An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93. [2nd ed.] 9780080495743, 0080495745

Table of contents :
Front Cover
An Introduction to Nonharmonic Fourier Series
Copyright Page
Contents
Preface to the Revised First Edition
Preface to the First Edition
Chapter 1. Bases in Banach Spaces
Chapter 2. Entire Functions of Exponential Type
Chapter 3. The Completeness of Sets of Complex Exponentials
Chapter 4. Interpolation and Bases in Hilbert Space
Notes and Comments
References
List of Special Symbols
Author Index
Subject Index.

Citation preview

AN INTRODUCTION TO NONHARMONIC FOURIER SERIES Revised First Edition

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An Introduction to Nonharmonic Fourier Series

Revised First Edition

Robert M. Young James F. Clark Professor of Mathematics Department of Mathematics Oberlin College Oberlin, Ohio

San Diego

San Francisco New York London Sydney Tokyo

Boston

This book is printed on acid-free paper. Copyright © 2001, 1980 by Academic Press 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. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida, 32887-6777. ACADEMIC PRESS A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK http://www.academicpress.com Library of Congress Catalog Card Number: 00-104370 International Standard Book Number: 0-12-772955-0 Printed in the United States of America 01 02 03 04 05 MB 9 8 7 6 5 4 3 2 1

To Linda

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CONTENTS

Preface to the Revised First Edition Preface to the First Edition

xi xiii

1 Bases in Banach Spaces 1 2 3 4 5 6 7 8 9 10

Schauder Bases Schauder’s Basis for C[a, b] Orthonormal Bases in Hilbert Space The Reproducing Kernel Complete Sequences The Coefficient Functionals Duality Riesz Bases The Stability of Bases in Banach Spaces The Stability of Orthonormal Bases in Hilbert Space

1 3 5 13 16 19 22 25 31 35

2 Entire Functions of Exponential Type Part One. The Classical Factorization Theorems 1 Weierstrass’s Factorization Theorem 2 Jensen’s Formula

46 46 50

viii

CONTENTS

3 Functions of Finite Order 4 Estimates for Canonical Products 5 Hadamard’s Factorization Theorem Part Two. Restrictions along a Line 1 2 3 4 5

The “Phragm´en–Lindel¨of” Method Carleman’s Formula Integrability on a Line The Paley–Wiener Theorem The Paley–Wiener Space

53 59 63 68 68 74 79 85 90

3 The Completeness of Sets of Complex Exponentials 1 2 3 4 5 6

The Trigonometric System Exponentials Close to the Trigonometric System A Counterexample Some Intrinsic Properties of Sets of Complex Exponentials Stability Density and the Completeness Radius

94 99 102 106 110 115

4 Interpolation and Bases in Hilbert Space 1 2 3 4 5 6 7 8 9

Moment Sequences in Hilbert Space Bessel Sequences and Riesz–Fischer Sequences Applications to Systems of Complex Exponentials The Moment Space and Its Relation to Equivalent Sequences Interpolation in the Paley–Wiener Space: Functions of Sine Type Interpolation in the Paley–Wiener Space: Stability The Theory of Frames The Stability of Nonharmonic Fourier Series Pointwise Convergence

122 128 135 139 142 149 154 160 165

CONTENTS

ix

Notes and Comments

171

References

199

List of Special Symbols

221

Author Index

223

Subject Index

229

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PREFACE TO THE REVISED FIRST EDITION

When An Introduction to Nonharmonic Fourier Series first appeared, there was little reason to believe that the theory of Riesz bases and frames in Hilbert space would one day play so prominent a role in both pure and applied harmonic analysis. The “wavelets revolution” of the past fifteen years helped change that. Now, the three classic treatises in the field — arguably those by Daubechies [1992], Meyer [1992], and Mallat [1999] — all attest to the fact that research in frame theory flourishes. Moreover, in the more specialized area of nonharmonic Fourier series, the celebrated work of Avdonin and Ivanov [1995] provides ample evidence that Riesz bases of complex exponentials remain of central importance in the theory of control. I am therefore deeply indebted to Bob Ross and the editorial staff of Academic Press for the opportunity to bring An Introduction to Nonharmonic Fourier Series up to date. To maintain the continuity of the original text, most of the new material has been incorporated in a greatly expanded set of Notes and References. In this way, the book remains true to its original goal — to provide an elementary introduction to a rich and multifaceted field, not an exhaustive account of all that is known.

xi

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PREFACE TO THE FIRST EDITION

The theory of nonharmonic Fourier series is concerned with the completeness and expansion properties of sets of complex exponentials fein t g in L p [, ]. Its origins, which are classical in spirit, lie in the celebrated works of Paley and Wiener [1934] and Levinson [1940]. In recent years, in response to the development of functional analysis and, in particular, to the growing interest in bases in Banach spaces, research in the area has flourished. New approaches to old problems have led to important advances in the theory. This book is an account of both the classical and the modern theories. Its underlying theme is the elegant interplay among the various parts of analysis. The catalyst in the present case is the Fourier transform, through which the classical Banach spaces are mapped into spaces of entire functions. In this way, problems in one domain can be examined via their transform image in the other. The book is designed primarily for the graduate student or mathematician who is approaching the subject for the first time. Its aim as such is to provide a unified and self-contained introduction to a multifaceted field, not an exhaustive account of all that is known. Accordingly, the first half of the book presents an elementary introduction to the theory of bases in Banach spaces and the theory of entire functions of exponential type. At the same time, an extensive set of notes touches on more advanced topics, indicates directions in which the theory can be extended, and should prove useful to both specialists and nonspecialists alike. Much of the material appears in book form for the first time. xiii

xiv

Preface to the First Edition

The only prerequisites are a working knowledge of real and complex analysis, together with the elements of functional analysis. By that I mean roughly what is contained in Rudin [1966]. On occasion, when more advanced tools of analysis are required, appropriate references are given. Apart from this, the work is essentially self-contained, and it can serve as a textbook for a course at the second- or third-year graduate level. The problems, which are of varying degrees of difficulty, are an integral part of the text. Some are routine applications of the theory, while others are important ancillary results — these are usually accompanied by an indication of the solution and an appropriate reference to the literature. A word about notation: Theorem 2.3 refers to Theorem 3 of Chapter 2. The labeling of all other results is self-explanatory. I am deeply indebted to Doug Dickson and Paul Muhly for their careful reading of the manuscript and for their sharp criticism and advice. I owe immeasurable thanks to Linda Miller, who proofread the entire book more times than I could possibly have hoped. ROBERT M. YOUNG

1 BASES IN BANACH SPACES

1 SCHAUDER BASES Let X be an infinite-dimensional Banach space over the field of real or complex numbers. When viewed as a vector space, X is known to possess a Hamel basis — a linearly independent subset of X that spans the entire space. Unfortunately, such bases cannot in general be constructed, their very existence depending on the axiom of choice, and their usefulness is therefore severely limited. Of far greater importance and applicability in analysis is the notion of a basis first introduced by Schauder [1927]. Definition. A sequence of vectors fx1 , x2 , x3 , . . .g in an infinitedimensional Banach space X is said to be a Schauder basis for X if to each vector x in the space there corresponds a unique sequence of scalars fc1 , c2 , c3 , . . .g such that 1  xD cn xn . nD1

The convergence of the series is understood to be with respect to the strong (norm) topology of X; in other words,   n      ci xi  ! 0 as n ! 1. x    iD1

1

2

BASES IN BANACH SPACES CH. 1

Henceforth, the term basis for an infinite-dimensional Banach space will always mean a Schauder basis. Example. The Banach space lp 1  p < 1 consists, by definition, of all infinite sequences of scalars c D fc1 , c2 , c3 , . . .g such that jjcjjp D  p 1/p < 1. The vector operations are coordinatewise. In each of  1 nD1 jcn j  these spaces, the “natural basis” fe1 , e2 , e3 , . . .g, where en D 0, 0, . . . , 0, 1, 0, . . ., and the 1 appears in the nth position, is easily seento be a Schauder basis. If c D fcn g is in lp , then the obvious expansion c D 1 nD1 cn en is valid. It is clear that a Banach space with a basis must be separable. Reason: If  fxn g is a basis for X, then the set of all finite linear combinations cn xn , where the cn are rational scalars, is countable and dense in X. It follows, for example, that since l1 is not separable, it cannot possess a basis. The “basis problem” — whether or not every separable Banach space has a basis — was raised by Banach [1932] and remained until recently one of the outstanding unsolved problems of functional analysis. The question was finally settled by Per Enflo [1973], who constructed an example of a separable Banach space having no basis. The negative answer to the basis problem is perhaps surprising in light of the fact that bases are now known for almost all the familiar examples of infinite-dimensional separable Banach spaces. PROBLEMS

1. Prove that every vector space has a Hamel basis. 2. Prove that every Hamel basis for a given vector space has the same number of elements. This number is called the (linear) dimension of the space. 3. Show that a Hamel basis for an infinite-dimensional Banach space is uncountable. 4. Show that the dimension of l1 is equal to c. (Hint: Show that the set f1, r, r 2 , . . . : 0 < r < 1g is linearly independent.) 5. Let X be an infinite-dimensional Banach space. (a) Prove that dim X  c. (Hint: Show that there is a vector space isomorphism between l1 and a subspace of X.) (b) Prove that if X is separable, then dim X D c. 6. The Banach space c0 consists of all infinite sequences of scalars which converge to zero (with the l1 norm). Show that the natural basis is a Schauder basis for c0 . 7. Exhibit a Schauder basis for the Banach space c consisting of all convergent 1 sequences of scalars (with the l norm). 8. An infinite series xn in a Banach space X is said to be unconditionally convergent if every arrangement of its terms converges to the same

SEC. 2

SCHAUDER’S BASIS FOR

C [a, b]

3

 element. It is said to be absolutely convergent if the series jjxn jj is convergent. Show that every absolutely convergent series in X is unconditionally convergent. What about the converse? said to be unconditional (absolute) if 9. A basis fxn g for a Banach space X is every convergent series of the form cn xn is unconditionally (absolutely) convergent. (a) Show that the natural basis is unconditional for the spaces lp , 1  p < 1, and c0 . Show also that it is absolute for lp only when p D 1. Is it absolute for c0 ? (b) Show that the sequence of vectors

1, 0, 0, 0, . . .,

1, 1, 0, 0, . . .,

1, 1, 1, 0, . . .,

...

forms a basis for c0 which is not unconditional.

2 SCHAUDER’S BASIS FOR C[a, b] One of the most important and widely studied classical Banach spaces is C[a, b], the space of all continuous functions on the closed finite interval [a, b], together with the norm jjfjj D max jfxj. The celebrated Weierstrass approximation theorem asserts that the polynomials are dense in C[a, b]: if f is continuous on [a, b], then for every positive number ε there is a polynomial P such that the inequality jfx  Pxj < ε holds throughout the interval [a, b]. For a given continuous function, a sequence of approximating polynomials can even be given explicitly. The most elegant representation is due to Bernstein. Let us suppose, for simplicity, that f is continuous on the interval [0, 1]. Then the nth Bernstein polynomial for f is n      k n Bn x D f x k 1  xnk , n D 1, 2, 3, . . . . k n kD0 As is well known, fx D lim Bn x n!1

uniformly on [0, 1] (see Akhiezer [1956, p. 30]). Since every polynomial can be uniformly approximated on a closed interval by a polynomial with rational coefficients, the preceding remarks show that the space C[a, b] is separable; in fact, it has a basis.

4

BASES IN BANACH SPACES CH. 1

Theorem 1 (Schauder). The space C[a, b] possesses a basis. Proof. We are going to construct a basis for C[a, b] consisting of piecewise-linear functions fn n D 0, 1, 2, . . .. This means that to each function f in the space there will correspond a unique sequence of scalars fcn g such that fx D

1 

cn fn x

nD0

uniformly on [a, b]. Let fx0 , x1 , x2 , . . .g be a countable dense subset of [a, b] with x0 D a and x1 D b. Set xa . f0 x D 1 and f1 x D ba When n  2, the set of points fx0 , x1 , . . . , xn1 g partitions [a, b] into disjoint open intervals, one of which contains xn ; call it I. Define  0 if x 2 /I fn x D 1 if x D xn linear elsewhere for n D 2, 3, 4, . . . . The sequence ff0 , f1 , f2 , . . .g will be the required basis. For each function f in C[a, b] and each positive integer n, we denote by Ln f the polygonal function that agrees with f at each of the points x0 , x1 , . . . , xn ; we denote by L0 f the function whose constant value is fx0 . Since f is uniformly continuous on [a, b], a simple continuity argument shows that Ln f ! f uniformly on [a, b]. Therefore, we can write f D L0 f C

1 

Ln f  Ln1 f.

nD1

We are going to show that there are scalars c1 , c2 , c3 , . . . such that Ln f  Ln1 f D cn fn

n D 1, 2, 3, . . ..

