂ , graph

f |∂σ 2 ⊂ . Thus f ∈ C (σ 2 , R). It is not difficult to show that is contractive in  · ∞ with contractivity s and that f interpolates . Remark 71. It is an immediate consequence of the proof of Proposition 83 that more general mappings vi can be considered, as long as the “join-up” condition (9.26) is satisfied (see also condition (B0 ) on p. 147). This more general setup will be considered in detail in Chapter 10, where it is used to define wavelet bases in L2 (Rn ). Remark 72 (Multidimensional Iterative Interpolation Functions). The iterative interpolation process was introduced in Section 5. The careful reader may have noticed that the results on the continuity of the iterative interpolation process also hold for X to be a closed discrete subgroup of Rn . Therefore an iterative interpolation process (f , X, T, p) can also be used to define fractal-like surfaces. Some properties of such surfaces generated by iterative interpolation processes, including continuity and differentiability, are discussed in more detail in Refs. [88, 160]. The reader who would like to learn more about surfaces generated by iterative interpolation processes is referred to these references.

3 Properties of fractal surfaces In this section some properties of fractal surfaces are discussed. Among those are oscillation, regularity and differentiability, Hölder continuity, and the box dimension. It will also be shown how moments of R-valued multivariate fractal functions can be recursively calculated.

344

Fractal Functions, Fractal Surfaces, and Wavelets

3.1 The oscillation of f In what follows it is assumed that X = σ n —that is, (X, w) is just an IFS on X—and that σn =

N 

σin ,

(9.27)

i=1

where each σin is similar to σ n . (That such a simplex exists follows from the theory of Coxeter groups. This claim will be verified in Chapter 10.) This then implies that if ui is of the form (9.4), with Ai being a similitude with scaling factor ai < 1; that is, Ai = ai i , where i is an isometry on Rn . Let us also assume that B is a similitude; that is B = s, where  is an isometry on Rm . To calculate the oscillation of f over σ n , a special class of covers of σ n is needed. Definition 142. Let ε > 0. An ε-cover Cε of a bounded set S ⊆ Rn is called admissible if it is of the form   √ (9.28) Cε = Bε (rα ): rα , rα  ∈ S ∧ |rα − rα  | ≥ ε/(2 n), ∀rα = rα  , where Bε (rα ) denotes the n-dimensional ball of radius ε centered at α ∈ S. At this point the definition of oscillation of a function over a set is recalled. Definition 143. Let X ⊆ Rn be the domain of a function f : Rn → Rm . The oscillation of f over B ⊆ X is defined as osc(f ; B) := sup f (x) − f (x ),

(9.29)

and the ε-oscillation of f over X is defined as  oscε (f ; X) := inf osc(f ; B),

(9.30)

x,x ∈B

B∈Cε

where the infimum is taken over all admissible ε-covers Cε of X. The next lemma states how fast the ε-oscillation of f increases as ε decreases to zero. Recall that M is the number of interpolation points (qj , zj ). Lemma 17. Suppose that 1. the set of interpolation points = {(qj , zj ): j ∈ NM } is not contained in any n-dimensional hyperplane of Rn+m ; 2.



n−1 > 1. i∈NN sai

Then lim εn−1 oscε (f ∗ ; σ n ) = +∞.

ε→0+

(9.31)

Fractal surfaces

345

Proof. Condition 1 implies that there exists an  x ∈ int σ n such that V := f ( x) − π( x) > 0, where π : Rn → Rm is the unique affine map such that π(qj ) = zj , for each of the n + 1 σ n be a closed and connected subset of int σ n such that vertices q1 , . . . , qn+1 of σ n . Let  σ n with f (x) − f (x ) ≤ V/2. whenever x ∈ σ n there is an x ∈  Let η > 0 be the distance between  σ n and ∂σ n . Let a := min{ai : i ∈ NN }, let 0 < ε < η a/2, and let ε be the collection of all finite codes i ∈  such that 2ε ≤ ηai ≤ 2ε/a

(9.32)

holds for i but no curtailment of it. As π( x) is in the convex hull of {f ∗ (qj ): j ∈ Nn+1 }, it follows that π( x) =



αj f ∗ (qj ),

j∈Nn+1

 where αj ≥ 0 and j∈Nn+1 αj = 1. Let xj ∈  σ n be such that f (qj ) − f (xj ) ≤ V/2 for j ∈ Nn+1 . Then σ n) ≥ osc(f ; 

n+1 

αj f ( x) − f (xj )

j=1

   n+1     x) − π( x) −  α (f (q ) − f (x )) ≥ Vf ( j j

j  ≥ V/2.   j=1  This inequality together with the fixed-point property of f gives     n+1    n  osc(f ; ui σ ) ≥ f (ui ( x)) − αj f (ui (xj ))    j=1 x)) − ( i π )(ui ( x)) ≥ ( i f )(ui (     n+1     − ( i π )(ui ( x)) − αj ( i f )(u i (xj ))    j=1  ⎞ ⎛   n+1     ⎝ ⎠ x) − π( x) − π( αj f (xj ) = Bi  f ( x) −    j=1 ≥

1 si V. 2

346

Fractal Functions, Fractal Surfaces, and Wavelets

−1 Here the notation ( i f )(x) = vi (u−1 i (x), f ◦ ui (x)) was used, where vi (x, y) is such that wi (x, y) = (u i (x), vi (x, y)) for all finite codes i in the associated code σ n ) ≥ osc(f ; ui σ n ), since f is continuous and ui σn space ε . Note that oscε (f ; ui is connected. Therefore using Eq. (9.32), one has

⎛ oscε (f ; σ n ) ≥ oscε ⎝f ; ⎛ ≥⎝



i∈ ε

⎞ si ⎠



⎞ ui σ n⎠ =

i∈ ε



 oscε f ; ui σn

i∈ ε

V . 2

 From Eq. (9.32) it also follows that (ai /ε)n−1 ≤ 2/ηa

n−1

=: γ −1 . Thus

1  γ   n−1  −n+1 γ   d−1   n−d  −n+1 si V ≥ = , ai si Vε a i si ai V ε 2 2 2 i∈ ε

i∈ ε

i∈ ε

 where d is the unique positive solution of i∈NN s ad−1 = 1. Note that d > n by i condition 2. for any cylinder set i. Now define a probability measure μ on  by μ(i) := si ad−1 i   . Therefore Since ε partitions , one has 1 = i∈ε μ(i) = i∈ε si ad−1 i oscε (f ; σ n ) ≥ (γ /2)(ak(n−d) V)ε−n+1 , where a = max{ai : iNN } and k = min{|i|: i ∈ ε }. Since k → ∞ as ε → 0+, the result follows. Lemma 17 will now be used to derive upper and lower bounds on the ε-oscillation of f over σ n . Theorem 139. Assume that the hypotheses of Lemma 17 are satisfied. Then there exist positive constants ε0 , k1 , and k2 such that k1 ε−δ ≤ oscε (f ; σ n ) ≤ k2 ε−δ

(9.33)

 for all 0 < ε < ε0 , where δ is the unique positive solution of i∈NN saδi = 1. Proof. Let i ∈ NN , let 0 < ε < 1, and suppose that the ball Bε/ai (r) is contained in σ n . The fixed-point property of f implies that osc(f ; ui Bε/ai (r)) =

sup

x,x ∈Bε (ui (r))

vi (x, f (x)) − vi (x , f (x))

≤ B osc(f ; Bε/ai (r)) +

2ε Ci . ai

(9.34)

Fractal surfaces

347

Let σin = {x ∈ ui σ n : dist(x, ∂ui σ n ) ≥ 2ε} (here dist(A, B) denotes the distance between the sets A and B). Note that σin = ∅ for ε small enough. To proceed, the following observation is needed: if Cε is an admissible ε-cover of a set S and x is a point not covered by Cε , then Cε ∪ Bε (x) is an admissible ε-cover of S ∪ {x}. Thus any admissible cover of a set S may be extended to an admissible cover of a superset of S. Let Cε/ai be an admissible ε/ai -cover of σ n . Applying ui to this cover yields an admissible ε-cover Cεi of ui σ n . Let Cεi := {Bε ∈ Cεi : Bε ∩ σin = ∅}. Note  i  n Cε is an admissible ε-cover of σi and may be extended to an that Cε := i∈NN i∈NN  n σi , as admissible ε-cover Cε of σ n by our adding ε-balls with centers in σ n \ i∈NN

described earlier. Therefore   oscε (f ; σ n ) ≤ osc(f , Bε ) = osc(f , Bε ) + Bε ∈Cε

Bε ∈Cε

It follows from Eq. (9.28) and voln (σ n \

 i∈NN

osc(f , Bε ).

Bε ∈Cε \Cε

σin ) ≤ 4ε voln−1 (

exists a positive constant c0 such that Cε \ Cε contains at most 





∂ui σ n ) that there

i∈NN c0 ε−n+1

ε-balls. Thus

osc(f , B) ≤ 2f c0 ε−n+1 .

Bε ∈Cε \Cε

Furthermore, if Bε ∈ Cεi , then Bε = ui Bε/ai for some Bε/ai ∈ Cε/ai . Hence 

osc(f ; Bε ) =

Bε∈Cε

N 



i=1 Bε/ai ∈Cε/ai

≤

N 



i=1 Bε/ai ∈Cε/ai

Note that by Eq. (9.28), Therefore oscε (f ; σ n ) ≤

N 

osc(f ; Bε/ai )

 2ε s osc(f ; Bε/ai ) + Ci  . ai



Bε/ai ∈Cε/ai (2ε/ai )Ci 



≤ c1 ε−n+1 for some c1 > 0.

s osc(f ; Bε/ai ) + c2 ε−n+1

i=1 Bε/ai ∈Cε/ai

for some c2 > 0. As the preceding inequality holds for any admissible ε/ai -cover, one has oscε (f ; σ ) ≤

 i∈NN

s oscε/ai (f ; σ ) + c2 ε−n+1 .

(9.35)

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Fractal Functions, Fractal Surfaces, and Wavelets

On the other hand, if Cε is an admissible cover of σin := ui σ n , i ∈ NN , then the fixed-point property of f (assuming s = 0) yields osc(f ; Bε/ai (u−1 i (rα ))) =

sup

x,x ∈Bε (rα )

−1  ∗  v−1 i (x, f (x)) − vi (x , f (x ))

≤ B−1  osc(f ; Bε (rα )) + 2B−1 Ci Ai ε −1 −1 −1 for all Bε (rα ) ∈ Cε , where v−1 i is such that wi (·, ∗) = (ui (·), vi (·, ∗)). Thus   N N N    n −1 n −1 (ε/ai ) − osc(f ; σ ) ≤ s oscε (f ; σi ) + c0 B Ci A ε−n+1 ; i=1

i=1

i=1

that is, oscε (f ; σ ) ≥ n

N 

s oscε/ai (f ; σ n ) − c1 ε−n+1

(9.36)

i=1

 for c1 = c0 i∈NN B−1 Ci Ai  > 0. Note that Eq. (9.36) holds trivially for s = 0. Hence combining Eqs. (9.35), (9.36) gives N 

s oscε/ai (f , σ n ) − c1 ε−n+1 ≤ oscε (f ; σ n )

i=1

≤

N 

s oscε/ai (f ; σ n ) + c2 ε−n+1 .

i=1

 Now set γ := i∈NN san−1 and let a := max{ai : iNN }. By Lemma 17, an ε0 > 0 can i be chosen small enough so that oscε (f ; σ n ) ≥ [2c1 /(γ − 1)]ε−n+1 for ε0 < ε ≤ ε0 /a. Then choose K1 > 0 small enough so that K1 ε0−δ ≤ [c1 /(γ − 1)]ε−n+1 and K2 > 0 large enough so that oscε (f , σ n ) ≤ [c2 /(1 − γ )]ε−n+1 + K2 ε−δ for ε0 ≤ ε ≤ ε0 /a. Now define functions ϕ, ϕ: (0, ε0 ] → R by

ϕ(ε) :=

 c1 ε−n+1 + K1 ε−δ γ −1

Fractal surfaces

349

and

ϕ(ε) :=

c2 1−γ



ε−n+1 + K2 ε−δ ,

respectively. It follows that for all ε0 ≤ ε ≤ ε0 /a, ϕ(ε) ≤ oscε (f ; σ n ) ≤ ϕ(ε). Note that ϕ(ε) =

N 

sϕ (ε/ai ) − c1 ε−n+1

i=1

and ϕ(ε) =

N 

sϕ (ε/ai ) + c2 ε−n+1 .

i=1

If aε0 ≤ ε ≤ ε0 , then ε0 ≤ ε/ai ≤ ε/a and oscε (f ; σ n ) ≤ ≤

N  i=1 N 

s oscε/ai (f ; σ n ) + c2 ε−n+1 sϕ (ε/ai ) + c2 ε−n+1 = ϕ(ε).

i=1

Similarly one shows that ϕ(ε) ≤ oscε (f ; σ n ) for aε0 ≤ ε ≤ ε0 . If ϕ(ε) ≤ oscε (f ; σ n ) ≤ ϕ(ε) holds for a ε0 ≤ ε ≤ ε0 , it must hold for a+1 ε0 ≤ ε ≤ ε0 . Therefore ϕ(ε) ≤ oscε (f ; σ n ) ≤ ϕ(ε) for all 0 < ε ≤ ε0 . Since δ > n − 1, there exist positive constants k1 and k2 such that k1 ε−δ ≤ oscε (f ; σ n ) ≤ k2 ε−δ .

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Fractal Functions, Fractal Surfaces, and Wavelets

Remark 73. The inequalities in Eq. (9.33) imply that log oscε (f ; σ n ) . ε→0+ − log ε

δ = lim

3.2 Box dimension of the projections f,j In this section a formula for the box dimension of a hidden-variable fractal surface is presented. For this purpose, let f ,j be the orthogonal projection of f onto Rn × 0 × · · · × R × · · · × 0, where the factor R is in the jth position. Recall that ui = Ai (·) + Di , where Ai is a similitude of norm ai , and that vi (·, ∗) = Ci (·) + s(∗) + Ei , with  being an isometry on Rm and s < 1. Denote by λ1 , . . . , λm and by h1 , . . . , hm the eigenvalues and orthonormal eigenvectors of , respectively. Let the eigenvalues of  be ordered in such a way that λ1 , . . . , λμ are all the distinct eigenvalues of , μ ∈ Nm . The canonical basis of Rm is denoted by {e1 , . . . , em }. Define ckj = hk , ej ,

(9.37)

where ·, · denotes the Euclidean inner product in Rm . Then f can be written as f (x) =

m 

bk (x)hk ,

with

bk (x) := f (x), hk .

