Wavelets: Theory, Algorithms, and Applications [1st Edition] 9780080520841

Table of contents :
Content:
Wavelet Analysis and Its ApplicationsPage ii
Front MatterPage iii
Copyright pagePage iv
ContributorsPages ix-xii
PrefacePages xiii-xviCharles K. Chui, Laura Montefusco, Luigia Puccio
Non-stationary Multiscale AnalysisPages 3-12Albert Cohen
The Spectral Theory of Multiresolution Operators and ApplicationsPages 13-31Peter Niels Heller, Raymond O. Wells Jr.
Multiresolution Analysis, Haar Bases and Wavelets on Riemannian ManifoldsPages 33-52Stephan Dahlke
Orthonormal Cardinal FunctionsPages 53-88T.N.T. Goodman, Charles A. Micchelli
Some Remarks on Wavelet Representations and Geometric AspectsPages 91-115Bruno Torrésani
A Matrix Approach to Discrete WaveletsPages 117-135Jaroslav Kautsky, Radka Turcajová
A Unified Approach to Periodic WaveletsPages 137-151Gerlind Plonka, Manfred Tasche
Spline Wavelets over R, Z, R/NZ, and Z/NZPages 155-177Gabriele Steidl
A Practice of Data Smoothing by B-spline WaveletsPages 179-196Susumu Sakakibara
L-Spline WaveletsPages 197-212Tom Lyche, Larry L. Schumaker
Wavelets and Frames on the Four-Directional MeshPages 213-230Charles K. Chui, Kurt Jetter, Joachim Stöckler
On Minimum Entropy SegmentationPages 233-269David L. Donoho
Adaptive Time-Frequency Approximations with Matching PursuitsPages 271-293Geoffrey Davis, Stéphane Mallat, Zhifeng Zhang
Getting Around the Balian-Low Theorem Using Generalized Malvar WaveletsPages 295-310Bruce W. Suter, Mark E. Oxley
Time Scale Energetic DistributionPages 311-322Guy Courbebaisse, Bernard Escudié, Thierry Paul
Some Mathematical Results about the Multifractal Formalism for FunctionsPages 325-361Stephane Jaffard
Fractal Wavelet Dimensions and Time EvolutionPages 363-381Matthias Holschneider
Multiscale Methods for Pseudo-Differential Equations on Smooth Closed ManifoldsPages 385-424Wolfgang Dahmen, Siegfried Prössdorf, Reinhold Schneider
Wavelet Methods for the Numerical Solution of Boundary Value Problems on the IntervalPages 425-448Silvia Bertoluzza, Giovanni Naldi, Jean Christophe Ravel
On the Nodal Values of the Franklin Analyzing WaveletPages 449-458Dimitri Karayannakis
Parallel Numerical Algorithms with Orthonormal Wavelet Packet BasesPages 459-493Laura Bacchelli Montefusco
Representation of the Atomic Hartree-Fock Equations in a Wavelet Basis by Means of the BCR AlgorithmPages 495-506Patrick Fischer, Mireille Defranceschi
Efficiency Comparison of Wavelet Packet and Adapted Local Cosine Bases for Compression of a Two-dimensional Turbulent FlowPages 509-531Mladen Victor Wickerhauser, Marie Farge, Eric Goirand, Eva Wesfreid, Echeyde Cubillo
Wavelet Spectra of Buoyant Atmospheric TurbulencePages 533-536, 536a, 536b, 536c, 536d, 536e, 536f, 536g, 536h, 537-541Meinhard E. Mayer, Lonnie Hudgins, Carl A. Friehe
Experimental Study of Inhomogeneous Turbulence in the Lower Troposphere by Wavelet AnalysisPages 543-559Aimé Druilhet, Jean-Luc Attié, Pierre Durand, Bruno Bénech, Leonardo de Abreu Sá
Applications of Wavelet Transform for Seismic Activity MonitoringPages 561-572Gabriella Olmo, Letizia Lo Presti
Mean Value Jump Detection: A Survey of Conventional and Wavelet Based MethodsPages 573-584Aline Denjean, Francis Castanié
Comparison of Picture Compression Methods: Wavelet, Wavelet Packet, and Local Cosine Transform CodingPages 585-621Mladen Victor Wickerhauser
Subject IndexPages 623-627
Wavelet Analysis and Its ApplicationsPage ibc1

Citation preview

Wavelet Analysis and Its Applications The subject of wavelet analysis has recently drawn a great deal of attention from mathematical scientists in various disciplines. It is creating a common link between mathematicians, physicists, and electrical engineers. This book series will consist of both monographs and edited volumes on the theory and applications of this rap­ idly developing subject. Its objective is to meet the needs of academic, industrial, and governmental researchers, as well as to provide instructional material for teaching at both the undergraduate and graduate levels. This is the fifth book of the series. It is a carefully edited volume t h a t covers seven active research areas in this field. Multiresolution analysis, being the most powerful tool for constructing wavelets, is the first topic. It consists of at least four interesting points of view, ranging from non-stationary schemes to spectral theory. The second topic of discussion is wavelet transforms, where not only the geometric and algebraic aspects are studied, but periodic wavelet transforms are also investi­ gated. Because spline functions are both local and refinable, they are particularly useful for representing wavelets. Spline-wavelets, as they are usually called, con­ stitute the third topic of study in this volume. Polynomial splines and L-splines are discussed in the one-variable setting, and the four-directional box spline is used to construct bivariate frames. To analyze a signal adaptively, other localization schemes are required. The fourth topic of discussion includes Donoho's minimum entropy segmentation, Mallat's matching pursuit approach, and generalized Malvar wavelets. The property of self-similarity is common to both fractals and scaling functions, and hence wavelets. To link these two subjects, two interesting papers, constituting the fifth topic, are included. To the reader who is concerned with com­ putations or applications, the last two topics are most important. A collection of five comprehensive articles on wavelet solution of PDE's, numerical computation, and algorithms, represents the sixth topic. Six contributions on applications constitute the seventh and concluding part of this volume. Topics included are: compression, analysis, and experimental study of turbulence; detection of seismic activities and mean-value jumps of a random process; as well as a comparison of four image com­ pression methods: wavelet, wavelet packet, local cosine transform, and an adaptive best basis scheme. The series editor is most grateful to the authors of these 28 chapters for their outstanding contributions. This is a volume in WAVELET ANALYSIS AND ITS APPLICATIONS CHARLES K. CHUI, SERIES EDITOR

Texas A&M University, College Station, Texas A list of titles in this series appears at the end of this volume.