For this purpose, we shall define a sequence of functions fg0 , g1 , g2 , . . .g recursively by the equations g0 D fx0 f0

and gn D gn1 C f  gn1 xn fn ,

n D 1, 2, 3, . . . .

The claim is that gn D Ln f, whence cn D f  Ln1 fxn . Since gn is a polygonal function whose only possible corners are at the points x0 , x1 , . . . , xn , it is sufficient to show that gn agrees with f at each

SEC. 3

ORTHONORMAL BASES IN HILBERT SPACE

5

of these points. This is trivial for n D 0, and we proceed by induction. Since fn xn  D 1, it follows that gn xn  D fxn ; if i < n, then fn xi  D 0, and it follows from the definition of gn , together with the induction hypothesis, that gn xi  D gn1 xi  D fxi . This establishes the claim. Accordingly, every function f 2 C[a, b] has at least one representation of the form 1  fD cn fn , nD0

and we have only to show that this representation is unique.Suppose then 1 that 1 some function g has two different representations, say nD0 an fn and nD0 bn fn . If N is the smallest value of n for which an 6D bn , then 1  nDN

an fn x D

1 

bn fn x

nDN

for every x. Choose x D xN . Since fn xN  D 0 whenever n > N, it follows that aN D bN . But this contradicts the choice of N, and hence an D bn for every n.  PROBLEMS

1. Give a probabilistic interpretation of the Bernstein polynomials (see Feller [1966, Chap. VII]). 2. Prove that the space C[a, b] is separable by showing that every continuous function on [a, b] can be uniformly approximated by polynomials with rational coefficients. 3. Let f be a continuous function on (1, 1). Prove that if there is a sequence of polynomials fP1 , P2 , P3 , . . .g such that Pn ! f uniformly on (1, 1), then f must itself be a polynomial. 4. Let f be a continuous function on [a, b]. Show  that there is a sequence of polynomials fP1 , P2 , P3 , . . .g such that f D 1 nD1 Pn and the series converges absolutely and uniformly on [a, b].

3 ORTHONORMAL BASES IN HILBERT SPACE In a separable Hilbert space† , a distinguished role is played by those Schauder bases that are orthonormal — the basis vectors are mutually perpendicular and each has unit length. An equivalent characterization of such bases is that they are complete orthonormal sequences. (Recall that a sequence † All Hilbert spaces are assumed to be infinite-dimensional.

6

BASES IN BANACH SPACES CH. 1

of vectors ff1 , f2 , f3 , . . .g in a Hilbert space is said to be complete if the zero vector alone is perpendicular to every fn .) It follows readily from this characterization that every separable Hilbert space has an orthonormal basis. The most important property of an orthonormal basis (as opposed to any other basis) is the simplicity of all basis expansions. If fe1 , e2 , e3 , . . .g is an orthonormal basis for a Hilbert space H, then for every element f 2 H we have the Fourier expansion fD

1  f, en en . nD1

The inner product (f, en ) is called the nth Fourier coefficient of f (relative to fen g). When the Pythagorean formula is applied to this series, the result is Parseval’s identity: 1  2 jf, en j2 . jjfjj D nD1

The validity of Parseval’s identity for every vector in the space is both necessary and sufficient for an orthonormal sequence to be a basis. Since the linear transformation f ! ff, en g from H into l2 preserves norms, it must also preserve inner products. Thus f, g D

1 

f, en g, en 

nD1

for every pair of vectors f and g; this is the generalized Parseval identity. Even if an orthonormal sequence fen g is incomplete, Bessel’s inequality is always valid: 1  jf, en j2  jjfjj2 nD1

whenever f 2 H. This shows, in particular, that the Fourier coefficients of each element of H form a square-summable sequence. The Riesz–Fischer theorem shows, that every square-summable sequence is obtained conversely, 2 jc j < 1, then there exists an element f in H for which in this way: if 1 nD1 n f, en  D cn ,

n D 1, 2, 3, . . . .  The proof is trivial: simply choose f D 1 nD1 cn en . We conclude that if fen g is a complete orthonormal sequence in H, then the correspondence f ! ff, en g between H and l2 is a Hilbert space isomorphism. It follows that from a geometric point of view, all separable Hilbert spaces are “indistinguishable”, that is to say, isomorphic.

SEC. 3

7

ORTHONORMAL BASES IN HILBERT SPACE

Example 1.

In l2 the “natural basis” is orthonormal.

Example 2.

In L 2 [, ], with the inner product   1 f, g D ftgt d t, 2 

the complex trigonometric system feint g1 1 constitutes an orthonormal basis. That the system is orthonormal is obvious; we prove that it is complete. Theorem 2. Proof.

The trigonometric system is complete in L 2 [, ].

The proof will establish even more. Suppose that   fteint d t D 0 

for some integrable function f defined on [, ] and n D 0, š1, š2, . . . . It is to be shown that f D 0 a.e. Set  t gt D fu d u 

for t 2 [, ]. Integration by parts shows that   gt  ceint d t D 0 

for every constant c and n D š1, š2, š3, . . .. Choose c so that this holds for n D 0 also, and put Ft D gt  c. Then F is continuous on [, ] and F D F. Weierstrass’s theorem on approximation by trigonometric polynomials guarantees that for each ε > 0 there is a finite trigonometric sum Tt D

n 

ck eikt

kDn

such that

jFt  Ttj < ε whenever

It follows that jjFjj2 D

1 2

ε  2

 

  



jFtj2 d t D

1 2



jFtj d t  εjjFjj,





jtj  .

FtFt  Tt d t

8

BASES IN BANACH SPACES CH. 1

so that

jjFjj  ε.

Since ε was arbitrary, F D 0, so that g D c and f D 0 a.e.



Consequently, every function f in L 2 [, ] has a unique Fourier series expansion 1  int O ft D fne 1

O (in the mean † ). Here fn denotes the nth Fourier coefficient of f relative to feint g, i.e.,   1 O fteint d t n D 0, š1, š2, . . .. fn D 2  By Parseval’s formula, 1 2



 

jftj2 d t D

1 

2 O jfnj .

1

O The mapping f ! ffng is a Hilbert space isomorphism between L 2 [, ] 2 and l . There is a simple but useful extension of Parseval’s identity that is worth mentioning. If f 2 L 2 [, ], let fO be the Fourier transform of f:   1 O fx D fteixt d t 1 < x < 1. 2  Proposition 1. ber A,

For every function f 2 L 2 [, ] and every real num1 

O C Aj2 D jjfjj2 . jfn

1

Proof.

Put gt D fteiAt . Then O C A D gn fn O

for every integer n. Since A is real, jjfjj D jjgjj, and the result follows from Parseval’s identity applied to g.  As an illustration, let us choose f to be the constant function 1. A simple calculation shows that O D sin x fx x † Pointwise convergence is of course much harder. A deep result of Carleson [1966] says that the Fourier series of an L 2 function converges (to the function) pointwise almost everywhere.

SEC. 3

9

ORTHONORMAL BASES IN HILBERT SPACE

for all real x. Setting A D t/, where t is real and not an integral multiple of , we obtain the important identity 1

 1 1 D . 2 n C t2 sin t 1

Example 3. The space H 2 (named after Hardy) consists of all functions f analytic in the open unit disk (in the complex plane) whose Taylor coefficients are square-summable, i.e., fz D

1 

cn z n ,

with

nD0

1 

jcn j2 < 1.

nD0

1

The inner product of two functions fz D nD0 an zn and gz D in H 2 is, by definition, 1  an bn . f, g D

1 nD0

bn zn

nD0

It is clear that H 2 can be identified with the (closed) subspace of L 2 [, ] spanned by the functions eint with n  0. Let en z D zn for jzj < 1 n D 0, 1, 2, . . .; then the en ’s form an orthonormal basis for H 2 . The natural mapping c0 , c1 , c2 , . . . !

1 

cn z n

nD0 2

2

between l and H is a Hilbert space isomorphism. Example 4. The space A2 consists of all functions f that are analytic in the open unit disk and have finite area norm  

1/2 2 jfzj d x d y . jjfjj D jzj n > k we have by hypothesis  n   m           jjckn  ckm xk jj  M  cin  cim xi   M2  cin  cim xi      iD1 iD1  n  m     2  D M  cin xi  cim xi  .   iD1

iD1

As m, n ! 1, this last expression tends to zero. Since xk 6D 0, it follows that for each k the sequence fckn g1 nD1 is a Cauchy sequence, and therefore the limit ck D lim ckn n!1

exists. It is now a routine matter to show that 1  ck xk . xD kD1

The details are left to the reader. We complete the proof by showing that this representation is unique.  It suffices to show that if 1 kD1 ck xk D 0, then ck D 0 for every k. For fixed k and n  k we have once again by hypothesis  n      ci xi   . jjck xk jj  M    iD1

Letting n ! 1, we find jjck xk jj D 0, and hence ck D 0 for every k.



22

BASES IN BANACH SPACES CH. 1

PROBLEMS

1. Show that the normed vector space Y defined in the proof of Theorem 3 is a Banach space. 2. Let fxn g denote Schauder’s basis for C[a, b] and let ffn g be the associated sequence of coefficient functionals. Compute jjfn jj. 3. Use the criterion of Theorem 4 to show that Schauder’s system is in fact a basis for C[a, b]. 4. Let fxn g be a basis for a Banach space X and fSn g the corresponding sequence of partial sum operators. Show that 1  supn jjSn jj < 1. 5. (Karlin) A basis fxn g for a Banach space X is said to be normal if jjxn jj D jjfn jj D 1 for every n. Prove that in a Hilbert space every normal basis is orthonormal. (Hint: Show that if fxn g is normal, then xn ? xm whenever n 6D m (see Problem 6, Section 3).) 6. A sequence fxn g of nonzero vectors in a Banach space X is said to be an orthogonal system if for each n n D 1, 2, 3, . . .) the linear span of fx1 , . . . , xn g is orthogonal to xnC1 (see Problem 6, Section 3). (a) Show that an orthogonal system is linearly independent. (b) Show that a complete orthogonal system is a basis. 7. A Banach space X is said to have the approximation property (in the sense of Grothendieck ) if the identity operator on X can be approximated, uniformly on every compact subset of X, by operators of finite rank. (a) Prove that every Banach space with a basis has the approximation property. (b) Let X be a Banach space with a basis and let Y be an arbitrary Banach space. Prove that every compact linear transformation from X into Y is the limit (with respect to the norm topology) of operators of finite rank. 8. Prove that the disk algebra A has the approximation property. (See Problem 7. In the notation of Problem 8, Section 5, A D AD, with D D fz: jzj < 1g.)

7 DUALITY Suppose that fxn g is a basis for a Banach space X and that ffn g is its associated sequence of coefficient functionals. What can we say about the sequence ffn g? Surely not that it is a basis for XŁ . For if XŁ is nonseparable, then it contains no basis at all (example: X D l1 ). Therefore, unless ffn g is complete, the most we can hope for is that it be a basis for its closed linear span. The hope is justified. The closed linear span of a sequence fxn g of elements from a normed vector space X will be denoted by [xn ]. Theorem 5. Let fxn g be a basis for a Banach space X and let ffn g be the associated sequence of coefficient functionals. Then ffn g is a basis for [fn ]

SEC. 7

23

DUALITY

and the expansion fD

1 

fxn fn

nD1

is valid for every f in [fn ]. Proof. The proof is based on the fact that the adjoint of the nth partial sum operator Sn is given by the formula SnŁ f D

n 

fxi fi

iD1

for every f 2 XŁ . The formula is valid because n

n

  fi xxi D fxi fi x SnŁ fx D fSn x D f iD1

iD1

for every x 2 X. It is to be shown that SnŁ f ! f for every f 2 [fn ]. Letus first assume that f is a finite linear combination of the fi , say fD m iD1 ci fi . Then for every n  m SnŁ f D

n  iD1

fxi fi D

m 

ci fi D f,

iD1

and hence trivially SnŁ f ! f. Iff is an arbitrary element of [fn ], then given ε > 0, we can find gD m iD1 ci fi such that jjf  gjj < ε/M C 1, where M D supn jjSn jj < 1. It follows that for all n  m jjSnŁ f  fjj  jjSnŁ f  SnŁ gjj C jjSnŁ g  gjj C jjg  fjj D jjSnŁ f  SnŁ gjj C jjf  gjj  jjSn jj C 1jjf  gjj < ε,