(9.38)

k=1

Moreover, f ,j (x) = f (x), ej . Let I(κ) be the set of all indices from Nm indexing the same eigenvalue of , κ ∈ Nμ , and let  bk (x)ckj . (9.39) dκ (x) := k∈I(κ)

Theorem 140. Suppose that 1. d ν ≡ 0, for some ν ∈ {1, . . . , μ}; n−1 s > 1. 2. i∈Nn ai

Then the box dimension d of graph f ,j is the unique positive solution of N 

ad−1 s = 1; i

i=1

otherwise d = n.

(9.40)

Fractal surfaces

351

Proof. To prove the theorem we make use of a special class Kε , ε > 0, of covers of graph f ,j . The covers Gε ∈ Kε are defined as follows: Gε = {Bε (rα ) × [yα + (k − 1)ε, yα + kε] : yα ≤ inf f ,j ∧ yα + nα ε ≥ sup f ,j Bε

Bε

∧ k ∈ Nnα ∧ nα ∈ N}, where {Bε (rα )} is an admissible ε-cover of σ n . Let Nε (f ,j ; σ n ) := inf{|Gε |c : Gε ∈ Kε }. Next it is shown that it suffices to consider covers from Kε to calculate the box Gε be an arbitrary minimal cover of graph f ,j dimension of graph f ,j . To this end, let  ε (f ,j ; D) the consisting of sets of the form Bε (r) × [z, z + ε], z ∈ R. Denote by N cardinality of this minimal cover. Then ε ( ,j ; σ n ) ≤ Nε (f ,j ; σ n ). N

(9.41)

But as {Bε (rα )} is an admissible cover of σ n , one also has that Bε/2√n (rα ) ∩ Bε/2√n (rβ ) = ∅,

rα = rβ ,

and that the number of balls Bε/2√n (rα ) contained in the ball Bε/2√n (rα ) is less than or √ equal to ξ := (4 n + 1)n . Thus any Bε (r) meets at most ξ elements of any admissible ε-cover of σ n . Hence ε (f ,j ; σ n ). Nε (f ,j ; σ n ) ≤ 3ξ N

(9.42)

Note that by Eq. (9.28) there exists a constant c > 0 such that |Cε |c ≤ cε−n for any admissible ε-cover of σ n . Furthermore, Eqs. (9.29), (9.30), and the definition of Nε (f ,j ; σ n ) yield ε −1 oscε (f ,j ; σ n ) − c1 ε−n ≤ Nε (f ,j ; σ n ) ≤ ε−1 oscε (f ,j ; σ n ) + c2 ε−n

(9.43)

for some positive constants c1 and c2 . As oscε (f ,j ; σ n ) ≤ oscε (f ; σ n ), Theorem 139 and Eq. (9.42) imply that Nε (fj∗ ; σ n ) ≤ k2 ε−d ,

(9.44)

where k2 > 0 and d is given by Eq. (9.40). Next a lower bound for Nε (f ,j ; σ n ) needs to be derived. To do this two technical lemmas have to be stated and proved. The first of these lemmas gives estimates on oscε . For this purpose a definition is needed.

352

Fractal Functions, Fractal Surfaces, and Wavelets

Definition 144. For f : σ n → R, define the upper ε-oscillation of f over σ n by oscε (f ; σ n ) := sup

⎧ ⎨ ⎩

⎫ ⎬

Bε ∈Cε

osc(f ; Bε ) , ⎭

(9.45)

where the supremum is taken over all admissible ε-covers Cε of σ n . Lemma 18. Let f : σ n → R and let ε > 0 be arbitrary. Then there exist positive constants K, k1 , and k2 such that: 1. oscε (f ; σ n ) ≤ oscε (f ; σ n ) ≤ K oscε (f ; σ n ); 2. k1 oscε (f ; σ n ) ≤ osccε (f ; σ n ) ≤ k2 oscε (f ; σ n ) for any c > 0. Furthermore, k1 and k2 depend only on f and c.

Proof. 1. Let Cε be an arbitrary admissible ε-cover of σ n and let Cε be an admissible ε-cover of σ n such that 

ω(f ; B ) ≤ 2 oscε (f ; σ n ).

B ∈Cε

For each B ∈ Cε , let S(B) := {B ∈ Cε : B ∩ B = ∅}. Then any B meets at most √ ξ= (4 n + 1)n elements of Cε . Furthermore, since S(B) ⊃ B ∩ σ n , one has oscε (f ; B) ≤  B ∈S(B) oscε (f ; B ) for any B ∈ Cε . Thus  B∈Cε

oscε (f ; B) ≤





B∈Cε B ∈S(B)

oscε (f ; B ) ≤ ξ



oscε (f ; B ) ≤ 2ξ oscε (f ; σ n ).

B ∈Cε

2. The proof of Theorem 139 implies that any Bc1 ε (r) meets at most  "n !

√ c n+1 2 1 +1 c2 elements of an admissible c2 ε-cover of σ n for any c1 , c2 > 0. Now the result follows as in part 1 of the proof.

Next a lower bound for oscε (dν , σ n ) is derived. To set up the induction, the next lemma is needed. Lemma 19. Suppose that conditions 1 and 2 of Theorem 140 are satisfied. Then lim εn Nε ( f ,j ; σ n ) = +∞.

ε→0+

Fractal surfaces

353

Proof. Since  is an isometry, |λnν | = 1 for all n ∈ N. For α, β ∈ span{(λnν )n∈N0 : ν ∈ Nμ }, define T 1 αn β n . T→∞ T

(α, β) := lim

n=0

Now it follows from hypothesis 1 in Theorem 140 and ((λnν ), (λnν  )) = δνν   that there exists an n0 ∈ N with ν dν (x)λnν0 ≡ 0. Assume without loss of generality that n0 = 0. Moreover, it can be assumed that z1 = · · · = zn+1 = 0 in Rm , with one of the vertices being the origin. To simplify the notation, the following definition is introduced. Definition 145. Let a, b ∈ Rn+1 and let π be a plane perpendicular to b containing the point P ∈ Rn+1 . Then proja;b,P (x) :=

x − x − P, b a, a, b

∀x ∈ Rn+1 .

As f ,j ≡ 0 and continuous, there exists a b ∈ Rn such that proj(b,0);(b,0),O (graph f ,j ) ⊃ C2ρ , where C2ρ denotes an n-dimensional cube of side 2ρ for some ρ > 0. (Here O is the origin in Rn+1 .) By the Poincaré recurrence theorem [161] there exists a subsequence {nk } ⊆ N such that |λν − λnνk | < ρ/2c for all ν ∈ Nμ , and c = max

#

$ n . Hence |d (x)|: x ∈ σ ν∈Nμ ν

        ρ ρ |f ,j (x) − (nk f ,j )(x)| =  dν (x)(λν − λnνk ) < dν (x) · ≤ . 2c 2 ν∈Nμ  ν∈Nμ Let i ∈ , |i| = nk , and a := wi (b, 0) − wi (0, 0). Then   proja;(b,0),P (graph f ,j |σin ) = proja;(b,0),P (ui (x), f ,j ◦ ui (x)): x ∈ σ n . After some algebra, this reduces to  x, b nk b,  f ,j (x), ej  , x− a, b

T

(i)

354

Fractal Functions, Fractal Surfaces, and Wavelets

where T (i) : Rn × Rm → Rn × Rm is defined by T (i) (x, y) := (ui (x), snk y + Ci x + Ei ). Note that the Jacobian of T (i) is given by Jac T (i) = an−1 s|i| , i

i ∈ .

Thus vol T nk (C2p ) = Jac T nk vol C2p . If Nε (f ,j ; σ n ) is the cardinality of a minimal ε-cover Gε ∈ Kε of graph f ,j , then for ε < 2ρa nk , ε Nε (f ,j ; σ ) ≥ n

n



 Jac T

(i)

vol C2ρ ≥

N 

|i| an−1 s i

vol C2ρ ,

i=1

i



and this last term tends to infinity as ε → 0+. Continuation of the proof of Theorem 140. Observe that f ,j =

μ 

dκ .

(9.46)

κ=1

Lemma 18 (condition 1) and Eq. (9.46) give oscε (f ,j ; σ n ) ≤

μ 

oscε (dκ ; σ n ) ≤ K

κ=1

μ 

oscε (dκ ; σ n ).

(9.47)

κ=1

Lemma 19, Eqs. (9.43), (9.47), and possibly reindexing yield lim sup εn−1 oscε (dν ; σ n ) = +∞. ε→0+

(9.48)

Taking the inner product of Eq. (9.38) with hk and using Eq. (9.39), we obtain the following functional equation for dν : % dν (ui (x)) = ci x + Ei ,



& hk ckj + sλν dν (x)

(9.49)

k∈I(κ)

for x ∈ σ n , i ∈ NN . Arguments similar to those that lead to Eq. (9.36) yield oscε (dν ; σ n ) ≥

 i

s oscε/ai (dν ; σ n ) − cε−n+1

(9.50)

Fractal surfaces

355

for some c > 0. By Lemma 18 (condition 1) and Lemma 19 there exists an ε0 > 0 such that oscε (dν ; σ n ) ≥ 2c/(γ − 1)  n−1 for ε0 ≤ ε ≤ ε0 /a, where γ := and a := min{|ai |: i ∈ NN }. Thus a i∈NN sai small enough constant K1 > 0 can be chosen so that

oscε (dν ; σ ) ≥ n

 c ε−n+1 + K1 ε−d+1 γ −1

(9.51)

for ε0 ≤ ε ≤ ε0 /a. Using induction and Eq. (9.50), one sees that Eq. (9.51) holds for all 0 < ε ≤ ε0 . In particular, oscε (dν ; σ n ) ≥ K1 ε−d+1

(9.52)

for 0 < ε ≤ ε0 . Let i ∈  be a finite code of length . Note that by Eq. (9.38), f ◦ ui (x) = vi (x, f (x)) = ci x + B f (x) + Ei for x ∈ σ n . Therefore Eqs. (9.38), (9.39) imply f ,j ◦ ui (x) − f ,j ◦ ui (y) = Ci (x − y), ej  +

μ 

s λκ (dκ (x) − dκ (y))

κ=1

for x, y ∈ σ n . Hence  μ    2C i oscε (f ,j ; ui σ n ) ≥ − εn+1 + s oscε/ai λκ dκ ; σ n , ai κ=1

and, by Lemma 18 (condition 2),  μ   2Ci  −n+1   n ε + k1 s oscε λκ dκ ; σ , oscε (f ,j ; ui σ ) ≥ − ai n

(9.53)

κ=1

where k1 = k1 (ai ). Hence oscε (f ,j ; σ ) ≥ −c ε n

−n+1

+ c oscε

 μ  κ=1

where c , c > 0.

 λκ dκ ; σ n

,

(9.54)

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Fractal Functions, Fractal Surfaces, and Wavelets

To obtain a lower bound for  oscε (f ,j ; σ n ), the lower bound for oscε (dν ; σ n ) is related to a lower bound for oscε ( κ∈Nμ λκ dκ ; σ n ). Note that since the Vandermonde determinant V(λ1 , . . . , λμ ) of λ1 , . . ., λμ , ⎛

1 λ1 λ21 .. .

⎜ ⎜ ⎜ V(λ1 , . . . , λμ ) = ⎜ ⎜ ⎝

1 λ2 λ22 .. .

μ−1

λ1

⎞ 1 λμ ⎟ ⎟ λ2μ ⎟ ⎟, .. ⎟ . ⎠ μ−1 λμ

... ... ...

μ−1

λ2

... ...

is nonzero, there exist constants α0 , . . . , αμ−1 such that  μ  μ−1   dν = α dκ . =0

(9.55)

κ=1

Therefore, by Eq. (9.52) and Lemma 18 (condition 1), the following inequalities hold: 0 < K1 ≤ lim inf ε−d+1 oscε (dν ; σ n ) ε→0+ ⎛ ⎞  μ  μ−1   −d+1  n = lim inf ε oscε ⎝ α λκ dκ ; σ ⎠ ε→0+

≤ lim inf ε→0+

≤ lim inf ε→0+

=0 μ−1 

|α |oscε

=0 μ−1 



λκ dκ ; σ n

κ=1

ε

−d+1

|α |K oscε

=0

≤ lim inf c ε→0+

ε

−d+1

κ=1

 μ 

μ−1 

ε−d+1 oscε

 μ 

 μ 

=0

κ=1

 λκ dκ ; σ n 

λκ dκ ; σ n ,

κ=1

where c = max{K|α |:  ∈ N0μ−1 }. Thus there exists an ε0 > 0 such that for each 0 < ε ≤ ε0 , there is some  ∈ N0μ−1 with ε

−d+1

oscε

 μ  κ=1

 λκ dκ ; σ n

≥

K1 . 2μc

(9.56)

Hence if we use Eqs. (9.54), (9.56), there exist K2 , K3 > 0 such that oscε (f ,j ; σ n ) ≥ K2 ε−n+1 + K3 ε−d+1 for 0 < ε ≤ ε0 . The result now follows from Theorem 139.