Wavelets: Theory, Algorithms, and Applications Edited by

Charles K. Chui Department of Mathematics Texas A&M University College Station, Texas

Laura Montefusco Departemento de Matematica Università di Messina Bologna, Italy

Luigia Puccio Departemento de Matematica Università di Messina Messina, Italy

®

ACADEMIC PRESS, INC. San Diego New York Boston London Sydney Tokyo Toronto

This book is printed on acid-free paper, fe) Copyright © 1994 by ACADEMIC PRESS, 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 photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Academic Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Wavelets : theory, algorithms, and applications / edited by Charles K. Chui, Laura Montefusco, Luigia Puccio p. cm. — (Wavelet analysis and its applications : v. 5) Includes index. ISBN 0-12-174575-9 1. Wavelets. I. Chui, C. K. II. Montefusco, Laura. III. Puccio, Luigia. IV. Series. QA403.3.W43 1994 515'2433-dc20 94-34457 CIP PRINTED IN THE UNITED STATES OF AMERICA 94 95 96 97 98 99 MM 9 8 7 6

5

4

3 2 1

Contributors Numbers in parentheses indicate pages on which authors'contributions JEAN-LUC ATTIÉ (543), Laboratoire F-31077 Toulouse, France BRUNO B E N E C H (543), Laboratoire F-31077 Toulouse, France

begin.

dAérologie,

Université

Paul

Sabatier,

d'Aerologie,

Université

Paul

Sabatier,

Consiglio

Nazionale

SILVIA BERTOLUZZA (425), Instituto di Analisi Numerica, delle Ricerche, 27100 Pavia, Italy FRANCIS CASTANIÉ (573), LEN7-GAPSE,

ENSEEIHT,

31 071 Toulouse, France

CHARLES K. CHUI (213), Department of Mathematics, Center for Approximation Theory, Texas A&M University, College Station, Texas 77843 ALBERT COHEN (3), CEREMADE, France

Université Paris IX Dauphine,

GUY COURBEBAISSE (311), ICPI-CPE-LTS,

75775 Paris,

69288 Lyon, France

ECHEYDE CUBILLO (509), Center for Theoretical Studies of Physical Clark Atlanta University, Atlanta, Georgia 30314

Systems,

STEPHAN DAHLKE (33), Institut für Geometrie und Praktische RWTHAachen, 52062 Aachen, Germany

Mathematik,

WOLFGANG DAHMEN (385), Institut für Geometrie und Praktische RWTH Aachen, 52056 Aachen, Germany

Mathematik,

GEOFFREY DAVIS (271), Computer Sciences Department, Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 LEONARDO DE ABREU SÀ (543), Instituto Nacional de Pesquisas Espacials, Jose dos Campos, Brazil MIREILLE DEFRANCESCHI (495), CEA/DSM/DRECAM/SRSIM, F-91191 Gif-sur-Yvette, France

Saö

Ce - Saclay,

ALINE DENJEAN (573), LEN7-GAPSE,

ENSEEIHT,

31 071 Toulouse, France

DAVID L. DONOHO (233), Department ford, California 94305

of Statistics,

Stanford

AlMÉ DRUILHET (543), Laboratoire F-31077 Toulouse, France

d'Aerologie,

Université

Paul

Sabatier,

PIERRE DURAND (543), Laboratoire F-31077 Toulouse, France

d'Aerologie,

Université

Paul

Sabatier,

ix

University,

Stan­

Contributors

X

BERNARD ESCUDIÉ (311), ICPI-CPE-LTS,

69288 Lyon, France

MARIE FARGE (509), Laboratoire de Météorologie Dynamique Normale Supérieure, 75231 Paris, France PATRICK FISCHER (495), CEA/DSM/DRECAM/SRSIM, Gif-sur-Yvette, France

du CNRS, Ecole

Ce - Saclay, F-91191

CARL A. FRIEHE (533), Department of Mechanical Engineering, California, Irvine, California 92717 ERIC GOIRAND (509), Mathematics Louis, Missouri 63130

Department,

Washington

T. N. T. GOODMAN (53), Department of Mathematical Dundee, Dundee DD1 4HN, Scotland, United

University of University,

Sciences, University of Kingdom

PETER NIELS HELLER (13), Aware, Inc., Cambridge, Massachusetts MATTHIAS HOLSCHNEIDER (363), CPT CNRS France

St.

Luminy,

02142

F-13288

LONNIE HUDGINS (533), Electronics Systems Division, Northrop Hawthorne, California 90251

Marseille, Corporation,

STEPHANE JAFFARD (325), CERMA, ENPC, La Courtine, 93167Noisy France; and CMLA, ENS Cachan, 94235 Cachan, France

le Grand,

KURT JETTER (213), FB Mathematik, Germany

Duisburg,

Universität Duisburg, D-47048

DMITRI KARAYANNAKIS (449), Department of Physics, University of Crete, Iraklion, Crete, Greece JAROSLAV KAUTSKY (117), School of Information Science and Technology, Flin­ ders University and Cooperative Research Center for Sensor, Signal, and Information Processing (CSSIP), Adelaide, SA 5001, Australia LETIZIA L O PRESTI (561), Department Torino, Italy TOM LYCHE (197), Institutt Norway

of Electronics,

for Informatikk,

Politecnico,

I-10129

University of Oslo, 0316 Oslo,

STÉPHANE MALLAT (271), Computer Sciences Department, Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 MEINHARD E. MAYER (533), Department Irvine, California 92717

of Physics, University of

CHARLES A. MICCHELLI (53), IBM Research Division, T J. Watson Center, Yorktown Heights, New York 10598

California, Research

Contributors

xi

LAURA BACCHELLI MONTEFUSCO (459), Department of Bologna, 40127 Bologna, Italy GIOVANNI NALDI (425), Dipartimento 27100 Pavia, Italy

of Mathematics,

di Matematica,

University

Università

di

Pavia,

GABRIELLA OLMO (561), Department of Electronics, Politecnico, 1-10129 Torino, Italy MARK E. OXLEY (295), Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, Ohio 45433 THIERRY PAUL (311), CEREMADE, France

Université Paris IXDauphine,

GERLIND PLONKA (137), Fachbereich D-18051 Rostock, Germany

Mathematik,

75775 Paris,

Universität

Rostock,

SIEGFRIED PRÖSSDORF (385), Institut für Geometrie und Praktische tik, RWTHAachen, 52056 Aachen, Germany

Mathema­

JEAN CHRISTOPHE RAVEL (425), Laboratoire dAnalyse Pierre et Marie Curie, 75252 Paris, France

Université

Numérique,

SUSUMU SAKAKIBARA (179), College of Science and Engineering, University, Fukushima-ken 970, Japan