SnŁ f

so that once again ! f.  Thus every element of [fn ] has at least one representation of the form f D 1 nD1 cn fn . Since fxn  D cn for every n, the representation is unique and ffn g is a basis for its closed linear span.  When X is a reflexive Banach space, ffn g is a fortiori complete. Theorem 6. If fxn g is a basis for a reflexive Banach space X, then the associated sequence of coefficient functionals ffn g is a basis for XŁ . Proof. In view of Theorem 5, we need only establish that ffn g is complete in XŁ . Suppose then that 2 XŁŁ and that fn  D 0 for n D 1, 2, 3, . . . . It is to be shown that D 0. Let P be the canonical imbedding of X into XŁŁ , i.e., Pxf D fx

24

BASES IN BANACH SPACES CH. 1

for f in XŁ and x in X. Since X is reflexive, it follows that D Px for some x; accordingly, fn x D 0 for n D 1, 2, 3, . . . . Since fxn g is a basis, we have  xD 1 f xx n D 0. Thus D 0.  nD1 n For the remainder of this section X will denote a fixed Hilbert space. Two sequences fxn g and fyn g in X are said to be biorthogonal if xm , yn  D υmn for every m and n. The Hahn-Banach theorem shows that for a given sequence fxn g a biorthogonal sequence fyn g will exist if and only if fxn g is minimal † (this means that each element of the sequence lies outside the closed linear span of the others). If this condition is fulfilled, then the biorthogonal sequence fyn g will be uniquely determined if and only if fxn g is complete. Suppose now that X is separable and that fxn g is a basis for X. The isomorphism between X and XŁ shows that to each coefficient functional fn there corresponds an element yn 2 X such that fn x D x, yn  for all x. Since fn xm  D υmn , we see that fxn g and fyn g are biorthogonal. Thus we have shown that every basis fxn g for a Hilbert space possesses a unique biorthogonal sequence fyn g. In terms of this biorthogonal pair, each vector x in the space can be uniquely represented in the form xD

1 

x, yn xn .

nD1

Combining Theorems 5 and 6, we see that fyn g is also a basis for X and that, by duality, 1  xD x, xn yn . nD1

Thus, in a Hilbert space the sequence biorthogonal to a basis is itself a basis. PROBLEMS

1. Let fxn g be a basis for a Banach space X and let ffn g be the associated sequence of coefficient functionals. Prove or disprove: if XŁ is separable, then ffn g is complete (and hence a basis) in XŁ . 2. A sequence fxn g of elements of a Banach space X is said to have ffn g, where fn 2 XŁ , as a biorthogonal sequence if fi xj  D υij for every i and j. (a) Show that fxn g has a biorthogonal sequence if and only if it is minimal. (b) Show that a biorthogonal sequence for fxn g is uniquely determined if and only if fxn g is complete in X. † The terms (topologically) independent and (topologically) free are also used.

SEC. 8

25

RIESZ BASES

(c) Show that fxn g is a basis for X if and only if it possesses a biorthogonal sequence ffn g such that 1 nD1 fn xxn converges to x for every x in X. 3. (Karlin) Let fxn g be a sequence of elements of a Banach space X and let ffn g be biorthogonal to fxn g. Prove that if ffn g is a basis for XŁ , then fxn g is a basis for X. 4. Show by an example that in a Hilbert space, a sequence biorthogonal to a complete sequence need not itself be complete.

8 RIESZ BASES The simplest and perhaps the most obvious way of constructing new bases from old is through an isomorphism of the underlying space. Thus, if fxn g is a fixed but arbitrary basis for a Banach space X and if the bounded invertible operator† T transforms fxn g into fyn g, that is, if Txn D yn

for

n D 1, 2, 3, . . . ,

then fyn g is also a basis for X. In fact, it is easy to see that fxn g and fyn g are equivalent in the following sense. Definition. Two bases fxn g and fyn g for a Banach space X are said to be equivalent if 1 

cn xn is convergent if and only if

1 

nD1

cn yn is convergent.

nD1

The property of being linked by an isomorphism of the space completely characterizes equivalent bases. Theorem 7. Two bases fxn g and fyn g for a Banach space X are equivalent if and only if there exists a bounded invertible operator T: X ! X such that Txn D yn for every n. Proof. The sufficiency is trivial. Suppose  then that fxn g and fyn g are 1 equivalent bases for X. If x 2 X, with x D nD1 cn xn , then the series 1 c y converges to an element Tx in X. The function T thus defined n n nD1 is clearly linear, one-to-one and onto, and Txn D yn for every n. To show that T is a bounded invertible operator, we define functions Tn by setting Tn x D niD1 ci yi . Then Tx D lim Tn x n!1

† “Invertible” means one-to-one and onto.

26

BASES IN BANACH SPACES CH. 1

for every x. Since each Tn is bounded (by Theorem 3), it follows that T is bounded (by the Banach–Steinhaus theorem). The open mapping theorem guarantees that T is invertible, and the result follows.  Theorem 8. In a Hilbert space equivalent bases have equivalent biorthogonal sequences. Proof. Let fxn g and fyn g be equivalent bases for a Hilbert space H and let ffn g and fgn g be their respective biorthogonal sequences. Observe to begin with that a sequence biorthogonal to a basis is also a basis, so that ffn g and fgn g are themselves bases for H. Let T be a bounded invertible operator on H such that Txn D yn n D 1, 2, 3, . . .. Then the adjoint operator TŁ is also bounded and invertible. Assertion: TŁ gn D fn n D 1, 2, 3, . . .. Indeed, for fixed n and all values of m we have TŁ gn , xm  D gn , Txm  D gn , ym  D υnm D fn , xm ; since fxm g is complete, the assertion follows. This proves that ffn g and fgn g are equivalent bases for H.  In a separable Hilbert space the most important bases are orthonormal. Second in importance are those bases that are equivalent to some orthonormal basis. They will be called Riesz bases, and they constitute the largest and most tractable class of bases known. Definition. A basis for a Hilbert space is a Riesz basis if it is equivalent to an orthonormal basis, that is, if it is obtained from an orthonormal basis by means of a bounded invertible operator. A Riesz basis ffn g for a Hilbert space is necessarily bounded, that is to say, 0 < inf jjfn jj  sup jjfn jj < 1. n

n

In fact, if ffn g is obtained from the orthonormal basis fen g by means of the bounded invertible operator T, then for every n 1  jjfn jj  jjTjj. jjT1 jj It follows readily from this that if ffn g is a Riesz basis, then the sequence of unit vectors ffn /jjfn jjg is also a Riesz basis. Reason: There exists a bounded invertible operator S for which Sen D

en jjfn jj

n D 1, 2, 3, . . .;

SEC. 8

27

RIESZ BASES

therefore, the operator TS is bounded and invertible, and TSen D

fn jjfn jj

n D 1, 2, 3, . . ..

The next theorem provides a number of important characteristic properties of Riesz bases. Theorem 9. Let H be a separable Hilbert space. Then the following statements are equivalent. (1) The sequence ffn g forms a Riesz basis for H. (2) There is an equivalent † inner product on H, with respect to which the sequence ffn g becomes an orthonormal basis for H. (3) The sequence ffn g is complete in H, and there exist positive constants A and B such that for an arbitrary positive integer n and arbitrary scalars c1 , . . . , cn one has A

n  iD1

 n 2 n       jci j2   ci fi   B jci j2 .   iD1

iD1

(4) The sequence ffn g is complete in H, and its Gram matrix fi , fj 1 i,jD1 generates a bounded invertible operator on l2 . (5) The sequence ffn g is complete in H and possesses a complete biorthogonal sequence fgn g such that 1  nD1

jf, fn j2 < 1 and

1 

jf, gn j2 < 1

nD1

for every f in H. Proof. 1 ) 2: Since ffn g is a Riesz basis for H, there exists a bounded invertible operator T that transforms ffn g into some orthonormal basis fen g, i.e., Tfn D en for n D 1, 2, 3, . . . . Define a new inner product f, g1 on H by setting f, g1 D Tf, Tg, † Two inner products are said to be equivalent if they generate equivalent norms.

28

BASES IN BANACH SPACES CH. 1

and let jj jj1 be the norm generated by this inner product. Then jjfjj  jjfjj1  jjTjj Ð jjfjj jjT1 jj for every f in H, so that the new inner product is equivalent to the original one. Clearly, fi , fj 1 D Tfi , Tfj  D ei , ej  D υij for every i and j. 2 ) 3: Suppose that f, g1 is an equivalent inner product on H relative to which the sequence ffn g forms an orthonormal basis. From the relations mjjfjj  jjfjj1  Mjjfjj

for every f,

where m and M are positive constants not depending on f, it follows that for arbitrary scalars c1 , . . . , cn one has  n 2 n n  1  2  1  2  jc j  c f  jci j .  i i i    M2 m2 iD1

iD1

iD1

Clearly, the sequence ffn g is complete in H. 3 ) 1: Let fen g be an arbitrary orthonormal basis for H. It follows by assumption that there exist bounded linear operators T and S such that Ten D fn and Sfn D en n D 1, 2, 3, . . .. Certainly, ST D I. Since ffn g is complete, we also have TS D I. Hence T is invertible, and ffn g is a Riesz basis for H. 1 ) 4: Let T be a bounded invertible operator on H that carries some orthonormal basis fen g into the basis ffn g. If A D aij  denotes the matrix of the (invertible) operator TŁ T relative to fen g, then aij D TŁ Tej , ei  D Tej , Tei  D fj , fi . Therefore, the Gram matrix of ffn g is the conjugate of A. 4 ) 3: Suppose that the Gram matrix of ffn g generates a bounded invertible operator on l2 . If fen g is an arbitrary orthonormal basis for H, then the transformation T: H ! H, defined by



   ci ei D ei fi , fj cj T i

i

j

whenever fci g 2 l2 , is obviously linear, bounded, and invertible. A straightforward calculation shows that  2



      ci fi  D T ci ei , ci ei ,    i

i

i

SEC. 8

29

RIESZ BASES

so that, in addition, T is a positive operator. (Recall that a bounded linear operator T on a Hilbert space is said to be positive if Tf, f  0 for every vector f in the space.) Since T is positive, it has a (unique) positive square root (see Riesz and Sz.-Nagy [1955, p. 265]); call it P. The equation above may then be put in the form  2 

2          ci fi  D P ci ei   ,      i

i

from which it follows at once that  2   1  2   jc j  c f  jjTjj jci j2 .     i i i   jjT1 jj i i i 1 ) 5: Let fgn g be the unique sequence in H biorthogonal to ffn g. By Theorem 8, fgn g is also a Riesz basis for H. Since every vector f in H has the two biorthogonal expansions fD

1 

f, gn fn

and f D

nD1

1 

f, fn gn ,

nD1

the result follows immediately from the definition of a Riesz basis. 5 ) 1: Consider the linear transformation from H into l2 defined by f ! ff, fn g. The reader will verify without great difficulty that this mapping is closed. By the closed graph theorem it is continuous, and hence there exists a positive constant C for which 1 

jf, fn j2  C2 jjfjj2

for all f.

nD1

Similarly, there exists a positive constant D for which 1 

jf, gn j2  D2 jjfjj2

for all f.

nD1

Fix an arbitrary orthonormal basis fen g for H, and define operators S and T on the linear subspaces spanned by the sequences ffn g and fgn g, respectively, by setting



    S ci fi D ci ei and T ci gi D ci ei . i

i

i

i

30

BASES IN BANACH SPACES CH. 1

By virtue of the two inequalities above, we have   

          ci fi   D  ci fi  S     i

and

i

  

          ci gi   C  ci gi  . T     i

i

Since both sequences ffn g and fgn g are complete, each of the operators S and T can be extended by continuity to a bounded linear operator on the entire space. If f D 6ai fi and g D 6bj gj are finite sums, a simple calculation shows that Sf, Tg D f, g; by continuity, this holds for every pair of vectors f and g. We have, accordingly, f, SŁ Tg D f, g, so that SŁ T D I. The existence of a right-inverse for SŁ implies that SŁ is onto, and hence that S is bounded from below (see Taylor [1958, p. 234]). Since the range of S is dense in H, we conclude that S is invertible. Thus the sequence ffn g forms a Riesz basis for H.  The class of Riesz bases is very large. It is extremely difficult to exhibit at least one bounded basis for a Hilbert space that is not equivalent to an orthonormal basis. We mention without proof the following example of such a basis in the space L 2 [, ]; it was discovered by Babenko [1948]. 1 Example. The sequence fjtj˛ eint g1 nD1 , with 0 < ˛ < 2 , is a bounded basis for L 2 [, ] that is not a Riesz basis. The appearance of simplicity is misleading; the example is exceedingly difficult.