Fractal surfaces

357

3.3 Box dimension of f ∈ RF (X ⊂ R2 , R) Now a formula for the box dimension of a recurrent fractal surface in R2 × R is presented. At this point the reader should recall the notation and terminology introduced in Section 6, and the construction of a multivariate recurrent R-valued affine fractal function f : X ⊂ R2 → R. Throughout this section it is assumed that the mappings ui , as defined in Eq. (9.3), are similitudes whose contractivity factors are given by ai , i ∈ NN , and that the connectivity matrix C of the underlying recurrent IFS is irreducible. Before the main result can be stated, a few remarks about covers have to be made and some technical lemmas must be proven. To this end, let G := graph f and Gi := graph f |σ 2 , i ∈ NN . Let Z := Bε (r) × i [z, z + ε], ε > 0, and let Cε (G) denote the class of all covers of G whose covering elements are of the form Z. That only this class of covers needs to be considered in the calculation of the box dimension of G should be clear. Let N (ε) := N (ε; G) := min{|Cε (G)|c : Cε (G) ∈ Cε }.

To calculate dimb G, an even more special class of covers needs to be introduced. A cover Kε (G) of G is called admissible iff it is of the form Kε (G) := {Bε (rα ) × [z + (k − 1)ε, z + kε]: k ∈ Nn } and |rα − rβ | > ε whenever α = β. The class of all admissible covers of G is denoted by Kε (G). Define N(ε) := N(ε; G) := min{|Kε (G)|c : Kε (G) ∈ Kε }. Lemma 20. N (ε) ≤ N(ε) ≤ 48 N (ε). Proof. Lemma 4 implies that at most 16 balls Bε (rα ) from an admissible cover meet any ball Bε (r). Thus any set of the form Z meets at most 3 · 16 of the elements of Kε (G∗ ). Now let Ni (ε) := min{|Kε (Gi )|c : Kε (Gi ) ∈ Kε }, i ∈ NN . The next result follows directly from the proofs of Lemma 14 and Theorem 104. Lemma 21. There exist positive constants A and B such that for 0 < ε < 1, ⎛ |s| ⎝ ai



⎞ Ni (ε/ai )⎠ −

j∈I(i)

⎛ A |s| ⎝ ≤ Ni (ε) ≤ ai ε2

 j∈I(i)

⎞ Ni (ε/ai )⎠ +

B . ε2

(9.57)

2 = σ 2. Here I(i) := {j ∈ NN : σj2 ⊂ τk(i) }, where k(i) is that index for which ui τk(i) i Lemma 22. Let B := diag(|s|ai )C have spectral radius r < 1. If there exists a k ∈ NN such that {(qj , zj ): qj ∈ τk2 } is not contained in a hyperplane of R3 , then

lim ε2 Ni (ε) = +∞.

ε→0+

Proof. Note that the irreducibility of the connection matrix C implies that Gi is not contained in any hyperplane of R2 , i ∈ NN . Let i denote the hyperplane through the

358

Fractal Functions, Fractal Surfaces, and Wavelets

points {(qij , zij ): j = 1, 2, 3}, where {qij : j = 1, 2, 3} are the vertices of σi2 . Denote by σi2 (η) that 2-simplex contained in σi2 whose η-body is σi2 . Now let v ∈ R3 and let P be a vertical plane in R3 . The projection onto P in the direction of v is denoted by projv,P . By the continuity of f and the noncoplanarity of G, one can choose an η > 0, a vi ∈ R3 parallel to i , and Pi such that projvi ,Pi G|σ 2 (η) i has area Ai > 0. Denote by  Ai the area of projvi ,Pi B1 (r) × [0, 1]. Then it follows that for any ε > 0, Ni (ε) ≥ (Ai / Ai )−2 .

Let e = (e1 , . . . , eN )t be a positive eigenvector of B and choose θ > 0 sufficiently Ai ≥ θ ei , i ∈ NN . small so that Ai / Next apply the map wi to G|σ 2 (η) for each j ∈ I(i). Since wi maps vertij

cal planes onto vertical planes and has Jacobian Jac(wi ) = sa2i , it follows that projwi (vj ),wi Pj G|σ 2 (η) has area equal to |s|ai Ai . Hence for 0 < ε < aη, with a := min j {ai : i ∈ NN }, one has Ni (ε) ≥ |s|ai

N  Aj  ε−2 ≥ θ ε−2 |s|ai cij ej = θ ε−2 rei .  A j i=1 j∈I(i)

Induction then gives Ni (ε) ≥ θ ε−2 rn ei

for 0 < ε < an η, n ∈ N, and i ∈ NN . Now the dimension theorem can be stated. Theorem 141. Suppose that the connection matrix C is irreducible and that there exists a k such that {(qj , zj ): qj ∈ τk2 } is not contained in a hyperplane of R3 and that the spectral radius r of diag(|s|ai ) is greater than 1. Then dimb graph f is the unique real number d such that )C) = 1; r(diag(|s|ad−1 i

(9.58)

otherwise dimb graph f = 2. Proof. Use of Lemmas 21–23 as well as the arguments given in the proof of Theorem 105 yields A1 ε−d + A2 ε−2 ≤ Ni (ε) ≤ A3 ε−d + A4 ε−2 for positive constants A1, . . . , A4 , and ε > 0. Hence dimb graph f = max{2, d}.

Fractal surfaces

359

2 } is contained in a hyperplane for all i ∈ N , then graph f is If {(qj , zj ): qj ∈ τk(i) N

piecewise coplanar and thus has box dimension equal to 2. If (X, w) is an IFS rather than a recurrent IFS—that is the connection matrix C is given by (cij ) = (1) for all i, j ∈ NN —Eq. (9.58) reduces to (9.59) below. Corollary 15. Suppose that {(qj , zj ): j ∈ NM } is not contained in any hyperplane of R3 and that i∈NN |s|ai > 1. Then the box dimension of graph f is the unique positive solution d of N 

|s|ad−1 = 1; i

(9.59)

i=1

otherwise dimb graph f = 2. Finally, a dimension result for affine fractal surfaces in R3 with coplanar interpolation points is stated. For this purpose, let xy be a set of N + 1 equally spaced points on ∂σ 2 from which N 2 subsimplices σi2 are constructed. The maps ui : σ 2  σi2 are then of the form

1  0 ±N (·) + Di ui = 0 ± N1 for i ∈ NN 2 . The mappings vi are given as in Eq. (9.25). Denote by f ∗ the bivariate fractal function generated by these maps. First, a lemma is needed. Lemma 23. Let Bε denote a ball in R3 of radius ε > 0. For ε > 0, let ε be the ε-prismatoidal set ε σ 2 × ε [0, 1], where ε A denotes the set {ε a: a ∈ A}. Let S be a bounded set in R3 and let N (Bε ) denote the minimum number of ε-balls Bε necessary to cover S. Then N (Bε ) ≤ N (ε ) ≤ 8 N (Bε ),

where N (ε ) is the minimum number of ε-prismatoidal sets needed to cover S. Proof. Every Bε can be covered by at most eight ε , and every ε can be covered by one Bε .   Theorem 142. Assume that i∈N 2 |si | > N and that |σi2 | is not contained in N

any hyperplane of

R3 .

Then the box dimension of

i∈NN 2 graph f ∗ is given

by

2

dimb graph f ∗

= 1 + logN

N 

|si |;

(9.60)

i=1

otherwise dimb graph f ∗ = 2. Proof. It suffices to prove the theorem for ε := εm := N −m , m ∈ N. To this end, let m ∈ N and let Um be a collection of covers of G∗ := graph f ∗ by εm -prismatoidal sets  = m (p, q) of the form projz=0  = |σp2 |, p ∈ NN 2m ,

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Fractal Functions, Fractal Surfaces, and Wavelets

σp2 := wi(p) σ 2 , with i(p) ∈  denoting a finite code of length p, and q ∈ NN (m,p) , where N (m, p) ∈ N is the number of such εm -prismatoidal sets over |σp2 |. Furthermore, it is required that m (p, q) and m (p, q + 1) intersect along their ∗ respective faces. Now let Um ∈ Um be a cover of G of minimal cardinality N (m) ∈ N. Observe that N (m) = p∈N 2m N (m, p). Let N



G∗p :=

m (p, q),

p ∈ NN 2m .

q∈NN (m,p)

It is immediate that G∗p is compact and connected. The image G∗pi of G∗p under the map wi , i ∈ NN 2 , is a compact set in σp2 × R  wi G∗ , it follows that G∗ ⊆ above the ith subsimplex σpi2 of σp2 . Since G∗ = 

 wi

 p

i∈NN 2

i∈NN 2



G∗p . The compact set G∗pi is contained entirely in the prismatoidal set

(N m+1 )−1 σ 2 × (N (m, p)|si | + |ci | + |di |) (N −m )[0, 1]. Hence if N (m + 1, p, i) denotes the number of εm+1 -prismatoidal sets above σpi2 , then N (m + 1, p, i) ≤ N (N (m, p)|si | + |ci | + |di |) + 1.

Note that N (m+1) = over p and i yields



⎡ N (m + 1) ≤ ⎣N

 i∈NN 2

N (m, p, i). Summing the preceding inequality

⎤

2

N 

p∈NN 2m

|si |⎦ N (m) + c1 N 2m+1 ,

i=1

with c1 :=



i∈NN 2 (|ci | + |di | + N)

> 0. If

 i∈NN 2

|si | ≤ N, then induction on m gives

N (m) ≤ [N (1) + c1 mN −1 ]N 2m ,

and thus lim sup m→∞

log N (m) ≤ 2. log N m

Hence dimb G∗ = 2 in this case. Also, if

 i∈NN 2

|σi2 | is contained in a hyperplane P of

R3 , then G∗ = P, and thus dimb G∗ = 2. If γ := i∈N 2 |si | > N, then, again by induction on m, N

Fractal surfaces

361

-



m−1 . c1 N N N (m) ≤ (Nγ ) N (1) + 1 + + ··· + γ γ γ ! " c1 ≤ (Nγ ) N (1) + , γ −N which implies that dimb G∗ ≤ 1 + logN γ . 

To obtain a lower bound for dimb G∗ when γ > N and

i∈NN 2

|σi2 | is not contained

in any hyperplane of R3 , one proceeds as follows: Choose i ∈ NN 2 and assume that si = 0. The inverse image of G∗p under w−1 i is contained in a prismatoidal set −1 (N −m+1 )σi2 × N −m (N (m, p)|s−1 i | + |ci si |N)[0, 1].

Hence in the same notation as before, −1 N (m − 1, p, i) ≤ |s−1 i |N (m, p, i) + (|si |N)(|ci | + |di |) + 1.

Summing over p and i yields N (m) ≥ (Nγ )N (m − 1) − c2 N 2m−1 ,

 with c2 := N i∈N 2 |s−1 i |(|ci | + |di |) + N > 0. Note that the preceding inequality N holds trivially for si = 0. Induction on m for all m0 ∈ Nm gives N (m) ≥ (Nγ )m−m0 (N (m0 ) − c2 N 2m0 (γ − N)−1 ).

By Lemma 24 one can choose m0 large enough to guarantee that N (m0 ) > c2 N 2m0 (γ − N)−1 .

Let c3 := (Nγ )−m0 (N (m0 ) − c2 N 2m0 (γ − N)−1 ) > 0. Then N (m) ≥ c3 (Nγ )m .

Hence lim inf m→∞

log N (m) ≥ 1 + logN γ . log N m

Thus dimb G∗ = 1 + logN γ for γ > N and

 i∈NN 2

hyperplane of R3 .

|σi2 | not contained in any

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Fractal Functions, Fractal Surfaces, and Wavelets

The proof of Theorem 142 used the following lemma, which is a special case of Lemma 22, and it is therefore left reader to establish its validity.  to the |σ 2 | is not coplanar, then Lemma 24. If γ > N and i∈NN 2

N 2m = 0. m→∞ N (m) lim

3.4 Hölder continuity Next it is shown that the bivariate fractal function f ∗ defined at the end of the previous section is Hölder continuous with exponent α = 3 − dimb graph f ∗ . Theorem 143. Let f ∗ be defined as before. Then for all 0 ≤ h < 1 and points (x, y), (x + h, y + h) ∈ σ 2 , |f ∗ (x + h, y + h) − f ∗ (x, y)| ≤ chα

(9.61)

for a positive constant c and α = 3 − dimb graph f ∗ = 2 − logN Proof. Let n ∈ N. For every finite code i(n) ∈  := the simplex w i(n) σ 2 , p ∈ NN 2n . Let (m, p) :=



NN N2

 i∈NN 2

|si |.

of length n denote by σp2

N −m σ 2 × N −m [k, k + N −m ]

k∈N0N (m,p)

be the collection of all sets of the above form that cover G∗ |σp2 . Let 0 ≤ h < 1 be given. Let m be the least integer such that there exists a finite code i(m) ∈  of length m so that (x, y) and (x + h, y + h) are both in the interior of ui(m) σ 2 . Then it follows from Theorem 142 that |f ∗ (x + h, y + h) − f ∗ (x, y)| ≤ N −m max{N (m, p): p ∈ NN 2m }. However, since N (m, p) ≥ cN −2m (N −m N −d ), where d = dimb G∗ , this implies that log |f ∗ (x + h, y + h) − f ∗ (x, y)| log cN −2m (N −m N −d ) ≥ log h log h log c log N −3m+md log c ≥ = + +3−d −m log h log N log h because h ≤ N −m . The fact that G∗ has Hölder exponent α = 3 − dimb G∗ has the following consequence.

Fractal surfaces

363

Proposition 84. Suppose that f ∗ is Hölder continuous with exponent α = 3 − dimb graph f ∗ . Then H3−α (graph f ∗ ) < +∞. Proof. Let 0 ≤ h < 1 and let m be the least integer so that h ≤ N −2m . Let σp2 be any of the N 2m subsimplices of σ 2 . Then G∗ |σp2 can be covered by at most N 2m (chα + 1) sets of the form N −m σ 2 × N −m [k, k + N −m ], k ∈ N. Thus

  ∗ dimb G∗ dimb G∗ HN −2m (G∗ ) ≤ N 2m ch3−dimb G + 1 N −m   ∗ dimb G∗ ≤ N 2m cN (−m)(3−dimb G ) + 1 N −m  dimb G∗ −2 = cN −m + N −m

dimb G∗ −2 c 1 ≤ + < +∞. N N ∗

Therefore, Hdimb G (G∗ ) < +∞.