Iwaki

REINHOLD SCHNEIDER (385), Institut für Geometrie und Praktische tik, RWTH Aachen, 52056 Aachen, Germany LARRY L. SCHUMAKER (197), Department of Mathematics, sity, Nashville, Tennessee 37240 GABRIELE STEIDL (155), THDarmstadt, stadt, Germany

JOACHIM STÖCKLER (213), FB Mathematik, Duisburg, Germany

Universität

Mathema­

Vanderbilt

Fachbereich Mathematik,

Meisei

Univer­

64289 Darm­

Duisburg,

D-47048

BRUCE W. SUTER (295), Department of Electrical and Computer Engineering, Air Force Institute of Technology, Wright-Patterson AFB, Ohio 45433 MANFRED TASCHE (137), Fachbereich D-18051 Rostock, Germany

Mathematik,

BRUNO TORRE SANI (91), CPT, CNRS-Luminy,

Universität

Rostock,

13288 Marseille, France

RADKA TURCAJOVÄ (117), School of Information Science and Technology, Flin­ ders University and Cooperative Research Center for Sensor, Signal, and Information Processing (CSSIP), Adelaide, SA 5001, Australia RAYMOND O. WELLS, J R . (13), Computational University, Houston, Texas 77251 EVA WESFREID (509), CEREMADE, France

Mathematics

Laboratory,

Université Paris-Dauphine,

Rice

5775 Paris,

Contributors

Xll

MLADEN VICTOR WICKERHAUSER (585), Department

of Mathematics,

Washing­

ton University, St. Louis, Missouri 63130 ZHIFENG ZHANG (271), Computer Sciences Department, Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

Preface Interest in wavelet analysis has been growing very rapidly in recent years. Advances in its theory, algorithms, and applications have greatly influenced the development of many disciplines of science and technology, including math­ ematics, physics, engineering, geosciences, meteorology, and will continue to influence other areas as well. However, while the rate of growth in each of these three directions of wavelet analysis has been phenomenal individually, the interaction among them is somewhat insufficient; and this creates an obstacle for the advancement of the field. Motivated by this need, we decided to invite several leading experts from each of the three areas and publish a wellbalanced volume based on their contributions. With financial support from the Consiglio Nazionale delle Ricerche of Italy, we organized the International Con­ ference on Wavelets, held in Taormina, Italy, during the period from October 14 to October 20,1993. The conference brought together experts and professionals specializing in wavelet theories, algorithms, and applications, from four conti­ nents. This volume consists mainly of articles by the invited speakers and par­ ticipants at the conference. Out of the 28 articles, 12 are invited papers by the main speakers, and 15 are selected from contributions by other participants. The author of the remaining paper, Donoho, was unable to accept our invitation to speak, but graciously agreed to contribute, which significantly strengthens the algorithmic area of this volume. The seven topics of this volume serve as a reader's guide and are not in­ tended to separate these somewhat closely related chapters into different areas. In addition, in order to avoid unnecessary discontinuities, chapters within each part are arranged according to subject matter, not alphabetically by contributor. Since the approach of mul tire solution analysis has proved to be the most fruitful, not only for constructing wavelets, but also for algorithmic development, the first of the seven parts in this volume is devoted to mul tiresolution and multiscale analyses. Cohen allows the finite two-scale sequences to change from one iteration to another in order to construct compactly sup­ ported scaling functions t h a t are infinitely differentiable. Heller and Wells dis­ cuss the spectral theory of multiresolution operators. In particular, they give certain asymptotic estimates of the Sobolev smoothness of rank 3 scaling func­ tions for large genus. Dahlke generalizes the notion of multiresolution analysis to certain Riemannian manifolds. On the other hand, Goodman and Micchelli investigate various questions related to the important properties of orthogon­ ality, compact support, finite bandwidth, symmetry, cardinal interpolation, and the two-scale relation t h a t induces a multiresolution analysis. The second part of this volume is devoted to the study of wavelet trans­ forms from different points of view. Geometric aspects of continuous wavelet transforms are discussed by Torrésani. For example, a scheme for constructing wavelet decompositions of functions in homogeneous spaces is introduced with respect to certain group action. An approach based mainly on matrix algebra is described by Kautsky and Turcajovâ. In particular, the properties of banded xm

xiv

Preface

block circulant matrices are used to study the conditions of orthogonality, biorthogonality, regularity, and symmetry of discrete wavelet transforms. Along this line, Plonka and Tasche discuss discrete wavelet transforms in the periodic setting. For example, they introduce a new approach to periodic wave­ let analysis based on Fourier techniques and propose certain efficient algo­ rithms for decompositions and reconstructions using FFT. Spline functions are refinable by knot insertions and removals. In particu­ lar, cardinal J3-splines are usually used to demonstrate the concept of multiresolution analysis. To give a firm mathematical setting, we begin the third part of this volume with a chapter by Steidl t h a t studies a unified approach to car­ dinal, periodic, and discrete spline wavelets with any integer dilation factor n ^ 2. This is done by using Fourier analysis on locally compact albelian groups. We then specialize to polynomial spline wavelets with a contribution by Sakakibara t h a t describes a method of implementation of the Chui-Wang spline wavelets for data smoothing. These polynomial spline wavelets are generalized by Lyche and Schumaker by using L-splines in the third chapter. They give an explicit construction of these compactly supported L-spline wavelets and apply their theory to develop multiresolution methods based on L-splines. On the other hand, Chui, Jetter, and Stöckler discuss the generalization to the higher dimensional setting by using box splines. In the bivariate case, a four-direc­ tional box spline is very attractive for surface modeling since it is most sym­ metric. However, since its integer translates do not constitute a Riesz basis, there is some difficulty in generating a stable wavelet basis with compact sup­ port. In this chapter, by using oversampling, the corresponding compactly sup­ ported semi-orthogonal wavelet with dilation matrix A = ( J _ J ) i s shown to generate a frame. To analyze a signal adaptively, other localization schemes are sometimes needed. The fourth part of this volume is devoted to the description of other mathematical tools for time-frequency analysis. The minimum entropy seg­ mentation approach is introduced by Donoho. In his chapter, he describes seg­ mented multiresolution analyses of a bounded interval t h a t lead to segmented wavelets t h a t are well adapted to discontinuities, cusps, etc. This approach emphasizes the idea of average interpolation and leads to the adaptive wavelet transform of a signal of length n with only 0(n log n) computations. In addition, Donoho describes an iterative approach, called segmentation pursuit, for edge identification. The adaptive method is described in the chapter by Davis, Mallat, and Zhang. A matching pursuit algorithm is introduced to compute a subop­ timal expansion of a signal in a redundant dictionary of waveforms to best match the structures of the signal. In addition, they describe an algorithm for isolating the coherent structures of a signal and demonstrate with an applica­ tion to pattern extraction in a noisy environment. Perhaps the best known adaptive wavelets are the so-called Malvar wavelets. Suter and Oxley give a complex-valued generalization and use these wavelets to get around the Balian-Low principle, obtaining something close to an exponential basis. The Wigner representation is another time-frequency localization tool. The final chapter in this part introduces an affine Wigner distribution and relates it with the affine Wigner operator. The authors, Courbebaisse, Escudié, and Paul, use