PROBLEMS

1. Prove that a sequence that is biorthogonal to a Riesz basis is also a Riesz basis. 2. Suppose that ffn g is a sequence of vectors in a Hilbert space H such that 6jf, fn j2 < 1 whenever f 2 H. Show that the mapping f ! ff, fn g has a closed graph. 3. Let ffn g be a Riesz basis for a Hilbert space H and let fgn g be biorthogonal to ffn g. Theorem 9 guarantees that there exist positive constants A and B such that   A jf, gn j2  jjfjj2  B jf, gn j2 for all f.

SEC. 9

31

THE STABILITY OF BASES IN BANACH SPACES

Show that one has the dual relation 1 1 jf, fn j2  jjfjj2  jf, fn j2 for all f. B A 4. Let ffn g be a basis for a Hilbert space H and let fgn g be biorthogonal to ffn g. We shall call ffn g a Bessel basis if 1 

cn fn

1 

is convergent only if

nD1

jcn j2 < 1;

nD1

we shall call ffn g a Hilbert basis if 1 

cn fn

is convergent if

1 

nD1

jcn j2 < 1.

nD1

(a) Show that ffn g is a Riesz basis if and only if it is both a Bessel basis and a Hilbert basis. (b) Show that ffn g is a Bessel (Hilbert) basis if and only if fgn g is a Hilbert (Bessel) basis. (c) Show that ffn g is a Bessel basis if and only if there exists a constant A > 0 such that 2  n n      2 jci j   ci fi  A   iD1

iD1

for arbitrary scalars c1 , . . . , cn (n D 1, 2, 3, . . .). (d) Show that ffn g is a Hilbert basis if and only if there exists a constant B > 0 such that  n 2 n      ci fi   B jci j2    iD1

iD1

for arbitrary scalars c1 , . . . , cn (n D 1, 2, 3, . . .). 5. Let ffn g be a basis for a Hilbert space H and let fgn g be its biorthogonal basis. Prove that ffn g and fgn g are equivalent if and only if ffn g is a Riesz basis. 6. Let ffn g be a basis for a Hilbert space H and let gn D n fn , where 0 < inf jn j  sup jn j < 1. n

n

Prove or disprove: ffn g and fgn g are equivalent.

9 THE STABILITY OF BASES IN BANACH SPACES Two mathematical objects that are in some sense “close” to each other often share common properties. In this and the remaining section of Chapter I, we

32

BASES IN BANACH SPACES CH. 1

will show that bases in Banach spaces form a stable class in the sense that sequences sufficiently close to bases are themselves bases. Part of the problem is to formulate broad and effective notions of “close”. Let fxn g be a basis for a Banach space X and let fyn g be a sequence of elements in X. When shall we say that fyn g is “close” to fxn g? Although there are many different criteria, they all have one element in common: each implies that the mapping xn ! yn for n D 1, 2, 3, . . . can be extended to an isomorphism T from X onto X that is in some sense “close” to the identity operator I. In this way, questions about the stability of bases can be reduced to questions about “small” perturbations of the identity operator. As we shall see, the operator approach provides a powerful tool in the solution of stability problems. The fundamental stability criterion, and historically the first, is due to Paley and Wiener [1934]. It is based on the elementary fact that a bounded linear operator T on a Banach space is invertible whenever jjI  Tjj < 1. (This is one of those striking instances in which linear operators behave like ordinary numbers: if j1  tj < 1, then surely t1 exists.) Theorem 10 (Paley–Wiener). Let fxn g be a basis for a Banach space X, and suppose that fyn g is a sequence of elements of X such that  n   n          ci xi  yi     ci xi       iD1

iD1

for some constant , 0   < 1, and all choices of the scalars c1 , . . . , cn n D 1, 2, 3, . . .. Then fyn g is a basis for X equivalent to fxn g. 1 Proof. It follows by assumption nD1 cn xn  yn  is 1 that the series convergent whenever the series nD1 cn xn is convergent. Define a mapping T: X ! X by setting 1

1   T cn xn D cn xn  yn . nD1

nD1

Evidently T is linear and bounded and jjTjj   < 1. Thus the operator I  T is invertible. Since I  Txn D yn for every n, the result follows.  Corollary. Let fxn g be a basis for a Banach space X and let ffn g be the associated sequence of coefficient functionals. If fyn g is a sequence of vectors

SEC. 9

33

THE STABILITY OF BASES IN BANACH SPACES

in X for which 1 

jjxn  yn jj Ð jjfn jj < 1,

nD1

then fyn g is a basis for X equivalent to fxn g.   Proof. Put  D 1 ci xi is an arbitrary finite nD1 jjxn  yn jj Ð jjfn jj. If x D sum, then          ci xi  yi  D  fi xxi  yi   jjfi xxi  yi jj       jjxjj D   ci xi  .

Since 0   < 1, the result follows from Theorem 10.



The following theorem is now immediate. Theorem 11 (Krein–Milman–Rutman). If fxn g is a basis for a Banach space X, then there exist numbers εn > 0 with the following property: if fyn g is a sequence of vectors in X for which jjxn  yn jj < εn ,

n D 1, 2, 3, . . . ,

then fyn g is a basis for X equivalent to fxn g. Proof. Let ffn g be the sequence of coefficient functionals associated with the basis fxn g. By  the corollary to Theorem 10, it suffices to choose εn small enough so that 1 nD1 εn jjfn jj < 1.  In the following two corollaries the strength of the Krein–Milman–Rutman theorem is forcefully revealed. Corollary 1. If X is a Banach space with a basis, then every dense subset of X contains a basis. Corollary 2. polynomials.

The space C[a, b] has a basis consisting entirely of

It is important for applications that the corollary to Theorem 10 be strengthened. For this purpose, we shall make use of a well-known result concerning “compact perturbations” of the identity operator. If T is a compact operator defined on a Banach space X and if KerI  T D f0g, then I  T is invertible.

34

BASES IN BANACH SPACES CH. 1

This is the Fredholm alternative (see Halmos [1967, Problem 140]); it lends support to the notion that a compact operator is in some sense “small”. Definition. A sequence fxn g of elements of a Banach space X is said to be !-independent if the equality 1 

cn xn D 0

nD1

is possible only for cn D 0 n D 1, 2, 3, . . .. Theorem 12. Let fxn g be a basis for a Banach space X and let ffn g be the associated sequence of coefficient functionals. If fyn g is complete in X and if 1 

jjxn  yn jj Ð jjfn jj < 1,

nD1

then fyn g is a basis for X equivalent to fxn g. Proof. Let us first show that the sequence fyn g is ω-independent. If it were 1 , c2 , c3 , . . . (not all zero) such that 1 not, then we could find scalars c 1 nD1 cn yn D 0. Choose N so large that nDNC1 jjxn  yn jj Ð jjfn jj < 1. It follows by the corollary to Theorem 10 that the sequence of vectors fx1 , . . . , xN , yNC1 , yNC2 , . . .g forms a basis for X. Accordingly, at least  one of the scalars c1 , . . . , cN does not vanish; suppose ck 6D 0. The equation 1 nD1 cn yn D 0 can then be solved for yk , showing that X D [yn ] D [yn ]n6Dk . Since 1  k  N, it follows that the codimension† of [yn ]n>N is at most N  1. But this is absurd: the quotient space X/[yn ]n>N is isomorphic to [x1 , . . . , xN ], and so has dimension N. This proves that fyn g is ω-independent. Define an operator T on X by setting Tx D

1 

fn xxn  yn .

nD1

It is clear that T isa bounded linear operator. We claim that T is compact. Indeed, if we put Tn x D niD1 fi xxi  yi , then by hypothesis jjT  Tn jj ! 0, so that T is the limit of operators of finite rank, and hence compact. Wecomplete the proof by showing that KerI  T D f0g. If I  Tx D 0, then 1 nD1 fn xyn D 0 and since fyn g is ω-independent, it follows that x D 0. † If Y is a closed subspace of a Banach space X, then the codimension of Y is defined to be the dimension of X/Y.

SEC. 10

THE STABILITY OF ORTHONORMAL BASES IN HILBERT SPACE

35

Thus the kernel of I  T is trivial. The proof is over: the Fredholm alternative shows that I  T is invertible and clearly I  Txn D yn for every n.  PROBLEMS

1. Show that if  D 1, then Theorem 10 is no longer valid. of vectors belonging to a Banach 2. Suppose that fxn g is a complete sequence    space X and that  ci xi  yi     ci xi  for some constant , 0   < 1, and arbitrary scalars c1 , . . . , cn n D 1, 2, 3, . . .. Show that fyn g is also complete. 3. (Retherford–Holub) Let fxn g be a basis for a Hilbert space H with jj D 1pfor every n, and let fyn g be biorthogonal to fxn g. Show that jjxn 1 jjyn jj  1 < 1, then fxn g is a Riesz basis. (Hint: Use Theoif nD1 rem 12 to show that fyn g is equivalent to fxn g, and then use Problem 5 of Section 8.)

10 THE STABILITY OF ORTHONORMAL BASES IN HILBERT SPACE Throughout this section H will denote a separable Hilbert space and fen g a fixed but arbitrary orthonormal basis for H. While it is of course true that every result of the preceding section applies equally well to orthonormal bases, the added structure of a Hilbert space provides additional and stronger stability criteria. We begin by reformulating Theorem 10. Theorem 13. Let fen g be an orthonormal basis for a Hilbert space H and let ffn g be “close” to fen g in the sense that      ci ei  fi    jci j2  for some constant , 0   < 1, and arbitrary scalars c1 , . . . , cn n D 1, 2, 3, . . .. Then ffn g is a Riesz basis for H. APPLICATION: THE STABILITY OF THE TRIGONOMETRIC SYSTEM

It follows from what has already been done that the trigonometric system 2 feint g1 1 is stable in L [, ] under “sufficiently small” perturbations of the integers. This means that if fn g is a sequence of real or complex numbers for which fn  ng is in some sense “small”, then the system fein t g1 1 will form a basis for L 2 [, ], in fact, a Riesz basis. Accordingly, every function f in

36

BASES IN BANACH SPACES CH. 1

L 2 [, ] will have a unique nonharmonic Fourier series expansion ft D

1 

cn ein t

in the mean,

1

 jcn j2 < 1. with The possibility of such nonharmonic expansions was discovered by Paley and Wiener [1934], and it was for this purpose that they formulated the criterion of Theorem 13. In the present setting that criterion takes the form     cn eint  ein t    < 1   jcn j2  1. whenever When shall the sequence fn  ng be considered “small”? Based on what has already been established, one might well suppose that the condition

n  n ! 0

as n ! š1

is, at the very least, necessary. Surprisingly, it is not. Theorem 14 (Kadec’s 14 -Theorem). If fn g is a sequence of real numbers for which jn  nj  L < 14 , n D 0, š1, š2, . . . , then fein t g satisfies the Paley–Wiener criterion and so forms a Riesz basis for L 2 [, ]. Proof.

whenever

It is to be shown that     cn eint  ein t    < 1  

jcn j2  1. Write eint  ein t D eint 1  eiυn t ,

where υn D n  n n D 0, š1, š2, . . .. The trick is to expand the function 1  eiυt   t   in a Fourier series relative to the complete orthonormal system f1, cos nt, sinn  12 tg1 nD1 (see Problem 9, Section 3) and then exploit the fact that jn  nj is not too large. Simple calculations show that when υ is real,    1 sin υ 1k 2υ sin υ iυt 1e D 1 cos kt C υ k 2  υ2  kD1 Ci

  1 sin k  t. 2 k  12 2  υ2 

1  1k 2υ cos υ kD1

SEC. 10

THE STABILITY OF ORTHONORMAL BASES IN HILBERT SPACE

37

 Let fcn g be an arbitrary finite sequence of scalars such that jcn j2  1. By interchanging the order of summation and then using the triangle inequality we see that     cn eint  ein t   A C B C C, 

where

and

      sin υn  int  A D  1 cn e  ,   υn n    1     1k 2υn sin υn  int  c BD e cos kt  , n   k 2  υ2n  n kD1     1    1  1k 2υn cos υn  int  CD t e c sin k   . n 1 2 2   2 n k  2   υn  kD1

Obvious estimates show that   1  sin L 2L sin L A 1 , , B L k 2  L 2  kD1 1

C

1  kD1

2L cos L . k  12 2  L 2 

1

But the series kD1 2L/k 2  L 2  and kD1 2L/k  12 2  L 2  are the partial fraction expansions of the functions 1/L  cot L and tan L, respectively (see Markushevich [1965, pp. 62, 64]), so that     cn eint  ein t    D 1  cos L C sin L.  The proof is over: L
0, n D 0, n D 0, ⎩ n C 14 , n < 0. Writing it down is easy; proving that it works is another matter (and hard). We defer the proof until Section 3.3. 2. The proof of Theorem 14 applies only when every n is real. Even earlier, however, it had been shown by Duffin and Eachus [1942] that the

38

BASES IN BANACH SPACES CH. 1

Paley–Wiener criterion is satisfied whenever the n are complex and log 2 , 

jn  nj  L
rk. The greatest lower bound of all positive numbers k for which this is true is called the order of the function and is denoted by . Thus, is the smallest nonnegative number such that Mr  er

Cε

for every positive value of ε, as soon as r is sufficiently large. It follows easily from the definition that the order of a nonconstant entire function is given by

PT. 1 SEC. 3

FUNCTIONS OF FINITE ORDER

55

the formula D lim sup r!1

log log Mr . log r

The order of a constant function is of course 0. Simple examples of functions of finite order include ez , sin z, and cos z, all p of which are of order 1, and cos z, which is of order 12 . An entire function of exponential type is of finite order at most 1; every polynomial is of order z 0; the function ee is of infinite order. Theorem 3 is easily modified for functions of finite order (the details are left to the reader). Theorem 5.