∗ 3.5 p-balanced measures and moment theory for f

The projection (σ 2 , u) of the IFS (σ 2 ×R, w) that generates graph f ∗ onto R2 is also an μ denote the p-balanced measure of (σ 2 × R, w) IFS whose attractor is σ 2 . Let μ and  2 and the p-balanced measure of (σ , u), respectively. Recall that G∗ = graph f ∗ . Let M(σ 2 ) and M(G∗ ) denote the measure spaces of σ 2 and G∗ , respectively, and let H: σ 2 → G∗ be the homeomorphism defined by (x, y) → (x, y, f ∗ (x, y)). Then the following relation between measures on σ 2 and G∗ holds. Theorem 144. The homeomorphism H: σ 2 → G∗ , as defined above, induces a contravariant homeomorphism M(H): M(G∗ ) → M(σ 2 ). Furthermore, (σ 2 ):  ∀ E∈B μ E = μH( E). If g ∈ L1 (σ 2 × R, μ), then /

/ G∗

gdμ =

σ2

/ g ◦ Hd μ=

H −1 σ 2

gd μ.

Proof. The proof is similar to that of Theorem 83 and is therefore omitted. The preceding theorem implies the following integral relation. Corollary 16. If the probabilities in the IFS (σ 2 × R, w) are chosen according to pi := area (σi2 ), i ∈ NN 2 , and if dA denotes the two-dimensional Lebesgue measure on R2 , then

364

Fractal Functions, Fractal Surfaces, and Wavelets

/

/ G∗

g dμ =

σ2

g ◦ H dA.

  Proof. With the above choice of probabilities, dA(u−1 i B) = dA(B) for all  B ∈ B(σ 2 ). This corollary, together with the stationarity of the p-balanced measure μ, can be used to calculate moments. Let g ∈ L1 (σ 2 × R, μ) and let pi = area (σi2 ), i ∈ NN 2 . Then / G∗

/

2

g(x, y, z) dμ(x, y, z) =

N 

pi

G∗

i=1

/

2

=

N 

pi

G∗

i=1

/ =

σ2

g ◦ wi (x, y, z) dμ(x, y, z)

g(ui (x, y), vi (x, y, z)) dμ(x, y, z)

g(x, y, f ∗ (x, y)) dA.

Also, /

/

2

G∗

g(x, y, z) dμ(x, y, z) =

N 

pi

σ2

i=1

g(ui (x, y), vi (x, y, f ∗ (x, y))) dA.

Hence / σ2

/

2

g(x, y, f ∗ (x, y)) dA =

N 

pi

i=1

σ2

g(ui (x, y), vi (x, y, f ∗ (x, y))) dA.

(9.62)

Recalling the definition of moments, Definition 117, we can now state and prove a moment theorem for affine fractal surfaces defined by functions f ∗ . To this end, let α, β, γ ∈ N0 . Theorem 145. Let G∗ be the graph of an affine fractal function f ∗ . Then the moments M α,β,γ f ∗ can be recursively and explicitly calculated from the lower-order moments. Proof. Let N± := ± N. M α,β,γ f ∗

/ :=

xα yβ f ∗∗γ dA

σ2

/ N  = (2N −2 )−1 2

i=1

σ2



x + xi N±

α

y + yi N±

β (Li (x, y) + si z)γ dA,

Fractal surfaces

365

with Li (x, y) := ki x+i y+zi , and where the xi , yi , zi , ki , and i are uniquely determined from Eqs. (9.3), (9.24). To simplify the notation the following abbreviations are introduced: 

:=

α−1 

a



a=0

:=

α a



a

x N±

c=0

:=

b

γ −1

c



xiα−a ,

 γ  ∗∗ si f c

c

β−1 

β b

b=0



γ −c Li ,



y N±

b

β−b

yi

,

2

:=

N 

i

(2N −2 )−1 .

i=1

Then M α,β,γ f ∗



   

    ∗ γ x α y β = + + + si f dA N± N± σ2 a c i b     M α,β,γ f ∗ + M a,b,c f ∗ + M a,b,γ f ∗ = /

i

+



a

i

M α,b,c f ∗

+

c

b

+







c

M α,b,γ f ∗ +



b

 a

c

M a,β,c f ∗

+



.

a

b

M a,β,γ f ∗

a

M a,b,γ f ∗ ,

c

b

or, on our transposing the first term on the right-hand side and substituting for



i,

" !    a,b,c ∗   a,b,γ ∗   α,b,c ∗ f + f + b c M  f a  b cM a bM    + c a M a,β,c f ∗ + a M a,β,γ f ∗ + b M α,b,γ f ∗ + c M a,b,γ f ∗ M α,β,γ f ∗ = .  2 γ −(α+β+γ +2) 1 − 12 N i=1 si N Note that the numerator in the preceding equation is strictly less than 1. Remark 74. In the case β := 0 and γ := 1, the moment formula for f ∗ reduces to something more appealing—namely, M α f ∗

N 2 α−1 α i=1

=

(xiα−a si /2N a+3 )M a f ∗ + Pα ,  2 α+3 1− N i=1 si /2N

a=0 a

where 2

Pα :=

N  i=1

(2N −2 )−1

/ σ2



x + xi N±

α Li (x, y) dA.

(9.63)

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Fractal Functions, Fractal Surfaces, and Wavelets

4 Fractal surfaces of class C k In this section a method to construct smooth fractal surfaces is introduced. This construction extends the result in Ref. [98] to the bivariate setting. Iteration of this process produces arbitrarily smooth surfaces. Let Q = [0, 1] × [0, 1], let e1 = (1, 0)t , e2 = (0, 1)t , and let  := {(m/N)e1 + (n/N)e2 : m, n ∈ Z} be a lattice in R2 . Suppose that for each lattice point (xj , yj ) ∈  ∩ Q a real number zij , i, j ∈ N0N is given. The set := {(xj , yj , zij ): i, j ∈ N0N } can be thought of as a given set of data or interpolation points on Q. Smooth affine fractal surfaces containing this interpolating set will be constructed. To this end, let uij : Q → Q be given by uij (x, y) =

1 N

0

  i−1  x N + j−1 , 1 y N N 0

(9.64)

and let vij : Q × R → R be such that 1. vij (·, ·, z) − vij (0, 0, z0 ) is a symmetric quadratic form for all z, z0 ∈ R. 2. vij (x, y, ·) − vij (x, y, z0 ) is a linear form for all (x, y) ∈ Q and z0 ∈ R.

In addition, it is required that vij (0, 0, z0,0 ) = zi−1,j−1 ,

vij (0, 1, z0,N ) = zi−1,j ,

vij (1, 0, zN,0 ) = zi,j−1 ,

vij (1, 1, zN,N ) = zi,j ,

(9.65)

i, j ∈ NN , and that the following join-up conditions are satisfied: for j ∈ NN and y ∈ [(j − 1)/N, j/N], vij (0, y, ϕ(0, y)) = vi−1,j (1, y, ϕ(1, y)), vij (1, y, ϕ(1, y)) = vi+1,j (0, y, ϕ(0, y)),

i = 2, . . . , N, i ∈ NN−1 ,

(9.66)

and for i ∈ NN and x ∈ [(j − 1)/N, j/N], vij (x, 0, ϕ(x, 0)) = vi,j−1 (x, 1, ϕ(x, 1)),

j = 2, . . . , N,

vij (x, 1, ϕ(x, 1)) = vi,j+1 (x, 0, ϕ(x, 0)),

j ∈ NN−1 .

(9.67)

Here ϕ denotes any C0 -function interpolating . Conditions (9.65)–(9.67) uniquely determine some of the coefficients Aij , . . . , Gij . For instance, if z0,0 = z0,N = zN,0 = zN,N = 0 and vij (x, y, z) := Aij x2 + Bij y2 + Cij z + Dij xy + Gij ,

Fractal surfaces

367

then Aij = zi,j−1 − zi−1,j−1 , Dij = (zij − zi,j−1 ) − (zi−1,j − zi−1,j−1 ),

Bij = zi−1,j − zi−1,j−1 Gij = zi−1,j−1

for i, j ∈ NN . If ϕ ≡ 0 on ∂Q, then Aij = Bij = Dij ≡ 0, and the join-up conditions are automatically satisfied. If ϕ(0, y) ≡ ϕ(1, y) and ϕ(x, 0) ≡ ϕ(x, 1), then one also needs that Aij = Ai,j−1 , Cij = Ci−1,j = Ci,j−1 ,

Bij = Bi−1,j , Eij = Fij = 0

for the join-up conditions to be satisfied. Now let C (Q) := {ϕ ∈ C(Q, R): ϕ(xj , yj ) = zij ∧ i, j ∈ N0N }. Define an RB operator : C (Q) → RQ by −1 ( ϕ)(x, y) := vij (u−1 ij (x, y), ϕ ◦ uij (x, y)),

(x, y) ∈ uij (Q).

(9.68)

Suppose, without loss of generality, that ϕ∞ ≤ 1 on Q and that s := max{|Cij |: i, j ∈ N0N } < 1. Theorem 146. maps C (Q) into itself, is well defined, and is contractive in the supremum norm with contractivity s. Proof. The results follow from the definition of , conditions (9.65)–(9.67), and the assumption on s. The unique fixed point of is the graph of a C0 -function f : Q → R that interpolates . The graph of f is again called a fractal surface. The reason for our choosing this particular form of the vij will become clear shortly. Let  f (x, y) :=  z0,0 +

/

x/ y

f (s, t) dt ds 0

0

for some  z0,0 ∈ R. Denote the integral operator uij (x, y) = (ξi (x), ηj (y)), where ξi (x) :=

1 i−1 x+ , N N

i ∈ N0N ,

0x0y 0

0 (·) dt ds

(x,y)

by I(0,0) (·), and let

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Fractal Functions, Fractal Surfaces, and Wavelets

and ηj (y) :=

1 j−1 y+ , N N

j ∈ N0N .

Then (xi−1 ,yj−1 ) (ξi (x),yj−1 )  z0,0 + I(0,0) (f ) + I(xi−1 (f ) f ◦ uij (x, y) =  ,0) (x

,η (y))

i−1 j + I(0,y i−1 )

(x

i−1 = z0,0 + I(0,0)

+

(ξ (x),η (y))

i j (f ) + I(xi−1 ,yj−1 ) (f )

,yj−1 )

(ξ (x),y

)

(x

,η (y))

i j−1 i−1 j (f ) + I(xi−1 (f ) + I(0,y ,0) j−1 )

(f )

1 (xi−1 ,yj−1 ) I (f ◦ uij ). N 2 (0,0)

As f ◦ uij = vij (·, f ), this gives 1 2 (xi−1 ,yj−1 ) (ξi (x),yj−1 ) (xi−1 ,ηj (y))  f ◦ uij (x, y) =  z0,0 + I(0,0) (f ) + I(xi−1 (f ) + I (f ) (0,yj−1 ) ,0) + =:

Cij (x,y) 1 (x,y) I(0,0) (f ) + 2 I(0,0) (vij |z=0 ) 2 N N

Cij  f (x, y) + Rij (x, y). N2

Hence  f is the unique fixed point of the operator : C1 (Q, R) → C1 (Q, R), −1 ϕ :=  vij (u−1 ij (·, ·), ϕ ◦ uij (·, ·)),

where  vij (x, y, z) = Rij (x, y, z) +

Cij z, N2

or, equivalently, graph f is the unique attractor of the IFS (Q × R,  w) with  w = { wij : Q × R → Q × R:  wij = (uij , vij ), i, j ∈ NN }. As the operator I(0,0) (·) is continuous,  f is continuous at its interpolating set (x,y)

 := {(xj , yj , zij ): i, j ∈ N0N }. To determine the zij , notice that uij (0, 0) = (xi−1 , yj−1 ) = zi−1,j−1 , and thus (xi−1 ,yj−1 )  f ◦ uij (0, 0) =  z0,0 + I(0,0) (f ) =:  zi−1,j−1 .

(9.69)

Fractal surfaces

369

Therefore Cij (1,1) 1 (1,1) (xi ,yj−1 ) (xi−1 ,yj ) I(0,0) (f ) + 2 I(0,0) (vij |z=0 ) + I(xi−1 ,0) (f ) + I(0,yj−1 ) (f ) 2 N N 1 (1,1) − z0,0 ) + 2 I(0,0) (vij |z=0 ) + ( zi−1,j − zi−1,j−1 ) + zi,j−1 . N

zi−1,j−1 +  zij =  =

Cij ( zN,N N2

(1,1) Hence the  zij can be expressed in terms of  z0,0 , Cij , and I(0,0) (vij |z=0 ), (i, j) = (0, 0). These results are now summarized in a theorem. Theorem 147. Suppose that graph f is a fractal surface generated by the IFS (Q × R, w), where wij = (uij , vij ) with

uij (x, y) =

1 N

0



1 N

0

x y







i−1 N j−1 N

+

and vij (x, y, z) = Aij x2 + Bij y2 + Cij z + Dij xy + Gij , such that max{|Cij |: i, j ∈ NN } < 1. Let  f (x, y) :=  z0,0 +

/

x/ y

for some z0,0 ∈ R.

f (s, t) dt ds 0

0

vij ), where Then graph f is the attractor of the IFS (Q × R,  w) with  wij = (uij , Cij 1 zi−1,j−1 + 2 z + 2  vij (x, y, z) =  N N

/

x/ y

vij (s, t, 0) dt ds, 0

i, j ∈ NN .

0

recursively and uniquely determined by  z0,0 , Furthermore, the  zij , (i, j) = (0, 0),0 are x0y which is a free parameter, Cij , and 0 0 vij (s, t, 0) dt ds. Moreover, ∇ f (x, y) = (gy (x), hx (y)), where / gy (x) =

/

y

f (x, t) dt 0

and

hx (y) =

x

f (s, y) ds. 0

∂ ∂ ∂ ∂ In addition, ∂x ∂y f = ∂y ∂x f = f . Proof. The last part of the theorem follows from calculus. The iteration of this procedure produces Cn -interpolating fractal surfaces for any n ∈ N.