Preface

xv

a family of coherent states in the derivation of their expression and give a computational scheme. Even before wavelets, fractals were already quite popular as a powerful image compression tool. The link between wavelets and fractals is selfsimilarity. The fifth part consists of two interesting contributions by two ex­ perts in this area. Jaffard investigates the mathematical validity of the multifractal formalism for functions, shows t h a t it always yields an upper-bound for the spectrum of singularities, and demonstrates its validity by studying the Riemann function. Holschneider gives a new definition of fractal dimension in terms of certain small-scale behavior of the ^-energy of wavelet transform. This is a generalization of the previous multifractal approaches and can be used to show t h a t the correlation dimension of the spectral measure determines the long-time behavior of the time evolution generated by bounded self-adjoint operator acting on some Hilbert space. We now come to numerical methods and algorithms. It is well-known t h a t wavelets provide a hierarchical structure similar to the multi-level approach in numerical pseudo-differential equations (PDEs). The first two papers in this sixth part of the volume are concerned with this point of view. Dahmen, Pròssdorf, and Schneider discuss the numerical solution of PDEs on smooth mani­ fold. In particular, some relevant concepts and basic properties of a multiscale framework t h a t facilitate an efficient and stable matrix compression for fast solutions of Galerkin-discretized linear systems are presented. The second chapter on numerical solutions is contributed by Bertoluzza, Naldi, and Ravel. They are concerned with boundary value problems on an interval and focus on the elliptic case. Both Galerkin and wavelet-collocation methods are described. The remaining chapters in this part are concerned with computations and al­ gorithms. Karayannakis derives both upper and lower bound estimates for the value of the Franklin wavelet at its exceptional node 3/2, confirming a conjec­ ture of Berkson. All calculations presented in this chapter can be carried out on an ordinary scientific pocket calculator. At the other end of the spectrum in calculations, Montefusco presents certain coarse-grained parallel algorithms for some basic problems of large-scale numerical linear algebra. It is shown that, starting from an efficient parallel wavelet packet matrix decomposition, these algorithms can reach good efficiency on distributed memory multiproces­ sors and require only few nearest neighbor communications. The last chapter in this part investigates the representation of the atomic Hartree-Fock equa­ tions in a wavelet basis by means of the Beylkin-Coifman-Rokhlin algorithm. The authors, Fischer and Defranceschi, demonstrate the effectiveness of their results by comparing with the Slater function, which provides the exact solu­ tion using the energy value as their criterion. The seventh and final part of this volume is devoted to applications of wavelets to three important areas. The first three chapters are in the area of turbulence. Wickerhauser, Farge, Goirand, Wesfreid, and Cubillo compare the efficiency of two rank-reduction methods for representing the essential fea­ tures of a two-dimensional turbulent vorticity field. The comparison is on effi­ ciency of capturing the enstrophy or square-vorticity, the faithfulness to the power spectra, and the precision in resolving coherent structures. The second

XVI

Preface

turbulence chapter by Mayer, Hudgins, and Friehe, presents a wavelet cospectral analysis of the boundary-layer flow in the atmosphere above ground. The cospectra show a bimodal structure similar to t h a t previously found in sheargenerated turbulence over ocean. The third chapter on turbulence deals with an experimental study of inhomogeneous turbulence in the lower-troposphere. The authors, Druilhet, Attié, de Abreu Sa, Durand, and Bénech, use the wavelet transform to analyze aircraft data for this purpose. Monitoring of seismic activ­ ities is the subject of the next contribution on wavelet applications. Olmo and Lo Presti demonstrate the effectiveness of the wavelet transform for estimation of arrival times of various seismic phases and for the capability as a de-noising process. The second detection application, presented by Denjean and Castanié, is on detections of mean-value jumps of a stationary random process. It surveys both the conventional method and the wavelet-based method. Perhaps the most well-known applications of wavelet techniques is to image compression. Wickerhauser gives a comprehensive comparison of these methods: wavelets, wavelet packets, local sines and cosines, and an adaptive best-basis method. Standard C algorithms for the local sine and cosine transforms are included in the last section of this interesting and useful chapter. The collection of graphics featured in this volume is impressive. There are more than 135 figures throughout the volume; nine of these are in color, placed in the color section. These figures supply a visual dimension to the authors' texts and frequently illustrate the results of their accomplishments. The reviewing and editing process, as well as producing the edited volume in a uniform style, requires an enormous amount of work. We are very grateful to the referees for their conscientious reviews and to the authors for their fine contributions. Many of the authors prepared their manuscripts in TgX and this helped a great deal in the editing process. Robin Campbell is to be thanked for typesetting some of the other papers. We are indebted to Margaret Chui for her most conscientious and diligent assistance in the editorial process. The fine quality of the volume owes much to her effort. We are thankful to Peter Renz and his editorial staff at Academic Press, Boston, for their helpful assistance and cooperation, and to N. Sivakumar for his valuable editorial comments. Finally, we would like to acknowledge the generous financial support of Consig­ lio Nazionale delle Ricerche of Italy.

College Station, Texas Bologna, Italy Messina, Italy June, 1994

Charles K. Chui Laura Montefusco Luigia Puccio

Non-stationary Multiscale Analysis Albert Cohen Abstract. Orthonormal bases of wavelets and wavelet packets yield lin­ ear, non-redundant time-scale and time-frequency decompositions for arbi­ trary functions and signals. Numerically, these decompositions are based on the iterative application of digital filter banks. Usually these filters are the same at every iteration. We show here the advantages of using filters that may vary from one iteration to the next one. An appropriate choice leads to C°° compactly supported wavelets and allows a better control of the time-frequency localization properties of wavelet packets. These results have been obtained jointly with N. Dyn of Tel-Aviv University and E. Sére of CEREMADE.