If fz is an entire function of finite order , then nr D Or Cε 

for every positive number ε. As a corollary, we have the following important result. Theorem 6. If fz is an entire function of finite order and if z1 , z2 , z3 , . . . are its zeros, other than z D 0, then the series 1  1 jz j˛ nD1 n

is convergent whenever ˛ > . Proof. We may suppose without loss of generality that the zeros of fz have been numbered so that 0 < jz1 j  jz2 j  Ð Ð Ð . Given ˛ > , choose ˇ so that < ˇ < ˛. Since fz is of order , it follows from Theorem 5 that nr  Ar ˇ for some constant A and all values of r. Take r D jzn j; then nr  n, and hence n  Ajzn jˇ for n D 1, 2, 3, . . . .  ˛/ˇ The result follows at once since 1 < 1.  nD1 1/n It is an important consequence of Theorem 6 that every entire function of finite order has a canonical factorization.

56

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

Corollary. If fz is an entire function of finite order , with zeros z1 , z2 , z3 , . . . , other than z D 0, then   1  z fz D zm egz E ,p , zn nD1 where the product is a canonical product of genus p and p  . 1Proof. The remarks following Theorem 1 show that the product nD1 Ez/zn , p converges uniformly on each bounded region of the plane whenever 1  1 < 1, pC1 jz j nD1 n

and this is certainly true as long as p C 1 > .



Much more is true. We shall prove subsequently (Theorem 9) that gz is a polynomial of degree no larger than . Definition. Let z1 , z2 , z3 , . . . be a sequence of complex numbers, none of which is zero. The greatest lower bound of positive numbers ˛ for which the series 1  1 jz j˛ nD1 n is convergent is called the exponent of convergence of the sequence fzn g and is denoted by . If the series is divergent for every ˛ > 0, then we set  D 1. For a “finite” sequence,  D 0 by definition. Example. The exponents of convergence of the sequences fen g,

fn1/ g,

and flog ng

are 0, , and 1, respectively. If is the exponent of convergence of the sequence fzn g, then the series 1/jzn j˛ converges if ˛ >  and diverges if ˛ < ; if ˛ D , 1/ then no conclusion can be drawn. For example, the sequences fn  g and 2 1/ 1/jzn j fn log n g have the same exponent of convergence , but diverges for the first sequence and converges for the second. Suppose now that fz is entire of finite order . Let  be the exponent of convergence of its zeros and let p be the genus of the corresponding canonical product. Our results thus far may be summarized by the following inequalities: by Theorem 6,   Hadamard;

PT. 1 SEC. 3

57

FUNCTIONS OF FINITE ORDER

by the definitions alone, p    p C 1. If  is not an integer, then p D [] (here [x] denotes the function “bracket x”, the largest integer not exceeding  x); if  is an integer, then there is an ambiguity: p D  when the series 1/jzn j diverges, while p D   1 when it converges. PROBLEMS

1. Prove that, for a given function, Mr is a continuous function of r. z 2. Determine p p Mr for each of the entire functions e , sin z, cos z, and sin z/ z. 3. Let fz be an entire function with f0 D 1. Prove that nr  log Mer, where nr is the number of zeros of fz in the closed disk jzj  r. 4. Let fz be an entire function such that Re fz < M for some constant M and all values of z. Show that fz is p a constant. p 5. Determine the order of each of the entire functions e2z , sin z/ z, and  1 zt2 0 e d t. 6. Show that the order of a product of two entire functions does not exceed the larger of the orders of the factors. 7. Prove that fz and f0 z are of the same order. z 8. Determine the exponent of convergence of the zeros of ee  1. 9. Prove that if fz has at least one zero, but is not identically zero, then  D lim sup r!1

10. If fz D

1 nD0

log nr . log r

an zn is an entire function of order , then D lim sup n!1

n log n log1/jan j

1

(the quotient is taken to be zero if an D 0). (Hint: Suppose first that < 1 and let the right-hand side of (1) be denoted by . (a)  . Using Cauchy’s estimate for the Taylor coefficients of fz, show that for each fixed number k greater than , one has jan j  r n er

k

n D 0, 1, 2, . . .

58

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

as soon as r is sufficiently large, i.e., r > rk. Show that the quantity on the right achieves its minimum value when r D n/k1/k , so that  n/k ek jan j  n large. n Conclude that k  and hence that  . (b)  . If k > , then 0

n log n k log1/jan j

for all sufficiently large values of n, say n > N, so that  n/k 1 jan j  for n > N. n Then 1  n/k  1 Mr  jan jr  jan jr C rn. n nD0 nD0 nDNC1 1 

Write

n

N 

n

1  n/k  1 r n D S1 C S2 , n nDNC1

where S1 contains those terms with n < 2rk and S2 the remaining terms. Show that for all values of r, S2  1, while for all positive values of ε,   1 n/k kCε 2rk S1 < r D Oer . n Conclude that  k C ε, and hence that  .) 11. Show that if the right-hand side of (1) in Problem 10 is finite, then fz is an entire function of order . 12. Determine the order of each of the following entire functions: 1 zn (a) ˛ > 0, nD0 n!˛ 1  ˛ n/˛ n (b) z ˛ > 0, nD1 n2 1 n zn , (c) nD0 e  1n zkC2n (d) Jk z D 1 kth Bessel function, nD0 kC2n 2 n!n C k! n 1 z ˛ > 0 Mittag–Leffler function. (e) nD0 01 C ˛n 13. Show that if ε D 0, then the conclusion of Theorem 5 no longer holds.

PT. 1 SEC. 4

59

ESTIMATES FOR CANONICAL PRODUCTS

4 ESTIMATES FOR CANONICAL PRODUCTS Hadamard’s inequality   is valid for all entire functions. For canonical products,  and are equal. Theorem 6 (Borel). The order of a canonical product is equal to the exponent of convergence of its zeros. Proof.

Let Pz D

1 



E

nD1

z ,p zn



be a canonical product of genus p and order formed with the zeros z1 , z2 , z3 , . . ., and let  be the exponent of convergence of these zeros. Since   , it is sufficient to show that  . In order to estimate the growth of Pz, we shall estimate the size of each of its factors. Fix z and write          z  log jPzj D C log E , p  D C , 1 2 zn r >2r r 2r n

n

where r D jzj and rn D jzn j. For the second sum

 2

we use the inequality

log jEu, pj  2jujpC1 , valid whenever juj   2

1 2

(by virtue of (3) of Section 1). We then have

  r pC1  2 D 2r pC1 rnp1 . r n r >2r r >2r n

If  D p C 1, then

n

 2

D Or pC1  D Or  .

If  < p C 1, then  C ε < p C 1 whenever ε is sufficiently small, and hence     2r pC1 rnp1 D 2r pC1 rnCεp1 rnε 2

rn >2r

< 2r

pC1

Cεp1

2r



rn >2r

rnε

D Or Cε .

rn >2r

For the sum



1,

suppose first that p > 0. Clearly,

log jEu, pj  log j1  uj C juj C

juj2 jujp CÐÐÐ C . 2 p

1

60

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

If juj  12 , then

jujk  2pk jujp

k D 1, . . . , p,

and hence log jEu, pj  log j1  uj C 2p jujp  2pC1 jujp , whenever p > 0 and juj  12 . Therefore, for each ε > 0, ⎛ ⎞ ⎛ ⎞     r p ⎠ D O ⎝r p D O⎝ rnCεp rnε ⎠ 1 r n r 2r r 2r ⎛

n

D O ⎝r p 2rCεp



⎞

n

rnε ⎠ D Or Cε .

2

rn 2r

If p D 0, then log jEu, 0j D Ojujε  for every positive number ε, and (2) remains valid when p is replaced by ε. The proof is over: combining (1) and (2), we see that log jPzj D Or Cε  for every ε > 0, so that  .



The simplest canonical product is of genus zero:  1  1   z 1 < 1. Pz D 1 with zn jz j nD1 nD1 n By Borel’s theorem, Pz is of order at most 1. However, a lot more can be said: Pz is actually an entire function of exponential type. We first state the following definition. Definition. Let fz be an entire function of exponential type. The exponential type † of fz is defined to be the number k D lim sup r!1

log Mr . r

The zero function has exponential type 0, by convention. † The notion of “exponential type” is not to be confused with that of “type”. An entire function of positive order is said to be of type  (relative to that order) if log Mr . r p p Thus, for example, the entire function sin z/ z is of order D it is clearly of exponential type 0.  D lim sup r!1

1 2

and type  D 1, whereas

PT. 1 SEC. 4

ESTIMATES FOR CANONICAL PRODUCTS

61

According to the definition, k is the smallest nonnegative number such that Mr  ekCεr for any given ε > 0 as soon as r is sufficiently large. The functions ez , sin z, cos z, and sin z/z are all of exponential type 1. Any entire function of order < 1 is of exponential type 0. Theorem 7. A canonical product of genus zero is an entire function of exponential type zero. Proof.

Let Pz be the canonical product  1  1   z 1 < 1. 1 with Pz D z jz j n nD1 nD1 n

We may suppose that the zn are numbered in nondecreasing order of magnitude. In this case  the sequence f1/jzn jg is nonincreasing, and the convergence of the series 1/jzn j implies that n ! 0 as n ! 1. jzn j It follows readily from this that if nr denotes the number of points z1 , z2 , z3 , . . . for which jzn j  r, then nr ! 0 as r ! 1 r

3

(see Problem 1). Writing 1/jzn j as a Stieltjes integral and then integrating by parts, we obtain  1  1 1  1 d nt nt d t. D D jz j t t2 0 0 nD1 n Hence the integral



1 0

nt dt t2

is convergent. We can now estimate the growth of Pz as follows: if r D jzj, then   1  r log jPzj  log 1 C jzn j nD1  1  r D log 1 C d nt t 0

4

62

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2



D

1

0



< 0

r

r nt d t integrate by parts tt C r  1 nt nt dt C r d t. t t2 r

From this estimate, together with (3) and the convergence of the integral (4), we readily obtain the asymptotic inequality log jPzj < εr for each fixed ε > 0, as soon as r is sufficiently large. This shows that Pz is of exponential type zero.  The next theorem shows that the maximum modulus of 1/Pz is very often of the same order of magnitude as the maximum modulus of Pz. Theorem 8. If Pz is a canonical product of order , then for each ε > 0, jPzj > er

Cε

5

on circles jzj D r of arbitrarily large radius. Proof. Let Pz be a canonical product of genus p and order , formed with the zeros z1 , z2 , z3 , . . .. Fix h, h > , and let Dn denote the disk jz  zn j  jzn jh ,

n D 1, 2, 3, . . . .

It is to be shown that (5) holds for each ε > 0 whenever z lies outside every disk Dn and r D jzj is sufficiently large, i.e., r > rε, h. Since the sum of the radii of these excluded disks is finite, the domain C  [Dn is non-empty and certainly contains circles jzj D r of arbitrarily large radius. Put rn D jzn j. The same method used to prove Theorem 6 shows that for every ε > 0,



     r p   r pC1  z   log jPzj  log 1    O O zn rn rn rn >2r rn 2r rn 2r     z D log 1    Or Cε  6 zn r 2r n

(remember that D ). If z lies outside every excluded disk and if rn  2r, then     1  z   r h1  2rh1 . n  z  n

PT. 1 SEC. 5

63

HADAMARD’S FACTORIZATION THEOREM

Therefore, for each ε > 0 and all sufficiently large values of r,     z   log 1    h C 1 Ð log2r Ð n2r zn r 2r n

 h C 1 Ð log2r Ð r Cε

by Theorem 5

> r C2ε .

7

Combining (6) and (7), we have the desired result.