Fractal surfaces and wavelets in Rn

10

Abstract In this chapter multivariate R-valued fractal functions are used to construct wavelet bases in Rn . To obtain nested subspaces, a prerequisite for the existence of a multiresolution analysis, foldable figures have to be chosen as the domains of the multivariate fractal functions. As was seen in Section 3.3, these foldable figures are in one-to-one correspondence with crystallographic Coxeter groups. At this point the reader is reminded of the notation and terminology as well as the results in Sections 3.2 and 3.3. First, fractal functions on foldable figures are constructed, and the concepts of interpolation and invariant subspaces are considered. Finally, a multiresolution analysis on L2 (Rn ) is presented and the corresponding wavelet basis is constructed. Section 5 then focuses on the relationship between fractal surfaces, affine Weyl groups, foldable figures, and wavelet sets. For this chapter, the following references were used: [11–18, 158, 159, 166, 171, 172, 229, 230].

1 Fractal functions on foldable figures Fractal surfaces defined over a given foldable figure are now constructed. For this purpose, let F be a foldable figure in Euclidean space E := Rn with the property that 0 is one of its vertices. Let S be the tessellation of Rn and H be the set of  be the affine reflection group generated by hyperplanes associated with F, and let W  is the group of affine isometries in En generated by the reflections rH for H; that is, W H ∈ H. Then the results in Sections 3.2 and 3.3 show the following properties of H : and W 1. H consists of the translates of a finite set of linear hyperplanes.   is simply transitive on S; that is, for any s, s ∈ S there exists a unique element rs s ∈ W 2. W mapping s onto s . 3. H ⊆ H for any  ∈ N, where H := {H: H ∈ H}.

Let  be a fixed positive integer and let X := F. Then X is also a foldable figure whose N subfigures Xi are in S. Note that the tessellation and the set of hyperplanes associated with X are S and  H, respectively. Furthermore, the affine reflection . The fact that the group W  is group generated by  H is an isomorphic subgroup of W simply transitive on S yields a set of similitudes ui : X → Xi , i ∈ NN , N =  n , in the following way. First, let u1 : X → X1 := F be given by u1 := 1 idRn . Define uj : X → Xj then by

Fractal Functions, Fractal Surfaces, and Wavelets. http://dx.doi.org/10.1016/B978-0-12-804408-7.00010-2 Copyright © 2016 Elsevier Inc. All rights reserved.

372

Fractal Functions, Fractal Surfaces, and Wavelets

uj := rXj X1 ◦ u1

(10.1)

for all j ∈ {2, . . . , N}. Let C∗N consist of all N-tuples λ := (λ1 , . . . , λN ) of continuous R-valued functions λi on X satisfying the following property: −1 (W) If ui X and uj X have a common face Eij , then λi (x) = λj (x) for all x ∈ u−1 i Eij = uj Eij , i, j ∈ NN .

For λ = (λ1 , . . . , λN ) ∈ C∗N and a fixed −1 < s < 1, let : C(X, R) × C∗N → RX be the Read-Bajraktarevi´c operator given by f :=

N  

 −1 λi ◦ u−1 χu i X . + sf ◦ u i i

(10.2)

i=1

Theorem 148. The Read-Bajraktarevi´c operator  defined by Eq. (10.2) is well defined and contractive in the supremum norm on C(X, R) with contractivity factor s. Hence it possesses a unique fixed point fλ ∈ C(X, R). Furthermore, the mapping θ : C∗N → C(X, R),

λ → fλ ,

(10.3)

is linear.  is simply transitive on Proof. Suppose Eij is a common face of ui X and uj X. Since W −1 S, it follows that uj := rXj Xi ◦ui . As rXj Xi is the identity on Eij , one has u−1 i (x) = uj (x) −1 −1 −1 for all x ∈ Eij . Thus λi ◦ u−1 i (x) + sf ◦ ui (x) = λj ◦ uj (x) + sf ◦ uj (x) for all m x ∈ ui X ∩ uj X. Hence for any f ∈ C(X, R ), f is well defined and continuous on X. Now let f , g ∈ C(X, Rm ). Then

    −1 (x) − g ◦ u (x))

f − g ∞ = sup s(f ◦ u−1  ≤ s f − g ∞ . i i x∈X i∈NN

Therefore, by the Banach fixed-point theorem,  has a unique fixed point fλ ∈ C(X, R). The linearity of θ follows directly from Eq. (10.2) and the uniqueness of the fixed point. The fixed point fλ ∈ C(X, R) is again called a multivariate fractal function and, for n > 1, its graph a fractal surface. N N  and recall that Bc (Rn ) Let C := r∈W  C∗ be the direct product of C∗ over W n denotes the linear space of nreal-valued functions bounded on compact subsets of R . For λ ∈ C, let fλ ∈ Bc (R ) be defined by fλ |

◦

rX

:= fλ(r) ◦ r−1 ,

, r∈W

where λ(r) := (λ(r)1 , . . . , λ(r)N ) is the “rth component” of λ.

(10.4)

Fractal surfaces and wavelets in Rn

373

1.5

1.0

0.5

0.2

0.4

0.6

0.8

1.0

Fig. 10.1 The graph of fλ for s := 34 .

Remark 75. The values of fλ on  H will be left unspecified so that fλ actually represents an equivalence class of functions. Fig. 10.1 shows the graph of a univariate fractal function fλ defined on the interval X := [0, 1] = [0, 12 ] ∪ [ 12 , 1] =: X1 ∪ X2 . Then u1 (x) = 12 x and u2 (x) = rX2 X1 ◦ u1 (x) = 1 − 12 x since rX2 X1 is the reflection about the line x = 12 . For the continuous functions λi , i = 1, 2, we used λ1 (x) = x3 and λ2 (x) = x2 . Condition (W) is satisfied since 1 ∈ {u1 X1 ( 12 )} = {u2 X2 ( 12 )} = {1}. See, in this context, also Example 23.

2 Interpolation on foldable figures In this section it is shown that continuous fractal functions constructed on a given foldable figure X can be used to interpolate a given set of data points. For this purpose, let W be the Coxeter group associated with the foldable figure X. If W is reducible, let W1 , . . . , WM denote its irreducible factors. Let Xm be the foldable figure corresponding M

to Wm , m ∈ NM . It is clear that X is isometrically isomorphic to X Xm . Without m=1

M

loss of generality, it may be assumed that X = X Xm and that the origin is in the m=1

set V of vertices of X. By Theorem 26, each Xm is a simplex in Rnm with nm + 1 vertices. Let m be the collection of affine maps λ: Rnm → R and let  := m∈NM m . Let  be a positive integer, let n := n1 +· · ·+nm , and let N =  n . Define u1 , . . . , uN and C∗N as in the previous section. Denote the set of vertices of ui X by Vi , i ∈ NN .

 Vi be a given interpolating set in Rn × Theorem 149. Let  := (v, zv ): v ∈ i∈NN N R. Then there exists a unique λ :=  (λ1 , . . . , λN ) such that each λi ∈ i and λ ∈ C∗ . Moreover, fλ (v) = zv for all v ∈ Vi . i∈NN

374

Fractal Functions, Fractal Surfaces, and Wavelets

Proof. First, suppose that M = 1 (ie, that W is irreducible). Hence X is a simplex, and so the vertices of X are geometrically independent. Thus given real numbers zv , v ∈ V, there exists a unique (affine) λ ∈  satisfying λ(v) = zv .

(10.5)

With use of properties of the tensor product, it follows inductively that there is a unique λ ∈  satisfying Eq. (10.5) in the M > 1 case. Now let λi , i ∈ NN , be the unique λi ∈  with the property

λi (v) = zui (v) − szv , v ∈ Vi , (10.6) i∈NN

and define λ := (λ1 , . . . , λN ). If Eij is a common face of ui X and uj X, then λi and λj agree on the vertices of −1 Xij := u−1 i Eij = uj Eij . Note that Xij is a foldable figure and so, by the unique interpolation property, λi ≡ λj on Xij . Hence λ ∈ CN .  Eqs. (10.2), (10.6) imply that fλ (v) = zv for all v ∈ Vi . i∈NN

Given a fractal function fλ defined over a foldable figure X, we can extend fλ to all of Rn by using the fact that the foldable figure X is a fundamental domain for its associated affine Weyl group and that it tessellates Rn by reflections in its ). For any foldable figure X  ∈ bounding hyperplanes (ie, under the action of W  such that X  = r(X). Define an extension S there exists a unique r ∈ W n f λ : R → R of fλ by f λ |X  := fλ ◦ r−1 . Similarly, we define an orthonormal affine fractal basis B (see Section 2.3) over X  by setting B := {b ◦ r−1 : b ∈ B}. Hence we obtain that

{b ◦ r−1 : b ∈ B}  r∈W

is an orthonormal affine fractal basis for L2 (Rn ).

 -invariant function spaces 3 Dilation- and W Let V be a linear space of real-valued functions on Euclidean space E := Rn , let  be  be a crystallographic Coxeter group. a positive integer greater than 1, and let W Definition 146. The linear space V is said to be dilation invariant (with scale ) iff  invariant iff f ∈ V implies f ◦ r ∈ V f ∈ V implies U f := f (·/) ∈ V. It is called W . for all r ∈ W , any Remark 76. Since the translations along the lattice are contained in W -invariant linear space V is also translation invariant with respect to the lattice W (ie, f ◦ τγ ∈ V for any γ ∈ ).

Fractal surfaces and wavelets in Rn

Definition 147. Let  :

375



C→



C be defined by

( λ)(r ◦ uj )i := λ(r)j ◦ ui + s(λ(r)i − λ(r)j )

(10.7)

for r ∈ W , i, j ∈ NN .  and, furthermore, that for any Remark 77. Eq. (10.1) implies that r ◦ uj ∈ W  there is some r ∈ W  and some j ∈ NN such that r = r ◦ uj . r ∈ W Theorem 150. Let λ ∈ C. Then U fλ = f λ .

(10.8)

Proof. First it is shown that for any r ∈ W and j ∈ NN , fλ(r) ◦ uj = f λ(r◦uj ) .

(10.9)

To see this, note that fλ(r) ◦ uj ◦ ui = λ(r) ◦ ui + sfλ(r) ◦ ui = λ(r) ◦ ui + s(λ(r) ◦ ui + sfλ(r) ) = λ(r) ◦ ui + s(λ(r) ◦ ui − λ(r) ◦ uj ) + s(λ(r) ◦ uj − sfλ(r) ) =  λ(ruj ) ◦ ui + sfλ(r) ◦ uj . Equality Eq. (10.9) now follows from the uniqueness of the fixed point of  λ . Finally, U fλ |r◦uj (X) = U fλ |r◦uj X = U fλ(r) ◦ r−1 |r◦uj X −1 = fλ(r) |uj X = fλ(r) ◦ uj ◦ u−1 j ◦ ( ◦ r)

= f λ(

r◦uj )

◦ (r ◦ uj )−1 .

Theorem 150 may be used to construct U -invariant function spaces from  invariant subspaces of C. For example, let d be the collection of all λ ∈ C∗N such that each component λi is a polynomial of degree at most d. It is not hard to verify that  d is  invariant. r∈W

4 Multiresolution analyses Next it is shown that the function spaces constructed previously form multiresolution analyses of L2 (Rn ). To this end, let  be a finite-dimensional subspace of C∗N such that   is  invariant. Define r∈W ⎧ ⎫ ⎨  ⎬ V0 := fλ : λ ∈  (10.10) ⎩ ⎭  r∈W

376

Fractal Functions, Fractal Surfaces, and Wavelets

and Vk , for 0 = k ∈ Z, by f ∈ Vk iff U −k f ∈ V0 . Let A := dim . Clearly V0 |X also has dimension A. Therefore one can construct an orthonormal basis {φ 1 , . . . , φ A } of V0 |X using, for instance, the Gram-Schmidt orthonormalization algorithm. Then } Bφ := {φ a ◦ r: a ∈ NA ∧ r ∈ W is an orthonormal basis of V0 . Let φ := (φ 1 , . . . , φ A )t . As V1 ⊂ V0 , there exists a sequence of A × A matrices {cr }r∈W , only a finite number of which are different from the zero matrix 0, such that  φ(x/) = cr (φ ◦ r)(x). (10.11)  r∈W

 is the semidirect product of with W , the basis Bφ can also Remark 78. Since W be generated by translates along the lattice of the enlarged set of scaling functions {φ ◦ r: r ∈ W }. Theorem 151. The sequence {Vk }k∈Z defined earlier is a multiresolution analysis . of L2 (Rn ) with respect to W Proof. Conditions (M1), (M4), and (M5) in Definition 129 required for a multiresolution analysis follow directly from the preceding construction. Conditions (M2), (M3), and (M6) are a consequence of Theorem 2.1 in Ref. [162]. For completeness we restate this theorem next using the notation and terminology developed in this monograph.  Theorem 152 (Ref. [162]). Let φ ∈ L2 (R) and let φp :=

∈Z |φ(· − )| be its periodization. Denote by L2 (R) the set of all Lebesgue measurable functions φ for which φp L2 ([0,1]n) < ∞. Suppose that φ ∈ L2 (R)A and that there exists a positive constant m such that m s

2 (Zn )A ≤ s  φ L2 (R)A . If there exists a biinfinite sequence c of A × A matrices whose elements are in 2 (Zn ) so that  φ= c φ(2 · − ),

∈Zn

then φ admits multiresolution. Proof. For details and the proof, see Ref. [162]. Next the wavelets associated with this multiresolution analysis are defined. For this purpose, denote the orthogonal complement of Vk in Vk−1 by Wk . First, a basis for W0 needs to be found. Note that the function space V−1 |X has dimension  n A, while V0 |X is an A-dimensional subspace of V−1 . Thus again using the Gram-Schmidt orthonormalization algorithm, one can construct an orthonormal basis of B = ( n −1)A functions ψ 1 , . . . , ψ B . Then

Fractal surfaces and wavelets in Rn

377

} Bψ := {ψ b ◦ r: b ∈ NB ∧ r ∈ W is an orthonormal basis of W0 . The functions ψ 1 , . . . , ψ B are now the wavelets, and Bψ is the wavelet basis. Note that the scaling functions φ 1 , . . . , φ A , as well as the wavelets ψ 1 , . . . , ψ B , are orthogonal and compactly supported, but not continuous. Also, Wk is orthogonal to W for k = . The wavelet spaces Wk form an orthogonal direct sum decomposition of L2 (Rn ). Let ψ := (ψ 1 , . . . , ψ B )t . Since W1 ⊂ V0 , there exists a sequence of B × A matrices {dr }r∈W  such that ψ(x/) =



dr (φ ◦ r)(x).