§1

Introduction

Wavelets and wavelet packets constitute useful tools for the decomposition of complicated functions into a small number of elementary waveforms t h a t are localized b o t h in time and frequency. We recall here t h e main ideas of their construction (a detailed review can be found in [11]). One s t a r t s with t h e d a t a of a multiresolut ion analysis, i.e., a nested se­ quence of approximation subspaces {0} -> . . . F_2 C V i i C V0 C Vi c V2 . . . -> L 2 ( R )

(1.1)

generated by a scaling function φ G VQ in t h e sense t h a t for each j e Z the family faj,k}kez = {2^2φ{7Ρχ — k)}kez is an orthonormal basis of Vj. Such a function can either be expressed as t h e solution of a "dilation equation" φ{χ) = 2 ] Γ hn(p(2x - n ) , (1.2) nGZ

or as t h e limit of a subdivision scheme, i.e., iterative refinements of an initial Dirac sequence by convolutions with the coefficients hn, since we have + 00

^ )

= [[^ο(2Λ),

(1.3)

fc=l Wavelets: Theory, Algorithms, and Applications Charles K. Chui, Laura Montefusco, and Luigia Puccio (eds.), pp. 3-12. Copyright © 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-174575-9

3

A. Cohen

4

with πΐο(ω) = Ση ^η^~%ηω (for this reason the function φ is said to be "refinable" and Equation (1.2) is also called "refinement equation"). The sequence hn and its Fourier series πΐο(ω) need to satisfy specific con­ straints that are related to the properties of the function φ: • If φ e L 1 , then Ση hn = rao(O) = 1. • φ is compactly supported in [p, q] if and only if hn = 0 for n < p or n > q. • When the two previous constraints are satisfied, the orthonormality of {φ(χ — k)}kez is ensured if and only if |m 0 (a;)| 2 + |mo(^ + 7r)| 2 :=l

(1.4)

and there exists a compact set K, congruent to [—π, π] modulo 2π, such that 77ΐ0(2-·?ω) φ 0 for all ω e K and j >0 (see [2]). From this framework, one derives an orthonormal wavelet by

ψ(χ) = 2^29ηφ(2χ-η),

(1.5)

n

with gn = (—l)nhi-n. The family {ifij^kez is then an orthonormal basis of the orthogonal complement Wj of Vj into Vj+χ. It characterizes the missing details between two successive levels of approximation. It follows that both {ipj,k}j,kez and &j,k}ktz U {^,fc}j>j,fcez are orthonormal bases of L 2 (R). The decomposition of V\ into Vo and Wo, as expressed by Equations (1.2) and (1.5), reveals a more general "splitting trick" : if {e n } n € z is an orthonor­ mal basis of a Hilbert space H, and if hn and gn are coefficients that sat­ isfy the previous constraints, then the sequences un = \/2Y^khk-2n^k and vn = >/2]Cfc 9k-2n^k are orthonormal bases of two orthogonal closed subspaces U and V such that H = U Θ V. Wavelet packets are obtained by using this trick to split the Wj spaces. More precisely, one can define a family {wn}neN by taking WQ = φ, w± = φ and applying the following recursion: w2n = 2 ^2 hkWn(2x - k) and w2n+i = 2 ^ k

gkwn(2x - k).

(1.6)

k

The families {wn(x — m)} mG z = {wm,n} for 2J < n < 2 J + 1 are the results of j splitting of the space Wj. Consequently, {wm,n}neN,m€Z is an orthonormal basis of L 2 (R). This is only a particular example of a wavelet packet basis. Many other bases can be obtained by splitting Wj more or less than j times. However, all the elements of these bases have the general form wjtmtn(*)

= 2j/2™n(2*x - m).

(1.7)

Such bases yield arbitrary decompositions of any discrete signal in the timefrequency plane, by means of fast algorithms. One assumes that the initial

Non-stationary Multiscale Analysis

5

data {sk}kez represents a function in Vj, i.e., f = ^kskwj,k,oThe splitting trick indicates that the coordinates of / in the basis {wm,n}mez,o 0, we have lim -cardi*; < n : ok > M} = 1.

(2.6)

η-κχ> TI

We now turn to the generalization in which one allows to use different filters at every splitting stage. §3 Non-stationary wavelets Let {mo(o;)}jb>o be a family of trigonometric polynomials of degree d{k) that satisfies the following properties:

|m5H| 2 + |mS(W + 7r)|2 = l,

(3.1)

Non-stationary Multiscale Analysis

7

™§(0) = 1,

(3.2)

M < ττ/2 =Φ πι%(ω) > C > 0,

(3.3)

and ^2-*d(Â;)0

(Condition (3.3) could be replaced by a weaker one: the existence of a compact set K congruous to [—π,π] modulo 2π such that πι^(2~ιω) > C > 0 for all &, / > 1 and ω G K, see [2] for more details) With these hypotheses the following results hold. Theorem 3.1. For all j < 0, the tempered distributions φ* defined by

^M=n™S+J(2-V>

(3.5)

k=l

are compactly supported L2 functions. The families {φ^(χ — k)}kez are or­ thonormal. Moreover, the spaces Vj generated by \2?Ι2φ*(??χ — k)}kez con­ stitute a "half multiresolution analysis'7 in the sense that Vj C V^+i and the projection Pj(f) of any I? function f onto Vj goes to f in L2 as j goes to +00. Theorem 3.2. If {πι^{ω)} = {m0,L(fc)(^)} is & subset of the family given by Equations (2.1) and (2.2), and if lim^-^+oo L(k) = +00, then the functions φ^ are smooth, i.e., contained in C°°. The proofs of these results are straightforward consequences of the results in [5]. The particular case of Theorem 3.2 where L(k) = k has also been considered by Berkolaiko and Novikov [1]. The proof of Theorem 3.1 goes in four steps: • Using Bernstein inequality, one remarks that for a fixed ω, the sequence Sk

= \m*+j(2~küü) - 1| < Cd(k)2~k\uj\

(3.6)

is absolutely summable and thus the infinite product in Equation (3.5) converges pointwise and in the sense of tempered distributions. By Equa­ tion (3.4), φ> is a tempered distribution with compact support. • As in the standard construction of wavelets (see [3]), one defines the approximants φ^ of φ* by n

0 Ì M = Π Wo +J (2- fc w) X[ _ 2 „ T , 2 „ 7r] (u;)

(3.7)

k=l

and checks by recursion, using Equation (3.1), that {φ°η(χ — k)}kez is an orthonormal system. By Fatou's lemma, it follows that φ* is in L 2 (R). • By Equation (3.3), we have \ω\ \φ>{ω)\

>C>0

(3.8)

8

A. Cohen and | 1 and consequently bL(ü;)| 2 = X : ( L ; 1 + i ) s i n ^ ( a ; / 2 ) L-\

= Σ(71+J)[2sin2(W/2)F23=0

3

0, it is possible to find JA > j such that, for all k > JA, s(L(k) — 1) > A. It follows that ψ*Α satisfies + 00 fc=l + 00