PROBLEMS

1. Prove: if fn g is a nondecreasing sequence of positive numbers and if nr denotes the number of n not exceeding r, then the statements limn!1 n/n D L and limr!1 nr/r D L are equivalent. 2. Determine the order of each of the canonical products 1  

1C

nD1

3. Prove that fz D and only if

z en

and

1  

1C

nD1

1 nD0

z  , n˛

with

˛ > 1.

an zn is an entire function of exponential type k if

k D lim sup

 n

n!jan j < 1.

n!1

4. Prove: fz and f0 z are of the same exponential type. 5. Give an example of an entire function of order 1 that is not of exponential type. of positive real numbers. Show each 6. Let r1 , r2 , r3 , . . . be a sequence  1 that for˛C1 ˛ positive number ˛, the series 1 dt nD1 1/rn and the integral 0 nt/t converge or diverge together.

5 HADAMARD’S FACTORIZATION THEOREM Equipped with the results of the preceding section, we can now establish the fundamental factorization theorem for entire functions of finite order. It is due to Hadamard who used the result in his celebrated proof of the Prime Number Theorem. It is one of the classical theorems in function theory. Theorem 9 (Hadamard). If fz is an entire function of finite order and if fz D zm egz Pz is its canonical factorization, then gz is a polynomial of degree no larger than .

64

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

Proof. Let  denote the exponent of convergence of the zeros of the canonical product Pz. Then Pz is of order  and   . Let ε be an arbitrary positive number. By assumption, log jfzj < r Cε as soon as jzj D r is sufficiently large, while, by Theorem 8, log jPzj > r Cε > r Cε on circles jzj D r of arbitrarily large radius. Combining these two inequalities, we see that    fz    < 2r Cε Re gz D log  m z Pz  on a sequence of circles jzj D rn with rn ! 1. Since gz is entire, the Borel–Carath´eodory inequality (see Titchmarsh [1939, p. 174]) shows that jgzj D Or Cε  on the same sequence of circles jzj D rn . Conclusion: gz is a polynomial of degree at most C ε (see the Remark following Theorem 4). Since ε was arbitrary, the result follows.  Remarks. 1. D max, deg g. Indeed, we now know that both  and the degree of gz do not exceed , so that max, deg g  . But the order of a product of two entire functions cannot exceed the larger of the orders of the factors (Problem 6, Section 3). Since zm Pz is of order  and egz is of order equal to the degree of gz, we have  max, deg g, and the result follows. 2. If is not an integer, then  D (for in this case the degree of gz is strictly smaller than ), and hence the form of the canonical product is uniquely determined: the genus p is equal to [ ]. If, on the other hand, is an integer, then there is an ambiguity: p may be equal to or  1. For example, the canonical products  1  1    z  z/n z 1 e and 1 n nlog n2 nD1 nD2 are both of order 1, but of different genus. The following impressive corollary is a direct consequence of Hadamard’s theorem. Corollary. An entire function of nonintegral order assumes every finite value infinitely many times.

PT. 1 SEC. 5

HADAMARD’S FACTORIZATION THEOREM

65

Proof. Since fz and fz  c have the same order for each constant c, it is enough to show that an entire function of nonintegral order has infinitely many zeros. But this is immediate from Remark 2 above since, in this case, the exponent of convergence must be positive.  Example 1.

Consider the entire function p sin  z fz D p .  z

It is of order D 12 and has a simple zero at z D n2 n D 1, 2, 3, . . .. Therefore, by Hadamard’s theorem, fz D c

1  

1

nD1

z  . n2

Since f0 D 1, it follows that c D 1. Observe that when z is replaced by z2 , we obtain the familiar formula  1   sin z z2 D 1 2 . z n nD1 Example 2.

The gamma function 0z is defined by the formula 1   z  z/n 1 D zez 1C e , 0z n nD1

where  is Euler’s constant,   1 1 1  D lim 1 C C C Ð Ð Ð C  log n . 2 3 n n!1 It is clear that 0z is meromorphic in the entire plane and has simple poles at z D 0, 1, 2, . . .. The constant  appearing in the exponent is chosen so that 01 D 1. To see this, simply observe that   1  N    1 1/n 1 1/n 1C e D lim 1C e n n N!1 nD1 nD1 N D lim N C 1e nD1 1/n D e . N!1

Let us put Pz D

1   nD1

1C

z  z/n e . n

66

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

Then Pz  1 is an entire function of order 1 and has (simple) zeros at z D 0, 1, 2, . . .. By Hadamard’s theorem, we can write Pz  1 D zeAzCB Pz. The values of A and B are easily determined. Taking the logarithmic derivative of both sides, we obtain 1   nD1

1 1  z1Cn n



D

 1   1 1 1 , CAC  z zCn n nD1

which reduces to A D 0. Setting z D 1, we find 1 D P0 D eB P1 D eB e , so that eB D e , and hence Pz  1 D ze Pz. It follows readily from this that 0z satisfies the difference equation 0z C 1 D z0z. Since 01 D 1, repeated application of the formula above shows that 0n D n  1! for every positive integer n. Thus, 0z is an extension of the factorial function to nonintegral values of the argument. From the relation sin z PzPz D , z we derive the important identity 0z01  z D

 . sin z

PROBLEMS

1. UsepHadamard’s theorem to determine the canonical factorization of cos z. 2. Show that  z z z z 1 ÐÐÐ. cos  sin D 1  z 1 C 4 4 3 5

PT. 1 SEC. 5

67

HADAMARD’S FACTORIZATION THEOREM

3. Let fz be an entire function of finite order , with zeros z1 , z2 , z3 , . . . other than z D 0. Let p be a nonnegative integer such that 1 

1/jzn jpC1 < 1,

nD1

and write fz D zm egz

1 



E

nD1

 z ,p . zn

(Note that the product is not necessarily a canonical product.) Does it still follow that gz is a polynomial? 4. Prove: if fz is an entire function of finite order , but not identically zero, then for each ε > 0, jfzj > er

Cε

on circles jzjDr ofparbitrarily large radius. 5. Show that 0 12 D . 6. Show that   1 1 Ð 3 Ð 5 Ð Ð Ð 2n  1 p 0 nC D  2 2n 7. Show that

n D 1, 2, 3, . . ..

 1    1 1 e1/2n D e /. 2n nD1

(Hint: Put fz D 00 z/0z. Show that   f0 z C f0 z C 12 D 2f0 2z, and then integrate.) 8. Prove that d dz



00 z 0z



D

1  nD0

1 . z C n2

9. Establish Legendre’s duplication formula,   p 02z D 22z1 0z0 z C 12 . 10. Show that 1/0z is not of exponential type. (Hint: Use Stirling’s formula to show that log Mr ¾ r log r as r ! 1.) 11. Prove Laguerre’s theorem: if fz is a nonconstant entire function, real for real z, of order less than 2, and with real zeros only, then (1) the zeros

68

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

of f0 z are also real and (2) f0 z vanishes once and only once between successive zeros of fz. (Hint: For the first assertion, show that  0  f z Im D 0 only when Im z D 0. fz For the second, show that   d f0 z < 0 when d z fz

Im z D 0.

PART TWO. RESTRICTIONS ALONG A LINE For the remainder of this chapter we shall be concerned mainly with the class of entire functions of exponential type. Our primary aim is to characterize those functions of this class that belong to L 2 along the real axis. These functions play an important role in both the theory and applications of entire functions. If an entire function of exponential type is known to satisfy additional growth conditions along a line, then more can be asserted about its growth in general and hence about the distribution of its zeros. We shall show, for example, that boundedness on one line carries with it boundedness on every parallel line; a similar assertion is true for functions belonging to L p on a given line. These results are in marked contrast with the behavior of entire functions in general, about which such assertions are not universally true. We begin by establishing a far-reaching generalization of the maximum modulus principle.

´ ¨ 1 THE ‘‘PHRAGMEN–LINDEL OF’’ METHOD Theorems that come under this collective heading establish the boundedness of an analytic function inside an infinite region, given that the function is known to be bounded on the boundary and not of too rapid growth inside. The fundamental result is that in which the unbounded region is an infinite sector. Theorem 10 (Phragm´en–Lindel¨of). Let fz be continuous on a closed sector of opening /˛ and analytic in the open sector. Suppose that on the bounding rays of the sector, jfzj  M,

PT. 2 SEC. 1

69

´ –LINDELOF ¨ ’’ METHOD THE ‘‘PHRAGMEN

and that for some ˇ < ˛, jfzj  er

ˇ

whenever z lies inside the sector and jzj D r is sufficiently large. Then jfzj  M throughout the sector. Proof. We can suppose without loss of generality that the given sector is symmetric with respect to the positive real axis and has its vertex at the origin. We introduce the auxiliary function gz D eεz fz, 

where ˇ <  < ˛ and ε > 0. Here z denotes that single-valued analytic branch of the multiple-valued function z D exp log z that takes positive values for positive real z. Setting z D rei , we have jgzj D eεr



cos 

jfzj.

On the bounding rays of the sector, cos  is positive, and hence jgzj  jfzj  M. On the arc jj  /2˛ of the circle jzj D r, jgzj  eεr



cos /2˛

jfzj < er

ˇ εr 

cos /2˛

as soon as r is sufficiently large. But this last expression approaches zero as r ! 1, so that, if r is sufficiently large, jgzj  M on this arc also. By the maximum modulus principle, jgzj  M throughout that part of the sector for which jzj  r and hence throughout the entire sector since r can be made arbitrarily large. Therefore, we conclude that 

jfzj  Meεjzj

everywhere within the sector. Since ε was arbitrary, the result follows.



The theorem is sharp in the sense that the conclusion is no longer valid when ˇ D ˛. Indeed, we need only consider the function fz D ez

˛

for j arg zj  /2˛;

fz is bounded on the rays of the sector but certainly not in the interior. Corollary. An entire function of order less than one that is bounded on a line must reduce to a constant.

70

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

We can now prove that an entire function of exponential type that is bounded on a line must be bounded on every parallel line. There is no loss of generality in supposing that the given line is the real axis. Theorem 11. Let fz be an entire function such that jfzj  AeBjzj for all values of z and jfxj  M for all real values of x. Then jfx C iyj  MeBjyj . Proof. and put

Suppose first that y > 0. Let ε be an arbitrary positive number gz D eiBCεz fz.

Then jgxj D jfxj  M for all real x, while giy ! 0

as

y ! 1.

Let N denote the maximum value of jgzj on the nonnegative imaginary axis. The Phragm´en–Lindel¨of theorem, applied separately to the first and second quadrants, shows that jgzj  maxN, M throughout the upper half-plane. A simple application of the maximum modulus principle then shows that N  M, and hence jgzj  M

whenever

Im z > 0.

Therefore, jfzj  eBCεy jgzj  MeBCεy throughout the upper half-plane, and the result follows by letting ε ! 0. The case in which y < 0 can be reduced to the first case by considering fz.  We note that the theorem is sharp in the sense that neither B nor M can be replaced by any smaller constant in the conclusion — simply consider fz D sin z. We note also that the conclusion holds under the apparently

PT. 2 SEC. 1

´ –LINDELOF ¨ ’’ METHOD THE ‘‘PHRAGMEN

71

weaker assumption that fz is an entire function of exponential type at most B. For in this case we have jfzj  AεeBCεjzj for each positive number ε and all values of z, and consequently jfx C iyj  MeBCεjyj . Now let ε ! 0. Theorem 11 shows that an entire function fz of exponential type that is bounded on the real axis is in fact “uniformly bounded” in every horizontal strip. If, in addition, we know that fx ! 0 as jxj ! 1, then more can be asserted. Theorem 12.

If fz is an entire function of exponential type and if fx ! 0 as jxj ! 1,

then fx C iy ! 0 as jxj ! 1 uniformly in every horizontal strip. Proof. Since fz is uniformly bounded in every horizontal strip, the result is an immediate consequence of Montel’s theorem (see Titchmarsh [1939, p. 170]).  The Fourier–Stieltjes integrals   

eizt d ωt,

where ωt is a function of bounded variation on the interval [, ], provide a large class of examples of entire functions of exponential type (no larger than ) that are bounded on the real axis. Special cases include the “almost periodic” exponential sums  cn ein z , n

with   n   and



jcn j < 1, as well as the integrals   eizt ft d t, 

with

 

jftj d t < 1.

72

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

The totality B of all entire functions fz of exponential type at most  that are bounded on the real axis, together with the norm jjfjj D

sup

1 0.)