(10.12)

 r∈W

As ψ is compactly supported, only a finite number of the coefficient matrices dr are nonzero. The orthonormality of Bφ and Bψ leads to simple decomposition and reconstruction algorithms. To this end, let f0 ∈ V0 . Then f0 = f1 + g1 , where f1 and g1 denote the orthogonal projections of f0 onto V1 and W1 , respectively. Expressing f0 =, f1 , and g1 in terms their respective bases yields f0 =



atr (0)φ ◦ r,

 r∈W

f1 =



atr (1)φ ◦ r,

 r∈W

and g1 =



btr (1)ψ ◦ r,

 r∈W

where the coefficients are row matrices of the appropriate size. Eqs. (10.11), (10.12) now yield the decomposition algorithm ar (1) =

  r  ∈W

ar r −1 (0)ctr



and br (1) =

 r  ∈W

as well as the reconstruction algorithm ar (0) =

  r  ∈W

ctr ar r −1 (1) + dtr br r −1 (1).

br r −1 (0)dtr ,

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Fractal Functions, Fractal Surfaces, and Wavelets

5 Wavelet sets and fractal surfaces In this section we establish a connection between the various concepts introduced in the foregoing presentations and the multiscale structure of wavelets. Our emphasis will be entirely on wavelet sets in Rn , and most of the material found here is taken from Refs. [163–166]. The reader interested in a more operator-theoretic formulation of wavelet sets is referred to Ref. [164]. For motivational purposes, we present first the one-dimensional setting (ie, wavelet sets on R) and then proceed to the higherdimensional scenario.

5.1 Translation/dilation wavelet sets To this end, we need to first introduce the following definition of wavelets. Definition 148. A dyadic orthonormal wavelet on R is a unit vector ψ ∈ L2 (R, m), where m denotes the Lebesgue measure, with the property that the set {2n/2 ψ(2n t − ): n, ∈ Z}

(10.13)

of all integral translates of ψ followed by dilations by arbitrary integral powers of 2 is an orthonormal basis for L2 (R, m). For later developments we require the following two operators. Let T denote the unitary translation and D denote the unitary dilation operator in B(L2 (R)), the Banach space of bounded linear operators from L2 (R) to itself, defined by (Tf )(t) := f (t − 1) and (Df )(t) :=

√ 2f (2t),

f ∈ L2 (R), t ∈ R.

Then we may write Eq. (10.13) more succinctly as 2n/2 ψ(2n t − ) = (Dn T ψ)(t) for all n, ∈ Z, t ∈ R. Note that TD = DT 2 . Next we introduce the unitary version of the Fourier transform on R and use it for the remainder of this chapter. For f , g ∈ L1 (R) ∩ L2 (R), let 1 (F f )(s) := √ 2π

 R

e−ist f (t) dt =:  f (s)

(10.14)

and (F

−1

1 g)(t) = √ 2π

 R

eist g(s)ds =  g(−t) =: g∨ (t).

(10.15)

Note that this particular form of the Fourier-Plancherel transform defines a unitary operator on L2 (R). In particular, we remark that

Fractal surfaces and wavelets in Rn

1 (F T)(f )(s) := √ 2π

379

 R

e−ist f (t − 1) dt = e−is (F f )(s) =: e−is f (s).

T := F TF −1 , then  T = Me−is . Hence F TF −1 f = e−is f =: Me−is f . If we define  Similarly, we obtain for any n ∈ Z  √ n √ −n 1  1 −n (F Dn f )(s) = √ e−ist 2 f (2n t)dt = 2 √ e−i2 st f (t) dt 2π R 2π R √ −n 2 (F f )(2−n s) = (D−n F f )(s). = Dn = D−n and therefore  D = D−1 . As above, if we set  Dn := F Dn F −1 , then  Wavelet sets belong to the theory of wavelets via the Fourier transform. Definition 149. A wavelet set in R is a measurable subset E of R for which √1 χE 2π is the Fourier transform of a wavelet. E := √1 χE is sometimes called s-elementary. The class of wavelet The wavelet ψ 2π sets was also investigated in Refs. [167, 168]. In their theory the corresponding wavelets are called minimally supported frequency wavelets. The prototype of a wavelet set is the Shannon set (see Fig. 10.2) given by ES = [−2π, −π) ∪ [π, 2π).

(10.16)

The orthonormal wavelet for ES is then given by ψS (t) = 2 sinc(2t − 1) − sinc t. To prove that ψS is indeed an orthonormal wavelet, note that the set of exponentials {ei s : ∈ Z}

1.0 0.5

–20

–10

10 –0.5 –1.0

Fig. 10.2 The Shannon wavelet.

20

380

Fractal Functions, Fractal Surfaces, and Wavelets

restricted to [0, 2π] and normalized by √1 is an orthonormal basis for L2 [0, 2π]. Write 2π ES = E− ∪ E+ , where E− = [−2π, −π), E+ = [π, 2π). Since {E− + 2π, E+ } is a partition of [0, 2π) and since the exponentials ei s are invariant under translation by 2π, it follows that

 ei s  √  : ∈Z 2π ES T = Me−is , this set can be written as is an orthonormal basis for L2 (ES ). Since  S : ∈ Z}. { T ψ Next, note that any “dyadic interval” of the form J = [b, 2b) for some b > 0 has the property that {2n J: n ∈ Z} is a partition of (0, ∞). Similarly, any set of the form K = [−2a, −a) ∪ [b, 2b)

for a, b > 0 has the property that {2n K: n ∈ Z} is a partition of R\{0}. To complete the proof we need to introduce one more item. Definition 150. Let U be a given unitary operator on a Hilbert space H. A nonempty subspace K of H is called a wandering subspace for U if U m (K) ⊥ U n (K),

∀m = n ∈ N.

 n If, in addition, H = n∈Z U (K), then we say that K is a complete wandering subspace of H for U. It follows that the space L2 (K), considered as a subspace of L2 (R), is a complete √ wandering subspace for the dilation unitary (Df )(s) = 2 f (2s). For each n ∈ Z, Dn (L2 (K)) = L2 (2−n K).  So n∈Z Dn (L2 (K)) is a direct sum decomposition of L2 (R). In particular, ES has this property. Thus

Dn

⎫ ⎧ n   ⎨ e2 i s  ⎬ ei s   √  : ∈Z = √  : ∈Z ⎩ 2π  −n ⎭ 2π ES 2 E S

is a basis for L2 (2−n ES ) for each n. Hence S : n, ∈ Z} = { S : n, ∈ Z} {Dn Dn T ψ T ψ is an orthonormal basis for L2 (R), as required.

Fractal surfaces and wavelets in Rn

381

It follows from the above arguments that sufficient conditions for E to be a wavelet set are   (W1) The set of normalized exponentials √1 ei s : ∈ Z , when restricted to E, constitutes 2π

an orthonormal basis for L2 (E). (W2) The family {2n E: n ∈ Z} of dilates of E by integral powers of 2 should constitute a measurable partition (ie, a partition modulo null sets) of R.

These observations now form the basis for the next definitions. Definition 151. Two measurable sets E, F ⊆ R are called translation congruent modulo 2π if there exists a measurable bijection φ: E → F such that φ(s) − s = n(s)2π,

∀s ∈ E,

and a unique n(s) ∈ Z, or, equivalently, if there is a measurable partition {En : n ∈ Z} of E such that {En + n2π: n ∈ Z} is a measurable partition of F. Two measurable sets G, H ⊆ R are called dilation congruent modulo 2 if there is a measurable bijection τ : G → H such that for each s ∈ G there exists an n = n(s) ∈ Z with τ (s) = 2n s, or, equivalently, if there is a measurable partition {Gn : n ∈ Z} of G such that {2n Gn : n ∈ Z } is a measurable partition of H. Example 60. E := [0, 2π] and F := [−2π, −π] ∪ [π, 2π] are translation congruent modulo 2π. The following lemma is proved in Ref. [163]. Lemma 25. Let f ∈ L2 (R), and let E = supp(f ). Then f has the property that {ei s f : ∈ Z} is an orthonormal basis for L2 (E) iff: • •

E is congruent to [0, 2π) modulo 2π. |f (s)| = √1 a.e. on E. 2π

Observe that if E is 2π-translation congruent to [0, 2π), then since {[0, 2π) + 2πn: n ∈ Z} is a measurable partition of R, so is {E + 2πn: n ∈ Z}. Similarly, if F is 2-dilation congruent to the Shannon wavelet set ES = [−2π, −π) ∪ [π, 2π), then since {2n ES : n ∈ Z} is a measurable partition of R, so is {2n F: n ∈ Z}.

382

Fractal Functions, Fractal Surfaces, and Wavelets

We say that a measurable subset G ⊆ R is a 2-dilation generator of a partition of R if the sets 2n G := {2ns: s ∈ G}, are disjoint and R \



n ∈ Z,

2n G is a null set.

n∈Z

Analogously, we say that E ⊆ R is a 2π-translation generator of a partition of R if the sets E + 2nπ := {s + 2nπ: s ∈ E}, are disjoint and R \



n ∈ Z,

(E + 2nπ) is a null set.

n∈Z

The next theorem gives necessary and sufficient conditions for a measurable set E ⊆ R to be a wavelet set. Before we state it, we need a definition. Definition 152. A fundamental domain for a group of (measurable) transformations G on a measure space (, μ) is a measurable set C with the  that {g(C): g ∈ G }  property  g(C) is a μ-null set and is a measurable partition (tessellation) of ; that is,  \ g∈G

g1 (C) ∩ g2 (C) is a μ-null set for g1 = g2 . Theorem 153. Let E ⊆ R be a measurable set. Then E is a wavelet set iff one of the following equivalent conditions holds 1. E is both a 2-dilation generator of a partition (modulo null sets) of R and a 2π-translation generator of a partition (modulo null sets) of R. 2. E is both translation congruent to [0, 2π) mod 2π and dilation congruent to [−2π, −π) ∪ [π, 2π) mod 2. 3. E is a fundamental domain for the dilation group Dn : n ∈ Z and at the same time a k : k ∈ Z. Here T fundamental domain for the translation group T2π 2π is translation by 2π along the real axis.

Proof. See Ref. [163]. Now we extend the above concepts and definitions to Rn . We will do this in a slightly more general setting. To this end, let X be a metric space and μ a σ -finite nonatomic Borel measure on X for which the measure of every open set is positive and for which bounded sets have finite measure. Let T and D be countable groups of homeomorphisms of X that map bounded sets to bounded sets and which are absolutely continuous in the sense that they map μ-null sets to μ-null sets. Furthermore, let G be a countable group of absolutely continuous Borel measurable bijections of X. Denote by B(X) the family of Borel sets of X. The following definition completely generalizes the definitions of 2π-translation congruence and 2-dilation congruence given above. Definition 153. Let E, F ∈ B(X). We call E and F G -congruent and write E ∼G F if there exist measurable partitions {Eg : g ∈ G } and {Fg : g ∈ G } of E and F, respectively, such that Fg = g(Eg ) for all g ∈ G , modulo μ-null sets.

Fractal surfaces and wavelets in Rn

383

This definition immediately entails the next two results. Proposition 85 1. G-congruence is an equivalence relation on the family of m-measurable sets. 2. If E is a fundamental domain for G, then F is a fundamental domain for G iff F ∼G E.

Proof. See Ref. [163]. Definition 154. We call (D, T ) an abstract dilation-translation pair if: 1. for each bounded set E and each open set F there exist elements δ ∈ D and τ ∈ T such that τ (F) ⊂ δ(E); and 2. there exists a fixed point θ ∈ X for D with the property that if N is any neighborhood of θ and E is any bounded set, there is an element δ ∈ D such that δ(E) ⊂ N.

The next result and its proof can be found in Ref. [164]. Theorem 154. Let X, B, μ, D, and T be as above. Let (D, T ) be an abstract dilation-translation pair, with θ being the fixed point of D. Assume that E and F are bounded measurable sets in X such that E contains a neighborhood of θ , and F has a nonempty interior andis bounded away from θ . Then there exists a measurable set G ⊂ X, contained in δ(F), which is both D-congruent to F and T δ∈D

congruent to E. The following is a consequence of Proposition 85 and Theorem 154 and is the key to obtaining wavelet sets. Corollary 17. With the terminology of Theorem 154, if in addition F is a fundamental domain for D and E is a fundamental domain for T , then there exists a set G which is a common fundamental domain for both D and T . To apply the above result to wavelet sets in Rn , we require the following two definitions. Definition 155. Let A ∈ Mn (R) be an n × n matrix with real coefficients. By an orthonormal (DA , T)-wavelet we mean a function ψ ∈ L2 (Rn ) such that   (10.17) | det(A)|n/2 ψ(An t − ): n ∈ Z, ∈ Zn , i = 1, . . . , n , where = ( 1 , 2 , . . . , n )t is an orthonormal basis for L2 (Rn ; m). (Here m is product Lebesgue measure.) If A ∈ Mn (R) is invertible (so in particular if A is expansive), then the operator defined by (DA f )(t) = | det A|1/2 f (At) for f ∈ L2 (Rn ), t ∈ Rn , is unitary. For 1 ≤ i ≤ n, let Ti be the unitary operator determined by translation by 1 in the ith coordinate direction. Then the set Eq. (10.17) can be written as {DkA T ψ: k, ∈ Zn }, with T := T1 1 · · · Tn n .