< J | |m(2-fca;)|L(fc-f^)-1 + 00

< Yl \m(2-kLü)\A/s < \Ρ{ω)\Α/° < C A ( l + |u;|)- A Since φ*{ω) = (pJA(2j~JAu) have

Y\jkt^3rao,L(fc+.?)(2_fcu;), it follows that we

\φ*(ω)\ 0, it follows that φΐ is a C°° function. Finally let us mention that the half-multiresolution analysis {Vj}j>o has the property of "spectral approximation." For all / G Hs and 0 < t < s one has \\Pjf - /Ha« < C(s, t)2-' U/H « . e u , / )

(3.12)

where ε(.7, / ) G [0,1] and tends to 0 as j goes to +oo (see [5]). §4 Non-stationary wavelet packets The previous results allow us to construct "non-stationary wavelet packets" with the following definition. Definition 4.1. Let {mo}jb>o be a family of positive trigonometric poly­ nomials that satisfies the set of Conditions (3.1)-(3.4) and define πι^ω) = β~χωπι$(ω + π). The non-stationary wavelet packets associated to {rao}fc>o are a sequence {wn}n>o of tempered distributions given by

ώ»(«) = Π < ( 2 " ^ ) . k=l

(4·1)

with n = Σ*>ο 2 * " ^ , ek e {0,1}. From Theorems 3.1 and 3.2, we can easily derive the following proposition.

10

A. Cohen

Proposition 4.2. The distributions {wn}n>o are compactly supported L2 func­ tions and the length of their support is independent ofn. The family wniTn(x) =wn(x — m),n>0,meZ, constitutes an orthonor­ mal basis of L2 (R). If {τη^(ω)} = {rao,L(fc)(^)} is a subset of the family given by Equations (2.1) and (2.2) and if limfc_>+00 L(k) — +oo, then the functions wn are in­ finitely smooth, i.e., contained in C°°. Proof: Prom the definition of wn, we see that for 0 < n < 2 J , the families {wn(x — m)}mez a r e the result of j splittings of the space Vj. It follows that these functions are finite combinations of the functions (pifóx — k). As a consequence, they are in L2. They are also in C°° as soon as the hypotheses of Theorem 3.2 are satisfied. From the denseness of the spaces Vj, it follows that the family wniTn(x) = wn(x — m), n > 0, m G Z, constitutes an orthonormal basis of L 2 (R). Finally, since mf has the same degree as m j , the support of wn cannot be larger than L = Y^,k>0 2~kd(k) < +oo. ■ Remark. Since the generating functions of the Vj spaces are different at every scale, one can no longer use a recursion formula such as Equation (1.6) to derive the non-stationary wavelet packets . However, fast algorithms can still be used to compute the non-stationary wavelet packet coefficients. The principle is exactly the same as in the standard case except that one uses different filters at every stage of the decomposition. Note that in the case where the degree d(k) of mfc is increasing as k goes to 4-oo, the decomposition starts at the finest scale with a long filter and ends at a coarser scale with a smaller filter. An opposite approach was used by Hess-Nielsen [13] to study the frequency spreading of wavelet packets. The choice of using longer filters for finer scales offers three important advantages: • Since the functions φ^ are in C°°, the reconstruction from the low fre­ quency components will have a smooth aspect. • In the algorithm, one needs to deal properly with the boundaries of the signal. Specific methods have been developed by Cohen, Daubechies, and Vial [4] to adapt the filtering process near the edges. These methods require, in particular, that the size of the filter is smaller than the size of the signal. This can only be achieved by using smaller filters in the coarse scales. • Finally, these techniques can be used to combine time and frequency lo­ calization properties as shown by Theorem 4.3 below. A positive result on frequency localization can be obtained with nonstationary wavelet packets, provided that the filterlength grows sufficiently fast. We assume, once again, that the hypotheses of Equations (3.1)-(3.4) are satisfied and that {τη^(ω)} = {m0,L(fc)(^)} is a subset of the family given by Equations (2.1) and (2.2).

Non-stationary Multiscale Analysis

11

Theorem 4.3. Assume that L(k) > Ck3+r for some r > 0. Then, for any ε > 0 there is Με > 0 such that, for all n > 1, then - card {k < n / ök > Με } < ε.

(4.2)

The proof of this result is detailed in [6]. It is rather technical, but es­ sentially exploits the fact that, as L(k) goes to -foo, the functionra0)z,(fc)(u;) converges pointwise to 1 on ] — π/2, π/2[, an immediate consequence of Lemma 3.3. References 1. Berkolaiko, S. and I. Novikov, Sur des presque-ondelettes indéfiniment différentiables àsupport compact, Dolci. Russ. Acal. Neuk. 326 (1992), 935938. 2. Cohen, A. Wavelets and digital signal processing, translated by R. Ryan, Chapman & Hall, 1994, to appear. 3. Cohen, A. and I. Daubechies, On the instability of arbitrary biorthogonal wavelet packets, SIAM J. Math. Anal. 24-5 (1993), 1340-1354. 4. Cohen, A. I. Daubechies and P. Vial, Wavelets and fast wavelet transforms on an interval, Appi, and Comp. Harmonic Anal. 1 (1993), 54-81. 5. Cohen, A. and N. Dyn, Nonstationary subdivision schemes and multiresolution analysis, CEREMADE, Université Paris-IX, Dauphine, 1994, pre­ print. 6. Cohen, A. and E. Sére, Time-frequency localization with nonstationary wavelet packets, CEREMADE, Université Paris-IX, Dauphine, 1994, pre­ print. 7. Coifman, R. R., Y. Meyer, and V. M. Wickerhauser, Size properties of wavelet packets, in Wavelets and Their Applications, G. Beylkin, R. R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. A. Raphael, and M. B. Ruskai (eds.), Jones and Bartlett, 1992, 453-470. 8. Coifman, R. R., Y. Meyer, and V. M. Wickerhauser, Entropy-based al­ gorithms for best basis selection, IEEE Trans, on Inform. Theory 38-2 (1992), 713-718. 9. Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. in Pure and Appi. Math. 41 (1988), 909-996. 10. Daubechies, I. and J. C. Lagarias, Two-scale difference equations I. Exis­ tence and global regularity of solutions, SIAM J. Math. Anal. 22-5 (1991), 1388-1410. 11. Daubechies, I. and J. C. Lagarias, Two-scale difference equations II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23-4 (1992), 1031-1079. 12. Dyn, N. and D. Levin, Stationary and non-stationary binary subdivision schemes, in Mathematical Methods in Computer-aided Geometric Design, T. Lyche and L. Schumaker (eds.), Academic Press, 1992, 209-216.