PT. 2 SEC. 1

73

´ –LINDELOF ¨ ’’ METHOD THE ‘‘PHRAGMEN

7. Let fz be continuous in a closed sector and analytic and bounded in its interior. Suppose that fz ! a as z ! 1 along one of the bounding rays, while fz ! b as z ! 1 along the other. Prove that a D b. (Hint: Consider the function (fz  12 a C b2 .) 8. Let fz be analytic in an open connected set . Suppose that for each boundary point of  and each ε > 0 there exists a neighborhood such that at every point z of  in this neighborhood one has jfzj < M C ε. Prove that jfzj  M throughout  (in fact, unless fz is a constant, jfzj < M). 9. Let fz be analytic and bounded in the right half-plane Re z > 0. Prove that if z1 , z2 , z3 , . . . are the zeros of fz in this half-plane, then the series   1  1 Re zn nD1 is convergent. (Hint: Prove that if jfzj  M whenever Re z > 0, then the stronger inequality    z1  z zn  z  ÐÐÐ M jfzj   z1 C z zn C z  holds for Re z > 0.) 10. Prove that if fz is an entire function of exponential type, bounded on the real axis, and if z1 , z2 , z3 , . . . are its zeros other than z D 0, then  1     Im 1  < 1.  z  n nD1 (Hint: Use the result of Problem 9.) 11. Use Cauchy’s integral formula to show that if fz is an entire function of exponential type that is bounded on the real axis, then f0 z is also bounded on the real axis. 12. Prove Bernstein’s inequality for functions of the form   eizt d ωt, fz D 

where ωt is of bounded variation on the interval [, ]. (Hint: Use the relations   iteizt d ωt f0 z D 

and iteit/2 D

1 4  1n int e  nD1 2n C 12

for

jtj  ,

74

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

to show that   1 4  1n 1 f xCnC . f x D  nD1 2n C 12 2 0

 int 13. (Bernstein) If ft D N is a trigonometric polynomial of nDN cn e degree N and if M is the maximum value of jftj (t real), then

jf0 tj  MN for all real values of t.

2 CARLEMAN’S FORMULA Jensen’s formula provides a simple relation between the modulus of an analytic function on a circle and the distribution of its zeros inside the circle. If additional information is available on the growth of the function along a line, then Carleman’s formula is of fundamental importance. Theorem 13 (Carleman). Let fz be analytic for Im z  0 and let zk D rk eik k D 1, . . . , n be its zeros in the region  D fz: Im z  0 and 1  jzj  Rg. Then

n   1

   rk 1  2 sin k D log jfRei j sin  d  r R R k 0 kD1   R 1 1 1 C  2 log jfxfxj d x C AR, 2 1 x2 R

where AR is a bounded function of R. Proof. Suppose to begin with that fz has no zeros lying on ∂ (the boundary of ). We consider the contour integral    1 1 1 ID  log fz d z, 2i ∂ R2 z2 where ∂ is assumed to be positively oriented and the integration begins at z D 1 with a fixed determination of the logarithm. The method of proof is to evaluate I in two different ways and then equate the results. First        1 1 1 1 d z f0 z z ID C log fz d z  C d z. 2i ∂ d z R2 z 2i ∂ fz R2 z

PT. 2 SEC. 2

75

CARLEMAN’S FORMULA

As we make one complete passage over the contour ∂, log fz increases by 2in (recall that n is the number of zeros of fz inside ∂), so that the first integral on the right-hand side is equal to   1 n C 1 . R2 The second integral is easily evaluated by the residue theorem — this gives 

IDn

   n  1 zk 1 C1 C C . R2 R2 zk kD1

Next, we evaluate I by integrating separately over each component of ∂. On the positive real axis, we obtain   R 1 1 1  2 log fx d x, 2i 1 R2 x and on the negative real axis, 1 2i



1

R



1 1  2 2 R x



log fx d x;

on the large semicircle, z D Rei , and we obtain   i logfRei  sin  d ; R 0 and, finally, on the small semicircle, we obtain a bounded function of R. The result now follows by equating the imaginary parts of both values of I. The restriction that fz be free of zeros on the contour can be eliminated by a simple continuity argument. The details are left to the reader.  An entire function fz of exponential type  > 0 that is bounded on the real axis is in many ways similar to sin z. It is clear, for example, that fz must have an infinite number of zeros. Reason: By Hadamard’s factorization theorem,  1   z fz D zm eazCb 1 ez/zn ; z n nD1 if the product were finite, then fz could not remain bounded on the real axis without reducing to a constant. It is a simple consequence of Carleman’s formula that the zeros of fz must cluster about the real axis. Specifically, we have the following result (cf. Problem 10, Section 1).

76

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

Theorem 14. If fz is an entire function of exponential type, bounded on the real axis, and if zn D rn ein n D 1, 2, 3, . . . are the zeros of fz other than z D 0, then the series 1  sin n rn nD1 is absolutely convergent. Proof.

Since jfxj is bounded on the real axis, the integrals   R 1 1  log jfxfxj d x x 2 R2 1

have a finite upper bound for all values of R > 1. In addition, since fz is of exponential type, there is a constant K such that log jfRei j  KR, as soon as R is sufficiently large, so that   1 2K . log jfRei j sin  d   R 0  Applying Carleman’s formula to fz in the upper and lower half-planes and then adding the result, we conclude that   r 2 j sin n j 1  n2 0 and continuous for Im z  0. 3. Let fz be an entire function of exponential type and suppose that  R log jfxfxj dx < M x2 1 for some constant M and all values of R > 1. Prove that if z1 , z2 , z3 , . . . are the zeros of fz other than z D 0, then  1     Im 1  < 1.  z  n nD1 (Hint: Set



x

gx D 1

log jftftj dt t2

and integrate by parts.) 4. Let fz be analytic for Re z  0 and suppose that for some positive number a, fz D Oeajzj  as z ! 1,

PT. 2 SEC. 3

79

INTEGRABILITY ON A LINE

uniformly in the sector. Show that fz is identically zero. (Hint: Consider the function gz D fz sin bz, where 0 < b < a.) 5. Let  b fz D ezt t d t, a

where t is a real integrable function on [a, b]. Show that if z1 , z2 , z3 , . . . are the zeros of fz other than z D 0, then   1     Re 1  < 1.  z  n

nD1

(Hint: Show that eaz fz is bounded for Re z  0 and ebz fz is bounded for Re z  0.) 6. (a) Show that M¨untz’s theorem need not hold if a < 0. (b) Prove that the set f1, t1 , t2 , . . .g is complete in C[0, 1] whenever fn g  is an increasing sequence of positive numbers for which 1/n D 1. (Hint: First show that the set ft1 , t2 , t3 , . . .g is complete in L 2 [0, 1] by considering the function 

1

tz t d t, 0

where  2 L 2 [0, 1] and Re z  0. Complete the proof by observing that for 0  t  1 and n D 1, 2, 3, . . .,  t        n  i  n1 i 1  ci t  D n  x  di x  d x  t  0

 n 0

1

1/2  2  n1    di x i 1  d x . x

3 INTEGRABILITY ON A LINE In this section we shall prove that an entire function of exponential type that belongs to L p along the real axis must also belong to L p along every line parallel to the real axis. In fact, much more is true. Theorem 16 (Plancherel–P´olya). If fz is an entire function of exponential type  and if for some positive number p,  1 jfxjp d x < 1, 1

80 then

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2



1 1



jfx C iyjp d x  epjyj

1

jfxjp d x.

1

Plancherel and P´olya [1938] have given an “elementary” proof of Theorem 16 based on the Phragm´en–Lindel¨of method, and it is this proof that we reproduce here. The proof will require two preliminary lemmas. Let gz be continuous in the closed upper half-plane, Im z  0, analytic in the interior, and suppose that gz does not reduce to a constant. Let a and p be positive real numbers and put  a Gz D jgz C tjp d t. a

It is clear that Gz is defined and continuous for Im z  0. Since jgzjp is subharmonic for Im z > 0 (see Rudin [1966, p. 329]), so is Gz. Lemma 1. Let gz be of exponential type in the half-plane Im z  0 and suppose that the following quantities are both finite: MD

sup

1 0; we choose the one whose real part is negative in the quarter-plane x > a, y  0. Put  a jgε z C tjp d t, Gε z D a

which is then defined and continuous in the upper half-plane Im z  0, and subharmonic in the interior. A simple calculation involving (1) and (2) shows that in the quarter plane x > a, y  0, 3/2

jgε zj  AeBjzjεjzCaj ,

3

PT. 2 SEC. 3

81

INTEGRABILITY ON A LINE

where  D cos 3/8, and jgε zj  jgzj. Hence Gε z  Gz

for x  0, y  0,

and in particular Gε x  M

for x  0

and Gε iy  N for

y  0.

Let z0 be a fixed but arbitrary point in the first quadrant. We shall apply the maximum principle to Gε z in the region  D fz: Re z  0, Im z  0, jzj  Rg, choosing R large enough so that (i) z0 2 , and (ii) the maximum value of Gε z on  is not attained on the circular arc jzj D R (this is possible by virtue of (3)). Since Gε z does not reduce to a constant, the maximum value of Gε z on  must be attained on one of the coordinate axes, and hence, in particular, Gε z0   maxM, N. Now let ε ! 0. This establishes the result for the first quadrant; the proof for the second quadrant is the same.  Lemma 2.

In addition to the hypotheses of Lemma 1, suppose that lim gx C iy D 0

4

y!1

uniformly in x, for a  x  a. Then Gz  M whenever

Im z  0.

Proof. It is sufficient to show that N  M. By virtue of (4), we see that the function Giy approaches zero as y ! 1, and so must attain its least upper bound N for some finite value of y, say y D y0 . If y0 D 0, then N D Giy0  D G0  M. If, on the other hand, y0 > 0, then the maximum principle shows that the least upper bound of Gz in the half-plane Im z  0 cannot be attained at the interior point z D iy0 . Therefore, by Lemma 1, N D Giy0  < maxM, N, and so N < M.



Theorem 16 can now be established.

82

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

Proof of Theorem 16. It is sufficient to prove the theorem when y > 0 and fz is not identically zero. Let ε be a fixed positive number and consider the function gz D fzeiCεz . It is a simple matter to verify that for each positive number a, the functions gz and Gz fulfill all of the necessary hypotheses of Lemmas 1 and 2. Consequently, if y > 0, then by Lemma 2  1 Giy  M < jgxjp d x. 1

This together with the definitions of gz and Gz implies  a  1 pCεy p e jfx C iyj d x < jfxjp d x. a

1

The proof is over: first let a ! 1, then let ε ! 0.



An important corollary to Theorem 16 asserts that fx ! 0 as jxj ! 1 whenever fz is of exponential type and belongs to L p along the real axis. The next theorem will prove even more. Theorem 17. Let fz be an entire function of exponential type  and suppose that for some positive number p,  1 jfxjp d x < 1. 1

If fn g is an increasing sequence of real numbers such that nC1  n  ε > 0, then

 n



jfn jp  B

1

1

jfxjp d x,

where B is a constant that depends only on p, , and ε. Proof.

Since jfjp is subharmonic, we know that the inequality  2 1 jfz0 C rei jp d  jfz0 jp  2 0

holds for all values of r (see Rudin [1966, p. 329]), and hence that  1 p jfzjp d x d y jfz0 j  2 υ jzz0 jυ

5

PT. 2 SEC. 3

83

INTEGRABILITY ON A LINE

for every z0 and every υ > 0 (multiply both sides of (5) by r and integrate between 0 and υ). Therefore,   1  jfn jp  2 jfn C zjp d x d y υ jzjυ



1 υ2



υ

υ



υ

υ

jfn C x C iyjp d x d y

  1  υ n Cυ jfx C iyjp d x d y. D 2 υ υ n υ

Take υ D ε/2. Then the intervals n  υ, n C υ are pairwise disjoint, and the last expression above is no larger than 1 υ2



υ



υ

1 1

jfx C iyjp d x d y.

Applying Theorem 16, we conclude that   υ  1  1 p pjyj p jfn j  2 e jfxj d x d y υ υ 1  1 DB jfxjp d x, 1

where B D Bp, , ε.



Remark. A sequence fn g of real or complex numbers is said to be separated if for some positive number ε, jn  m j  ε whenever n 6D m. The theorem is easily modified to allow for complex n ’s, provided they are separated and sup j Im n j < 1. n

The details are left to the reader. Corollary.

If fz is an entire function of exponential type and if  1 jfxjp d x < 1 1

for some positive number p, then fx ! 0 as jxj ! 1.

84

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

PROBLEMS

1. Give an example of an entire function that belongs to L 1 along the real axis but does not belong to L 1 along any other line. 2. Prove that if fz is an entire function of exponential type zero and 1 p jfxj d x < 1 for some positive value of p, then fz is identically 1 zero. 3. Verify the Remark following Theorem 17. 4. Prove: if fz is an entire function of exponential type  and  1 jfxjp d x < 1 1

for some positive value of p, then 

jfx C iyj  Ce

1

jyj

1/p

jfxj d x p

1

,

where C is a constant depending only on p and . (Hint: For every υ > 0,  yCυ  1 1 jfx C iyjp  2 jfs C itjp d s d t. υ yυ 1 5. Show that the space Ep p  1,  > 0 consisting of all entire functions fz of exponential type at most  for which  1 1/p jjfjjp D jfxjp d x 0, then  υ υ 0 p jf z0 j  C jfz0 C x C iyjp d x d y, υ

υ

where C depends only on p and υ. To prove this, apply inequality (5) to the function fz0 C z  fz0  gz D z

PT. 2 SEC. 4

85

THE PALEY –WIENER THEOREM

inside a circle of radius r centered at z D 0 and obtain  2 2p jfz0 C rei jp d . jf0 z0 jp  p r 0 Now multiply both sides by r pC1 and integrate with respect to r between 0 and υ. (With Theorem 18 of the next section, the problem becomes trivial for p D 2; see the remarks preceding Theorem 19.))