384

Fractal Functions, Fractal Surfaces, and Wavelets

Definition 156. A (DA , T)-wavelet set is a measurable subset E of Rn for which the inverse Fourier transform of (m(E))−n/2 χE is an orthonormal (DA , T)-wavelet. Two measurable subsets H and K of Rn are called A-dilation congruent, in symbols H ∼δA K, if there exist measurable partitions {H : ∈ Z} of H and {K : ∈ Z} of K such that K = A H modulo Lebesgue null sets. Moreover, two measurable sets E and F of Rn are called 2π-translation congruent, written E ∼τ2π F, if there exist measurable partitions {E : ∈ Zn } of E and {F : ∈ Zn } of F such that F = E +2π modulo Lebesgue null sets. We remark that this generalizes to Rn the previous definition of 2π-translation congruence for subsets of R. Observe that A-dilation by an expansive matrix together with 2π-translation congruence is a special case of an abstract dilation-translation pair as introduced in Definition 154. Let D := Ak : k ∈ Z be the dilation group generated

by powers of A, and let T := T2π : ∈ Zn  be the group of translations generated by the translations T2π along the coordinate directions. Let E be any bounded set and let F be any open set that is bounded away from 0. Let r > 0 be such that E ⊆ Br (0). Since A is expansive, there is an ∈ N such that A F contains a ball B of radius large enough so that B contains some lattice point 2kπ together with the ball BR (2kπ) of radius R > 0 centered at the lattice point. Then E + 2kπ ⊆ A F. That is, the 2kπ-translate of E is contained in the A -dilate of F, as required in condition 1 of Definition 154. For condition 2 of Definition 154, let θ = 0, and let N be a neighborhood of 0, and let E be any bounded set. As above, choose r > 0 with E ⊆ Br (0). Let

∈ N be such that A N contains Br (0). Then A− is the required dilation such that A− E ⊆ N. Note that if W is a measurable subset of Rn that is 2π-translation congruent to the N

Tj that n-cube E := X [−π, π), it follows from the exponential form of  i=1

   T1 1  Tn n (m(W))−1/2 χW : = ( 1 , 2 , . . . , n ) ∈ Zn T2 2 · · ·  is an orthonormal basis for L2 (W). Furthermore, if A is an expansive matrix and B is the unit ball of Rn , then with FA := A(B) \ B the collection {Ak FA : k ∈ Z} is a partition of Rn \ {0}. Consequently, L2 (FA ), considered as a subspace of L2 (Rn ), is a complete wandering subspace for DA . Hence L2 (Rn ) is a direct sum decomposition of the subspaces {DkA L2 (FA ): k ∈ Z}. Clearly any other measurable set F  ∼δA FA has this same property. The following theorem gives the existence of wavelet sets in Rn . For the proof, see Ref. [165]. Theorem 155. Let n ∈ N and let A be an expansive n × n matrix. Then there exist (DA , T)-wavelet sets. Example 61. An example of a (fractal) wavelet set in R2 is shown in Fig. 10.3. The exact construction parameters are given in Ref. [163]. For more examples and constructions of wavelet sets, we encourage the reader to consult, for instance, Refs. [165, 169–173].

Fractal surfaces and wavelets in Rn

385

2p

−2 p

2p

−2 p

Fig. 10.3 A wavelet set in R2 .

5.2 Reflection/dilation wavelet sets Finally, we put the concepts of fractal hypersurface, foldable figure, affine Weyl group, and wavelet together and introduce a new type of wavelet set called a dilation-reflection wavelet set. The idea is to adapt Definition 154, replacing the group of translations T in  whose fundamental domain the traditional wavelet theory by an affine Weyl group W is a foldable figure C, and to use the orthonormal basis of affine fractal hypersurfaces constructed in the previous four sections. In Definition 154, we take X := Rn endowed with the Euclidean affine structure and distance, and for the abstract translation group T we take the affine Weyl group  generated by a group of affine reflections arising from a locally finite collection of W affine hyperplanes of X. , which is also a foldable figure. Recall Let C denote a fundamental domain for W that C is a simplex (ie, a convex connected polytope) which tessellates Rn by reflections about its bounding hyperplanes. Let θ be any fixed interior point of C. Let A be any real expansive matrix in Mn (R) acting as a linear transformation on Rn . In the case where θ is the origin 0 in Rn , we simply take DA to be the usual dilation by A and the abstract dilation group to be D = {DkA : k ∈ Z}. For a general θ , define DA,θ to be the affine mapping Dθ (x) := A(x − θ ) + θ , x ∈ Rn and Dθ = {DkA,θ : k ∈ Z}. ) is an abstract dilation-translation pair in the sense of Proposition 86. (Dθ , W Definition 154. Proof. The proof is given in Ref. [166] but for completeness we repeat it here. By the definition of D, θ is a fixed point for Dθ . By a change of coordinates we may assume without loss of generality that θ = 0 and consequently that D is multiplication by A on Rn .

386

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Let Br (0) be an open ball centered at 0 with radius r > 0 containing both E and C. Since F is open and A is expansive, there exists a k ∈ N sufficiently large so that Dk F contains an open ball B3r (p) of radius 3r and with some center p. Since C tiles , there exists a word w ∈ W  such that w(C) ∩ Br (p) has Rn under the action of W positive measure. (Note here that Br (p) is the ball with the same center p but with smaller radius r.) Then w(Br (0)) ∩ Br (p) = ∅. Since reflections (and hence words in ) preserve diameters of sets in Rn , it follows that w(Br (0)) is contained in B3r (p). W Hence w(E) is contained in Dk (F), as required. This establishes condition 1 of Definition 154. Condition 2 follows from the fact that θ = 0 and D is multiplication by an expansive matrix in Mn (R). Now we extend the definition of a (DA , T)-wavelet set in Rn to this new setting.  acting on Rn with fundamental Definition 157. Given an affine Weyl group W domain a foldable figure C, given a designated interior point θ of C, and given an )-wavelet set is a measurable subset E of Rn expansive matrix A on Rn , a (DA,θ , W satisfying the properties:  1. E is congruent to C (in the sense of Definition 152) under the action of W. 2. W generates a measurable partition of Rn under the action of the affine mapping D(x) := A(x − θ) + θ.

) to (DA , W ). In the case where θ = 0, we abbreviate (DA,θ , W  The next result establishes the existence of (DA,θ , W )-wavelet sets. The proof is a direct application of Theorem 154 and can be found in Ref. [166]. )-wavelet sets for every choice of Theorem 156. There exist (DA,θ , W  A, and θ . W, In the case of a dilation-translation wavelet set W, the two systems of unitaries are D := {DkA : k ∈ Z}, where A ∈ Mn (R) is an expansive matrix, and T := {T : ∈ Zn }. An orthonormal wavelet basis of L2 (Rn ) is then obtained by W := (m(W))−1/2 χW and taking our setting ψ    W : k ∈ Z, ∈ Zn . DkA T ψ

(10.18)

, the affine Weyl group For the systems of unitaries D := {DkA : k ∈ Z} and W associated with a foldable figure C, one obtains as an orthonormal basis for L2 (Rn ) 

  , DkI Br : k ∈ Z, r ∈ W

(10.19)

where Br = {b ◦ r: b ∈ B} is an affine fractal hypersurface basis as constructed in Section 2.3.

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Index Note: Page numbers followed by f indicate figures. A Abel integral equation, 238 Address function, 96 Address structure, 99 Admissible covers, 357 Affine fractal basis, 340–341, 340f Affine fractal functions box dimension, 243 contractive IFS, 243 functional inequalities, 241 nonoverlapping ε-cover, 240 upper and lower functions, 241–242 Affine fractal hypersurface. See Affinely generated fractal hypersurface Affine fractal surfaces in Rn+m affinely generated fractal surfaces, 337–343 construction, 332–334 fractal hypersurfaces in Rn+1 , 334–337 Affinely generated fractal hypersurface affine fractal basis, 340–341 construction, 337–340 coplanar boundary data, 341–343 Affine mappings, 251 Affine Weyl groups and foldable figures Bourbaki’s theorem, 37 Coxeter/Dynkin diagrams, 39 crystallographic, 39 irreducible root systems, 37, 38f unit cube and prism, 39, 40f Algebraic manipulation, fractels, 234–236 B Banach fixed-point theorem biaffine fractal functions, 211–212 local fractal functions, 161 mathematical preliminaries, 9 RB operator, 146 recurrent sets, 83, 86

Banach space fractal function wavelets, 284 mathematical preliminaries, 8–9 quasi-Banach space, 41 Basic wavelet theory decomposition and reconstruction algorithm, 280–283 dilation equation, 272–277 multiresolution analyses, 262–272 RB operator, 277–280 wavelet decompositions, 262 Besicovitch–Ursell theorem, 58 Besov spaces fractal functions, 226–228 function spaces, 43–52 Biaffine fractal functions Banach fixed-point theorem, 211–212 bilinear filtering, 210 bilinear fractal interpolant, 215 bilinear interpolation, 210 dimension, 251–259 IFS, 214 Lipschitz constant, 213–214 self-referential functional equations, 215 taxi-cab metric, 212–213 Biaffine mappings, 251 Bilinear filtering, 210 Bilinear fractal interpolant, 215 Bilinear interpolation, 210 Billingsley dimension, 111–112 Biorthogonal scaling vectors, 325 Birkhoff’s ergodic theorem, 131 Bivariate fractal function, 362–363 Borel field, 16 Borel measurable bijections, 382 Borel probability measure ergodic theorems and entropy, 134 Lyapunov dimension, 136–137 Bourbaki’s theorem, 37

398

Box dimension of projections geometrical observations, 124 metric dimensions, 106–107 OSC, 126–127 Perron–Frobenius theorem, 125–126 recurrent IFS, 122–126 row-stochastic irreducible matrix, 124–125 ˇ Brouwer-Cech dimension dimension, 102 C Calderón’s formula, 49 Canonical/adjoint pairing, 8 Cantor set, 54 Cartesian product fractels, 237 hidden-variable fractal functions, 180–181 topological dimensions, 102 Cauchy net, 8 Cauchy–Schwarz inequality hidden-variable fractal functions, 251 mathematical preliminaries, 4, 251 Cauchy’s formula, 238 Chebyshev norm, 13 Classical fractal sets Cantor set, 54 Hausdorff measures and Hausdorff dimension, 54–57 Lebesgue measure, 54 Weierstraß-like fractal functions, 57–58 Code space hidden-variable fractal functions, 182–183 IFS, 63 mathematical preliminaries, 22 transformations, fractal sets, 97 Continuous local fractal interpolation function, 220 Contractive IFS affine fractal functions, 243 hidden-variable fractal functions, 183 local, 77 local fractal functions, 162 RB operator, 150, 156–157 recurrent sets, 93 Contravariant homeomorphism, 363 Coplanar boundary data, 341–343

Index

Countable Markov chain, 23 Coxeter/Dynkin diagrams, 39 Coxeter groups, 344 D Daubechies–Lagarias theorem, 279 Daubechies scaling function decomposition and reconstruction algorithm, 282–283 dilation equation, 277, 283 piecewise fractal functions, 325–326 RB operator, 280f Decomposition and reconstruction algorithm Daubechies scaling functions, 282–283 inverse Mallat transform, 281–282 Mallat transform, 280–281 Schwarz’s inequality, 283 Sobolev space, 282–283 Whitney–Jackson type, 282–283 Dekking’s construction, 79–80 DGHM multiwavelet, 321–322 Digraph recursive fractal, 96  Dilation-and W-invariant function spaces, 374–375 Dilation congruent modulo 2, 381 Dilation equation compactly supported wavelets, 273 Daubechies scaling functions and wavelets, 277, 283 Fourier transform, 276–277 Galerkin methods, 283 RB operator, 277–280 trigonometric polynomial solution, 274 Dimension, fractal functions affine, 239–245 biaffine, 251–259 hidden-variable, 248–251 recurrent, 245–248 Dimension theory box dimension of projections, 122–127 metric, 104–111 probabilistic, 111–114 self-affine fractal sets, 114–122 topological, 101–104 Discrete wavelet transform, 267 Dragon fractals, 84–85, 85f Dyadic interpolation, 171 Dyadic orthonormal wavelet, 378

Index

Dynamical systems and dimension ergodic theorems and entropy, 129–135 Lyapunov dimension, 135–142 E Entropy. See Ergodic theorems and entropy Ergodic theorems and entropy attractor, 129 Birkhoff’s ergodic theorem, 131 Borel probability measure, 134 diffeomorphism, 131 (n-dimensional) dynamical system, 129 entropy of the partition, 133 Fréchet differentiable, 130 Kolmogorov’s definition, 132 logistic map, 130 Oseledec’s multiplicative ergodic theorem, 130, 132 Pesin’s formula, 134 tangent space, 131 Euclidean space/unitary space, 4 Existence theorem, 25, 28 Extrema of fractal functions, 193–197 F Finitely generated shift invariant space, 263 Foldable figures fractal functions, 371–373 interpolation, 373–374 Fourier–Plancherel transform multiresolution analysis, 268–269 translation/dilation wavelet sets, 378–379 Fourier transform dilation equation, 276–277 fractal function wavelets, 296 integral transforms, 190 orthogonal fractal function wavelets, 313–315 Fractal bases affine fractal function, 165f Gram orthonormalization procedure, 166 hyperfinite splines, 164 Lagrange type, 163–164 Fractal functions Besov spaces, 226–228

399

biaffine, 210–216 class Ck , 204–209 extrema, 193–197 fractal bases, 163–166 hidden-variable, 180–185 indefinite integrals, continuous fractal functions, 207–209 integral transforms, 189–191 iterative interpolation functions, 169–178 Lebesgue spaces, 216–218 Lipschitz continuity, 192–193 local, 157–163 moment theory, 187–189 notation and terminology, 186 Peano curves, 197–203 RB operator, 145–157 recurrent, 178–180 recurrent sets, 166–169 smoothness spaces Cn and Hölder spaces Cs , 218–224 Sobolev spaces Wm,p , 224–226 Triebel–Lizorkin spaces, 228–230 Weierstraß-like, 57–58 Fractal function wavelets Banach space, 284 density condition, 294–295 Fourier transform, 296 linear isomorphism, 286–287 multiresolution analysis, 300–302 parameterized spline space, 284 piecewise, 287f Poisson summation formula, 295–297 rescaling functions, 285–286 Riesz basis, 298, 300 separation condition, 294–295 translation and dilation invariance, 295 Fractal hypersurfaces in Rn+1 , 334–337 Fractal interpolation functions, 152 Fractal surfaces class Ck , 366–369 hidden-variable, box dimension, 350–356 Hölder continuity, 362–363 oscillation of f  , 344–350 p-balanced measures and moment theory, 363–365 recurrent IFS, box dimension, 357–362