12

A. Cohen

13. Hess-Nielsen, N., Frequency localization of generalized wavelet packets, Wash. Univ., St. Louis., 1993, preprint. 14. Micchelli, C. A. and H. Prautzsch, Uniform refinements of curves, Linear Algebra and Applications, 114-115 (1989), 841-870. 15. Rioul, O., Simple regularity criteria for subdivision schemes, SIAM J. Math. Anal. 23-6 (1992), 1544-1576. 16. Sére, E., Localisation fréquentielle des paquets d'ondelettes, Revista Mate­ matica Iberoamericana, 1993, to appear. 17. Villemoes, L., Sobolev regularity of wavelets and stability of iterated filter banks, in Proc. Int. Conf. on Wavelets and Applications, Y. Meyer and S. Roques (eds.), Frontiers, 1992. Albert Cohen CEREMADE-Université Paris IX Dauphine 75775 Paris Cedex 16, France [email protected]

T h e Spectral Theory of Multiresolut ion Operators and Applications

Peter Niels Heller and Raymond 0 . Wells, Jr.

Abstract. For any wavelet matrix of rank m and genus g there is an associated wavelet system with a square integrable scaling function and (m-1) wavelet functions; the fundamental scaling equation has mg coeffi­ cients. This paper concerns itself with a family of multiresolution opera­ tors associated with such a wavelet matrix and the corresponding wavelet system. These operators act on continuous even periodic functions and depend quadratically on the coefficients of the wavelet matrix; they have been studied for rank 2 wavelet systems by Lawton, Cohen, and Eirola in different contexts and under different names (transition operator, waveletGalerkin operator, etc.). Lawton and Cohen independently found that a rank 2 wavelet system yields an orthonormal basis if and only if 1 is a nondegenerate eigenvalue of the multiresolution operator. Eirola showed for the rank 2 case that the spectral radius of this operator could be used to explicitly compute the Sobolev smoothness of the scaling function (and hence the wavelet system), and that moreover, this spectral radius could be computed in terms of the spectral radius of a finite-dimensional operator. He found asymptotic formulas for this spectral radius and hence for the Sobolev smoothness as the genus (and number of coefficients in the scaling equation) grows large. This paper reviews the above results and outlines a generalization of Eirola's work to arbitrary rank wavelet systems. In par­ ticular we give asymptotic estimates of the Sobolev smoothness of rank 3 scaling functions for large genus by evaluating the kernel of the multireso­ lution operator at fixed points of an automorphism of the unit circle.

§1

Introduction

In this article we explore the notion of the multiresolution operator, its spectral theory, and applications. This operator (also called the transition operator) has appeared as a fundamental tool in several aspects of wavelet theory, such as t h e Lawton-Cohen theorem on wavelet orthonormal bases [2,18,19], and the work of Eirola [8] and others [1,3,4,5,23] on the Sobolev smoothness of wavelet scaling functions. After introducing the multiresolution operator and Wavelets: Theory, Algorithms, and Applications Charles K. Chui, Laura Montefusco, and Luigia P u c c i o ( e d s . ) , p p . 13—31. Copyright Θ 1994 by A c a d e m i c Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-174575-9

lo

14

P. Heller and R. Wells

observing its connection with the convolution and downsamphng operations of multirate signal processing, we present a review of the work of Lawton, and that of Eirola. Throughout the paper we work in the setting of rank m wavelet systems (m is the integer dilation factor, not necessarily 2). This involves some generalization of previous work, and yields initial results on the differentiability of wavelet scaling functions for rank m > 2. In particular, we find that the minimal support rank 3 scaling functions with N vanishing wavelet moments have Sobolev smoothness which grows only logarithmically with N. §2 Wavelets and multiresolut ion operators We are considering here rank m wavelets of compact support [9,13], defined by a rank m wavelet matrix a = (ar)k) which is an m x mg matrix satisfying the orthogonality condition 22

a>r,kO>r',k+rnl = m6ry

f c = m6r}0 .

(2.2)

If we look at the Fourier transforms (frequency responses in signal processing terminology) m

k

of the wavelet sequences (filters), the conditions in equations (2.1)-(2.2) can be restated as m— 1

Σ

ΑΛω + 2nk/m) Ar> [ω + 2nk'/m) = —Sry

,

(2.3)

fc,fc'=0

and Ar(0) = Srto .

(2.4)

Associated to the wavelet matrix is a system of scaling functions and wavelets; the rank m scaling function φ associated with the scaling sequence ao,fc is the solution to the dilation equation

Φ(χ) = Σ

α

ο^Φ{τηχ - k) .

(2.5)

k

This dilation equation has a unique compactly supported solution in I? P| L 1 ; its Fourier transform φ satisfies oo 3=1

Spectral Theory of Multiresolution Operators

15

The corresponding family of wavelets ißrj,k (1 < r < m — 1) is defined from φ by ΨΛΧ) = Σ ατ^Φ(τηχ - k) k

and

Vv,j,fc = τη~;ΐ'2'φΓ(Ύη~ΐχ — k) .

Lawton [17] proved that the collection {ipr,j,k(x)} form a tight frame for L 2 (R). We are now in a position to ask critical questions about the family of functions {Vv,i,fc(aO}j such as "When is this tight frame an orthonormal basis?" and "How differentiable are the individual ^r,j,fc?" Since the wavelets Vv,j,fc are linear combinations of dilates of the scaling function φ, these questions reduce to the study of φ: "When do the {φ(χ — k)} form an orthonormal basis for the space Vb = Span {φ(χ — fc)}?" and "How differentiable can we make (#)?" It is in answering these questions that one is led to the (rank m) multires­ olution operator associated with the scaling sequence {a*;}. (Because of the special role of the scaling sequence ao,fc, we will often abbreviate our notation and refer to the scaling sequence as {a^} and its Fourier transform as Α(ω).) This operator is defined in the frequency domain as the map taking a function u on [0,2π) to the function

m χ

. A ί ω + 2π(τη — 1) ' ' + ΙΑ' where we have extended the domain of definition of u by its natural periodization. The multiresolution operator is defined in the time domain as the operator taking a sequence un to (fu)n

= m^2a>kaïv>rntn+k-i ·

(2.6)

(The operator T has been referred to variously as the transition operator and the wavelet- Galerkin operatoria the literature; we take the liberty of introduc­ ing the name multiresolution operator because it more clearly illustrates the central role of this operator in understanding wavelet bases and multiresolu­ tion analysis.) If the function u(u) has the Fourier series expansion ]T n unetnu} then equation (2.6) expresses the relationship between the Fourier coefficients un of u and those of fu; i.e., (Tu)n is the n-th Fourier coefficient of Tu. 2.1 Multiresolution operators and multirate signal processing The multiresolution operator has a natural meaning in the context of multirate signal processing: it represents convolution followed by downsampling - the two operations essential to moving between scales of a multiresolution analysis