400

Fractel algebra, 234–236 Cartesian products and function composition, 237 integrals and derivatives, 237 linear, 232 RB operator, 233–234 self-referential, 232 trivial fractel, 231 Fréchet differentiable, 130 Fréchet metric, 22 Free semigroup endomorphism, 89–90 Frobenius representation, 80 Function spaces Besov and Triebel–Lizorkin spaces, 43–52 Hölder spaces, 41–42 Lebesgue spaces, 40–41 Sobolev spaces, 42–43 G Galerkin methods, 283 GHM scaling vector, 310–313 Gram orthonormalization procedure, 166 Gram–Schmidt orthogonalization, 341 Gram–Schmidt orthonormalization, 375–376 Graph-directed fractal constructions Mauldin–Williams fractal/digraph recursive fractal, 96 Mauldin–Williams graph, 94–95 similarity constant, 94 strictly contracting, 94 H Haar system, 265 Hahn–Banach extension theorem IFS, 66 mathematical preliminaries, 11 Hausdorff–Besicovitch dimension, 55–56 Hausdorff dimension Ergodic theorems and entropy, 134–135 Lyapunov dimension, 139 metric dimensions, 104, 106–107, 109 probabilistic dimensions, 111 recurrent sets and Mauldin–Williams fractals, 122 self-similar fractals, 114–115

Index

Hausdorff measures Hausdorff dimension, 55 Lipschitz function, 56 s-dimensional, 55 Sierpi´nski gasket/Sierpi´nski triangle, 56, 57f Hausdorff metric IFS, 59–60 local IFS, 78 recurrent sets, 81, 168 Hidden-variable fractal functions Cartesian product, 180–181 code space, 182–183 contractive IFS, 183 dimension, 248–251 Lipschitz constant, 183 Peano curve, 185 Sierpi´nski triangle, 184 supremum norm, 181 Hilbert cube, 103–104 Hilbert’s construction, 198 Hilbert space mathematical preliminaries, 8 multiresolution analysis, 265 Hölder continuity, 362–363 Hölder exponent orthogonal fractal function wavelets, 310 RB operatorr, 279 Hölder spaces, 41–42 Homeomorphic address structures, 100 Homomorphism, 25 Hyperfinite splines, 164 I Inner product space, 4 Integral transforms affine fractal functions, 191 Fourier transform, 190 Intermediate space, 47 Interpolation bilinear, 210 foldable figures, 373–374 functions, 321–322 points, 344–346 spaces, 47 Inverse Mallat transform, 281–282

Index

Inverse system of topological spaces, 32 Involuntary automorphism, 3 Irreducible root systems, 37, 38f Iterated function systems (IFS) attractor, 60 biaffine fractal functions, 214 classical Cantor set, 61 code space, 63 contractive/hyperbolic, 59 Hahn–Banach extension theorem, 66 Hausdorff metric, 59–60 Hutchinson metric, 62 iterated Riemann surfaces, 73–76 Lipschitz constant, 62 local, 76–79 Lyapunov dimension, 137–138 moment theory and, 67–69 point fibered, 65 recurrent, 69–73 Sierpi´nski triangle, 61 Iterated Riemann surfaces algebraic function, 75 coordinate transition functions, 74 definition, 76 holomorphic, 74 Riemann sphere, 74 Iterative interpolation functions Dubuc’s interpolation process, 170–171 dyadic interpolation, 171 fundamental interpolating function, 172 Koch curve, 171, 171f linearity, 170 scalar multiplication, 170 topological group, 169 translation, 171 Iterative interpolation process, 343 J Jordan curve, 167–168 K Kiesswetter’s curve, 166, 167f Kiesswetter’s fractal function, 166, 167f Knot points, 218–219 Koch curve, 171, 171f

401

Kolmogorov’s definition, 132 Kronecker delta, 30 L Lagrange type fractal base, 163–164 Lebesgue measure classical fractal sets, 54 moment theory, 187–188 Lebesgue space filling curve, 199–200 function spaces, 40–41 local fractal functions, 216–218 mathematical preliminaries, 19 Lebesgue–Stieltjes integral, 17 Linear fractels, 232 Lipschitz constant biaffine fractal functions, 213–214 hidden-variable fractal functions, 183 IFS, 62 Lipschitz continuity, 192–193 Local attractor, 77 Local fractal functions Banach fixed-point theorem, 161 Besov spaces, 226–228 bounded local fractal function, 159, 160f contractive local IFS, 162 Lebesgue spaces, 216–218 metric linear spaces, 157 RB operator, 158 self-referential equation, 159 smoothness spaces Cn and Hölder spaces Cs , 218–224 Sobolev spaces Wm,p , 224–226 Triebel–Lizorkin spaces, 228–230 Local IFS affine fractal surfaces, 336 attractor, 77 contractive, 77 definition, 76 Hausdorff metric, 78 Lyapunov dimension Borel probability measure, 136–137 definition, 135–136 Hausdorff dimension, 139 IFS, 137–138

402

M Mallat transform, 280–281 Markov chains countable, 23 mathematical preliminaries, 27 topological, 87 Markov process mathematical preliminaries, 20 recurrent IFS, 70 Mathematical preliminaries algebra, 23–40 analysis and topology, 3–15 function spaces, 40–52 measures and probability theory, 15–23 Matrix mask, 264 Mauldin–Williams fractal, 96, 121–122 Mauldin–Williams graph, 94–95 Menger–Urysohn dimension, 101–102 Metric dimensions box dimensions, 106–107 dimension function, 104 Hausdorff dimension, 104, 106–107, 109 h-Hausdorff measure, 105 Minkowski dimensions, 108 net measures, 105 Minimally supported frequency wavelets, 379 Minkowski dimensions, 108 Moment theory affine, 188 homeomorphism, 187 IFS and, 67–69 Lebesgue measure, 187–188 p-balanced measures and, 363–365 Monomorphism, 25 Multidimensional iterative interpolation functions, 343 Multiresolution analysis dilation operator, 262 discrete wavelet transform, 267 finitely generated shift invariant space, 263 Fourier–Plancherel transform, 268–269 fractal function wavelets, 300–302 fractal surfaces and wavelets, 375–378 Haar system, 265 Hilbert space, 265

Index

matrix mask, 264 multiwavelet, 267 Parseval’s identity, 271–272 Riesz basis, 265 r-regular, 266 scaling function, 263 semiconvolution product, 263 translation operator, 263 two-component Kronecker delta, 268 wavelet series/wavelet decomposition, 267 wavelet spaces, 266 Multivariate fractal function foldable figures, 372 real-valued, 335, 339 N Norm/strong topology, 5 Null-free endomorphism, 167 O Open set condition (OSC) box dimension of projections, 126–127 self-similar fractals, 115 strong, 118 Orthogonal fractal function wavelets affine fractal functions, 318 aymmetric/anti-symmetric wavelets, 321, 322f coefficient matrices, 305 convex function, 308–309 DGHM multiwavelet, 321–322 Fourier transforms, 313–315 GHM scaling vector, 310–313 Hölder exponent, 310 interpolation functions, 321–322 multiwavelet construction, 315 right-shift map, 306–307 scaling factors, 304–305 scaling functions, 309–310, 310f zero-set of polynomial, 306–308 Orthonormal affine fractal basis, 374 Oseledec’s multiplicative ergodic theorem, 130, 132 P Parameterized spline space, 284 Parseval’s identity, 271–272

Index

p-balanced measures and moment theory, 363–365 Peano curves definition, 197 generator and polygon, 198, 199f hidden-variable fractal functions, 185 Hilbert’s construction, 198 isodimensional orthogonal projection, 198 isosceles triangle, 199, 199f Lebesgue’s space-filling curve, 199–200 nth polygonal approximations, 197–198 periodic extension, 197–198 second approximation, 203, 203f sixth-and seventh-approximating polygons, 200–201, 201f Perron–Frobenius theorem box dimension of projections, 125–126 mathematical preliminaries, 15 recurrent fractal functions, 245–246 recurrent iterated function systems and dimension, 121 recurrent sets and Mauldin–Williams fractals, 122 Pesin’s formula, 134 Piecewise fractal functions biorthogonal scaling vectors, 325 Daubechies scaling function, 325–326 local dimension, 327 shift-invariant space, 324 Poincaré recurrence theorem, 353–354 Poisson summation formula, 295–297 Polynomial B-splines Bn , 223–224 Probabilistic dimensions Billingsley dimension, 111–112 Hausdorff dimension, 111 IFS, 111 reverse inequality, 113 Q Quasi–Banach space Lebesgue spaces, 216–217 mathematical preliminaries, 41 R Read–Bajraktarevi´c (RB) operator affine functions, 277–278 Banach fixed-point theorem, 146

403

continuous, compactly supported, and orthogonal (fractal) wavelets, 150 contractive IFS, 150, 156–157 Daubechies–Lagarias theorem, 279 Daubechies scaling function, 280f Daubechies theorem, 279 dilation equation, 277–280 foldable figures, 372 fractal interpolation functions, 152 fractels, 233–234 Hölder exponent α, 279 interpolation pair, 152 interpolation property, 154f, 155f local fractal functions, 158 self-referential equation, 147 vector-valued fractal function, 153 Recurrent fractal functions box dimension, 247–248 Perron-Frobenius theorem, 246 positive eigenvector, 246 Recurrent IFS attractor, 71, 73 box dimension, 357–362 communication relation, 72 definition, 69 Markov process, 70 Tychonov’s theorem, 72–73 Recurrent iterated function systems and dimension nonoverlapping, 120 Perron–Frobenius theorem, 121 Recurrent sets Banach fixed-point theorem, 83, 86 connection matrix, 89 contractive IFS, 93 definition, 82 Dekking’s construction, 79–80 dragon fractals, 84–85, 85f free semigroup endomorphism, 89–90 Frobenius representation, 80 Hausdorff metric, 81 necessity, 88 subshifts of finite type/topological Markov chains, 87 sufficiency, 88 Recurrent sets and Mauldin–Williams fractals essentially mixing, 122 Hausdorff dimension, 122

404

Recurrent sets and Mauldin–Williams fractals (Continued) Perron–Frobenius theorem, 122 semigroup endomorphism, 121 Recurrent sets as fractal functions Hausdorff metric, 168 Jordan curve, 167–168 Kiesswetter’s fractal function/Kiesswetter’s curve, 166, 167f null-free endomorphism, 167 Reflection dilation wavelet set, 385–386 Reflection groups and root systems Coxeter group, 33 root lattice/coroot lattice, 35 Weyl chambers of E, 35–36, 36f Riemann sphere, 74 Riesz basis fractal function wavelets, 298, 300 multiresolution analysis, 265 Riesz representation theorem, 19 S Schauder fixed-point theorem, 11 Schwarz’s inequality, 283 s-dimensional Hausdorff measures, 55 Self-affine fractal sets definition, 114 recurrent iterated function systems and dimension, 120–121 recurrent sets and Mauldin–Williams fractals, 121–122 self-affine fractals, 118–119 self-similar fractals, 114–118 special orthogonal group SO(n), 114 Self-referential equation biaffine fractal functions, 215 fractels, 232 local fractal functions, 159 RB operator, 147 Self-referential functions, 232 Self-similar fractals concatenation of codes, 116 Hausdorff dimension, 114–115 OSC, 115 strong OSC, 118 Semigroup, 24 Shannon wavelet, 379–380, 379f, 381

Index

Shift-invariant space, 324 Sierpi´nski gasket, 56, 57f Sierpi´nski triangle hidden-variable fractal functions, 184 IFS, 61 2-simplex and associated partition, 339, 339f Smoothness spaces Cn and Hölder spaces Cs binary partition, 218–223 vanishing endpoint conditions for Si, 223–224 Sobolev spaces decomposition and reconstruction algorithm, 282–283 function spaces, 42–43 Sobolev spaces Wm,p , 224–226 Space-filling curve. See Peano curves Strong law of large numbers, 18 Subshifts of finite type, 87 Sup(remum) norm, 13 Symmetric/anti-symmetric wavelets, 321, 322f T Taxi-cab metric, 212–213 Tensor product fractal surfaces, 329–331 Topological dimensions ˇ Brouwer-Cech dimension, 102 Cartesian product theorem, 102 coincidence theorem, 103 Hilbert cube, 103–104 Menger–Urysohn dimension, 101–102 Topological Markov chains, 87 Topological vector spaces, 8 Topology, 5 Tops code space, 97 Tops function, 97 Totally disconnected IFS, 96 Transformations, fractal sets address function, 96 address structure, 99 fractal transformation, 98 homeomorphic address structures, 100 tops function and tops code space, 97 totally disconnected IFS, 96 Translation congruent modulo 2π , 381

Index

Translation/dilation wavelet set, 378–384 Triebel–Lizorkin spaces fractal functions, 228–230 function spaces, 43–52 Trivial fractel, 231 Two-component Kronecker delta, 268 Two-dimensional Lebesgue measure, 363 Tychonov’s theorem mathematical preliminaries, 7 recurrent IFS, 72–73 U Uniformities/entourages, 7 Uniqueness theorem, 28 Univariate fractal function, 372, 373f Upper ε-oscillation, 352 Upper semicontinuous/lower semicontinuous function, 7

405

V Vague convergence, 18 Vandermonde determinant, 356 Vector-valued fractal function, 153 W Wandering subspace, 380 Wavelet set in R2 , 384, 385f Wavelet sets and fractal surfaces reflection/dilation, 385–386 translation/dilation, 378–384 Wavelet spaces, 266, 302 Weak law of large numbers, 19 Weierstraß-like fractal functions, 57–58 Whitney–Jackson type, 282–283  W-invariant function spaces, 374–375 Z Zygmund spaces, 45–46