16

P. Heller and R. Wells

(hence the name multiresolution operator). In the time domain, we can inter­ pret the multiresolution operator as performing convolution of a sequence with the autocorrelation ä * a, followed by downsampling by m (keeping every ra-th sample). For further discussion of this from a signal processing perspective, see [6]. Given an arbitrary sequence {hn}, the frequency domain representation of the map un —* (h * w) m n (also written (h * u) [ m) is

Φ) -.H(!L)U (Ü) + H ( ί ϋ ± ^ „ (^±^λ + \mJ \mJ \ m J \ m J +H U+2 1) U U;+2 1)

- { T' ) {

T~ )

■

(2 7)

-

We use the notation TH for this map and refer to if as the symbol of the operator T//, in analogy with the theory of pseudodifferential operators. If H is a trigonometric polynomial (i.e., {hn} is a finite sequence), then TH is a bounded linear operator on the Banach space C[0, π] of continuous functions on [0, π] with the sup norm. Since the multiresolution operator takes the form of equation (2.7) with \A\ playing the role of H, we can write TjA|2 for the multiresolution operator, and Γα*α for its time domain form. 2.2 Vanishing moments and the Daubechies coefficients An early theorem on wavelets ([7], p. 153) states that if a wavelet ipr(x) is N times continuously differentiable and has sufficient decay to be in L 1 , then Vv(£) must vanish to order TV at ξ = 0. In order for this to happen, Ar must vanish to order N at ω = 0, which leads us to the following definition. Theorem 2.1. A rank m wavelet system is said to have degree N if any of the following equivalent conditions hold: (d/d£)ni>r(0) = 0, for n = 0, 1, . . . , N - 1 and r = 1, 2, . . . , m - 1 . (d/du;)nAr(0) = 0 , forn = 0, 1, . . . , N - 1 and r = 1, 2, . . . , m - 1 . \d/dJ)nA0 (32*) = 0, forn = 0, 1, . . . , N - 1; k = 1, 2, . . . , m - 1 . Y^i(k + ra/)nao,fc+m/ is independent of k for k = 0, 1, . . . , m — 1 and n = 0, 1, . . . , J V - 1 . (v) Α0(ω) = ( Ι + β ^ + , . .+β* (τη-1)ω ) Ν )Ng(uj); then (2.14) T > / M = i i 1 - cosu;)"TRflM . m Hence, Tj^ is an invariant subspace for Tp, and moreover if f is an eigenvector for Tp in FJ^N with eigenvalue μ, then Τα9{ω)

m

2N

Ρ9(ω),

so that g is an eigenfunction for TR with eigenvalue

m2Nμ.

Remark. Part (iii) establishes a one-to-one correspondence between eigenfunctions of Tp\jrJN and eigenfunctions of TR\SJ_N. This will enable us to reduce the study of the spectrum of Tp to that of TR in the work to follow. Proof: To prove (i), given u G Sj and P as in equation (2.13), extend Tpu to be an even function on [—π,π], and expand it in a Fourier series on this interval. The sum over m terms in the definition of Tp offsets the dilation by ^ so that no fractional powers of sine or cosine appear. The orthogonality of {sinfco;,cosfco;}ensures that all but a finite number of coefficients in the Fourier expansion of Tpu vanish, and we find that Tpu G Sj. (ii) is strongly related to Theorem 2.1. As in that theorem, the factorization I-el |m(l implies that the partial moments Ρ(ω) =

.2iV

(2.15)

R{u))

-eiu))

I

are equal to a number Mn independent of A:, k = 0, 1, . . . , m — 1 for n = 0, 1, . . . , 2AT — 1. Following [7,12], if we consider the polynomial se­ quence en = (jn)j=-j,...tj £ £ji w e find that

enT =

m-^^^2(l)Mn.kek k=0

Since Mo = 1 with our normalizations, we see that each element en of Ej is a generalized left eigenvector of Tp with eigenvalue m~n for n = 0, 1, . . . , 2 Α Γ - 1 . (iii) follows by the factorization in equation (2.15) of P: Τρ\ΐ-βί»\2Ν9(ω)

Τρ/(ω) = m(l 1 m

2N

.

piu)/m\

m

e™\™TRg{u>).

1-e

ίω/τη

2ΛΓ

*(-) + "

20

P. Heller and R. Wells §3 The Lawton-Cohen-Gopinath Theorem on wavelet orthonormal bases

With the definition and elementary properties of the multiresolution operator in hand, we can now begin to apply it to problems of wavelet theory. The first question we asked was "When do the integer translates of 0, {φ(χ — &)}, form an orthonormal basis for the space VQ defined as their span?" In other words, when does the sequence hk defined by hk '= / φ(χ)φ(χ — k)dx satisfy hk = ök,o ? By applying the scaling relation (2.5) to the definition of hk, we find hk = \_]ajai

/ Φ(τηχ — 3)Φ(™>% — I — mk)dx J

ù,i

2_^ajüih2k+i-j-

= 3,1

Simply put, h = Tà*ah . However, since the scaling coefficients α^ satisfy the orthogonality relation /

J

a>kO>k+rnl = £o,Z 5

k

then Τα*αδο,ι = δ0ίι * (α * ä) i m = (a * ä) j m = 0. Therefore, it is sufficient to check that fj converges to / uniformly on B. Since Q is a fundamental region, the translated versions of XQ clearly form a partition of unity, z.e., 1~ £ JeK

XQ

(J(x)) .

(2.7)

Employing Equation (2.7), we get

\m - fi(x)\ = \m - Σ / dA~j °J_1) (*)) *Q (J W*))) JeK

= | Σ (/(*) - / i(A~j ° J'1)^))) XQ (J (A3(*))) I JeK

< Σ | (/(*) - / ^A~j ° J_1)(^))) | XQ ( J (Ai(*))) ■ JeK The sum on the right does not vanish if x e A~j

(J-\Q))

holds. Then we have d(x,A~j

(J-^z)))

< di8Lm(A-j ( J " 1 Q)) ^^diam^-^Q))

< c[ diam (Q) , where we used the facts that A'1 is a contraction and that J - 1 is an isometry. Since the function / is compactly supported and continuous, it is uniformly continuous, so we can find, for a given e > 0, a number δ > 0 such that

d(x,A-j

(j-\z)))

implies

f(A-i(J-\z)))-f(x)\