4 THE PALEY–WIENER THEOREM It is a relatively simple matter to exhibit a large class of examples of entire functions fz of exponential type for which  1 jfxj2 d x < 1. 1

If  2 L 2 [A, A], then



fz D

A A

teizt d t

is such a function: a straightforward application of Morera’s theorem shows that fz is entire; the estimate  A  A yt Ajyj jfzj  jtje d t  e jtj d t A

A

shows that fz is of exponential type at most A; and Plancherel’s theorem 1 A shows that 1 jfxj2 d x D 2 A jtj2 d t < 1. It is a remarkable fact that there are no other examples — every entire function of exponential type that belongs to L 2 along the real axis is obtained in this way. This is the content of the celebrated “Paley–Wiener Theorem”. Theorem 18 (Paley–Wiener). Let fz be an entire function such that jfzj  CeAjzj

1

for positive constants A and C and all values of z, and  1 jfxj2 d x < 1. 1

Then there exists a function  in L 2 [A, A] such that  A teizt d t. fz D A

2

86

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

Proof (Boas). Let t be the Fourier transform of fx, i.e.,  1 1 t D fxeixt d x, 2 1 where the integral is to be interpreted as a limit in the mean in the L 2 sense. Then  2 L 2 1, 1 and, by the Fourier inversion formula,  1 teixt d t. fx D 1

Accordingly, we need only show that t vanishes almost everywhere outside the interval [A, A]. This will establish (2) for all real values of z and hence for all complex values as well since both sides of (2) are entire. Let T be a positive number. We shall consider the contour integral  I D fzeizt d z, 

where t is fixed and  consists of the upper three sides of the rectangle [T, T] ð [0, T]. Since the integrand is an entire function of z for each value of t, we have  ID

T

T

fxeixt d x

by Cauchy’s integral theorem. We are going to show that I ! 0 as T ! 1 whenever jtj > A. Since  T fxeixt d t ! t in the mean as T ! 1, T

it will follow that t D 0 almost everywhere outside [A, A]. Suppose first that t < A. Integrating over each of the three sides of , we readily obtain  T  T ty tT e jfT C iyj d y C e jfx C iTj d x jIj  0



C

T

T

ety jfT C iyj d y  I1 C I2 C I3 .

0

1 We shall estimate the size of each of these three integrals. Since 1 jfxj2 d x is finite, it follows that fx ! 0 as jxj ! 1 (by the corollary to Theorem 17), and hence, in particular, that jfxj has a finite upper bound M on the real axis. By Theorem 11,

jfx C iTj  MeAT

PT. 2 SEC. 4

THE PALEY –WIENER THEOREM

87

for all real values of x, and hence I2  2TMetCAT . Since t < A, the exponent above is negative, and we conclude that I2 ! 0 as T ! 1. Turning next to I1 (I3 is treated in the same way), we write  R  T  I1 D C ety jfT C iyj d y R > 0. 0

R

Theorem 12 shows that for each fixed R, fT C iy ! 0 as T ! 1, uniformly in y, 0  y  R. Consequently,  R ety jfT C iyj d y ! 0 as T ! 1. 0

T It remains only to show that R ety jfT C iyj d y can be made arbitrarily small by choosing R and T sufficiently large. Again appealing to Theorem 11, we have  T  T M etCAT  etCAR , ety jfT C iyj d y  M etCAy d y D tCA R R

and since t C A is negative, the last term approaches zero as R and T approach infinity. We have thus shown that I ! 0 as T ! 1 whenever t < A. The case t > A is treated similarly, with  in the lower half-plane. The details, which are obvious, are left to the reader.  It is important to note that condition (1) may be replaced by the seemingly weaker assumption that fz is entire of exponential type at most A. For, if this is the case, then jfzj  CεeACεjzj for every positive number ε, and the proof of Theorem 18 shows that the Fourier transform of fx must vanish almost everywhere outside the interval [A  ε, A C ε], and so almost everywhere outside [A, A]. The following useful corollary is now readily established. Corollary. If fz is an entire function of exponential type A, belonging to L 2 on the real axis, then jfzjeAjyj ! 0 as jzj ! 1.

88

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

The Paley–Wiener theorem has a wide range of important and varied applications, several of which are illustrated by the following examples. We begin with an integral representation for entire functions of exponential type that are bounded along the real axis. Example 1.

If fz 2 B , then 

fz D f0 C z





teizt d t,

where  2 L 2 [, ]. Proof.

It is clear that the function fz  f0 z

is entire of exponential type at most  and belongs to L 2 along the real axis. Now apply Theorem 18.  Example 2 (Bernstein’s inequality). If fz 2 B and jfxj  M for all real values of x, then jf0 xj  M. Proof.

The inequality is easily established for functions of the form   eizt d ωt, 3 

where ωt is of bounded variation on the interval [, ] (see Problem 12, Section 1). Suppose now that fz is an arbitrary function belonging to B . If we put gε z D fz

sin εz εz

ε > 0,

then gε z is entire of exponential type at most  C ε and belongs to L 2 along the real axis. By virtue of the Paley–Wiener theorem, gε z is of the form (3) with  C ε in place of . Since jgxj  M, it follows that jg0ε xj  M C ε for every real x. But g0ε x ! f0 x as ε ! 0, and the result follows.



As a final application, we shall establish an interesting quadrature formula for a certain class of entire functions of exponential type.

PT. 2 SEC. 4

89

THE PALEY –WIENER THEOREM

Example 3. If fz is an entire function of exponential type at most 2 1 and 1 jfxj d x < 1, then 1 



fn D

nD1

1

1

fx d x.

Proof. Since fx is integrable over 1, 1, so is f0 x (Problem 7, Section 3) and fx ! 0 as jxj ! 1. Under these conditions, the Poisson summation formula  1 is applicable: if F is the Fourier transform of f, i.e., if Ft D 1/2 1 fxeixt d x, then 1 

1 

fn D 2

nD1

F2n.

nD1

1 But 1 jfxj2 d x < 1, and the Paley–Wiener theorem implies that Ft D 0 for every t with jtj  2 (note that Ft is continuous). Conclusion: 

1 

fn D 2F0 D

nD1

1 1

fx d x.



PROBLEMS

A

1. Let fz D A teizt d t, where  2 L 2 [A, A], and suppose that  does not vanish almost everywhere in any neighborhood of A. Show that fz is of order 1 and exponential type A. 2. Establish the corollary to Theorem 18. 3. Prove the following “gap” theorem for entire functions: if fz is an entire function of exponential type, bounded on the real axis, and if fn  0 D 0 for a sequence of positive integers n for which  n

1 D1  even n

and

 1 D 1,   odd n n

then fz must reduce to a constant. (Hint: Put gz D fz  f0/z. A Then gz D A teizt d t, with  in L 2 [A, A]. Consider the functions  0

A



tz t C t d t

A

and 0

tz t  t d t.

90

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

5 THE PALEY–WIENER SPACE The totality of all entire functions of exponential type at most  that are square integrable on the real axis will henceforth be known as the Paley–Wiener space and will be designated by P. Clearly, P is a vector space under pointwise addition and scalar multiplication; it is also an inner product space with respect to the inner product  1 f, g D fxgx d x. 1

Since the Fourier transform is an isometry, the Paley–Wiener theorem shows that P is a separable Hilbert space, isometrically isomorphic to L 2 [, ]. The isomorphism between P and L 2 [, ] has far-reaching consequences. At present, we shall investigate some of the ways in which known properties of L 2 [, ] can be transformed easily into nontrivial assertions about P. This will then serve as the basis for a more penetrating analysis of nonharmonic Fourier series in L 2 [, ]. If f belongs to P and has the representation   1 fz D teizt d t, 1 2  with  in L 2 [, ], then Plancherel’s theorem shows that  1   1 jfxj2 d x D jtj2 d t D jjjj2 . jjfjj2 D 2  1 When Parseval’s identity is applied to , we obtain the important formula jjfjj2 D

1 

jfnj2 .

1

By taking the Fourier transform of eint n D 0, š1, š2, . . ., we see that the set of functions fsin z  n/z  ng1 1 forms an orthonormal basis for P. Accordingly, each function f in P has a unique expansion of the form fz D

1  nD1



cn

sin z  n , z  n

jcn j2 < 1. The convergence of the series is understood to be in the with metric of P. But convergence in P implies uniform convergence in each horizontal strip. This is an immediate consequence of the following useful estimate: jfx C iyj  ejyj jjfjj . 2

PT. 2 SEC. 5

91

THE PALEY –WIENER SPACE

(Proof: Simply use (1) together with the fact that jjjj D jjfjj.) If we now set z D n, it follows that cn D fn, and we have thereby obtained the cardinal series for f: 1  fn fz D sin z 1n . z  n nD1 Thus, a function in P can be recaptured from its values at the integers. The cardinal series was first introduced by Whittaker [1915]. It plays an important role in the mathematical theory of communication, where it is known as the sampling theorem (see Shannon [1949] and Shannon and Weaver [1964]). The Paley–Wiener representation yields a trivial proof that P is “closed” under differentiation (cf. Problem 7, Section 3). Indeed, if fz is given by (1), then   1 f0 z D itteizt d t 2  and tt 2 L 2 [, ]. Moreover, we have the following sharp estimate for the norm of f0 z: jjf0 jj  jjfjj. (Proof: jjf0 jj D jjttjj  jjjj D jjfjj.) Theorem 19. The Paley–Wiener space is a functional Hilbert space of entire functions. Its reproducing kernel K is given by Kz, w D and the integral representation  fz D

sin z  w , z  w

1

1

ft

sin t  z dt t  z

is valid for every function f belonging to the space. Proof. Inequality (2) shows that “point-evaluations” on P are bounded linear functionals. Thus, P is a functional Hilbert space. If fz 2 P, then the Paley–Wiener theorem shows that fw D t, eiwt  for some function  in L 2 [, ] and all values of w. But the Fourier transform is an isomorphism between L 2 [, ] and P, and hence   sin z  w . fw D fz, z  w

92

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE CH. 2

Since Kz, w is the unique function on C ð C for which fw D fz, Kz, w, it follows that Kz, w D sin z  w/z  w. Corollary.



If f 2 P and z D x C iy, then jfzj2  jjfjj2

sin 2y . 2y

PROBLEMS

1. Show that if f 2 P, then  jyj jjfjj  e

1

1/2

jfx C iyj dx 2

1

 ejyj jjfjj .

2. Let fz be an entire function of exponential type at most , and suppose that fx D Ox ˛ , 0 < ˛ < 12 . Show that fz D f0 C

 1n fn  f0 f0 0 sin z C z sin z .  nz  n n6D0

3. Obtain the cardinal series for functions of the form   1 fz D teizt d t, 2  with  integrable on [, ]. (Hint: Replace  by its Fourier series over [, ] and integrate term-by-term.) 4. Show that the function fz D

1  1n sin z  n nD1

n

z  n

belongs to the Paley–Wiener space but that zfz is not bounded on the real axis (cf. Example 1, Section 4). 5. Establish the corollary to Theorem 19.

3 THE COMPLETENESS OF SETS OF COMPLEX EXPONENTIALS

In this chapter we shall discuss a few of the fundamental completeness properties of sets of complex exponentials fein t g over a finite interval of the real axis. The most extensive results in this direction were obtained by Paley and Wiener [1934] and Levinson [1940]. At the same time, we will be laying the groundwork for a more penetrating investigation of nonharmonic Fourier series in L 2 . By making an appropriate change of variables, we may consider the system fein t g over an interval [A, A] symmetric with respect to the origin. The spaces of interest will be L p [A, A], for 1  p < 1, and C[A, A]. As we have seen, for the system fein t g to be incomplete in C[A, A], it is necessary and sufficient that there exist a nontrivial entire function fz, zero for every n , and expressible in the form  A eizt d ωt, 1 fz D A

where ωt is of bounded variation on [A, A]. When L p [A, A] is considered, fz takes the form  A fz D eizt t d t, 2 A

with  in L [A, A]. Here q denotes the conjugate exponent: 1/p C 1/q D 1. The remarks above establish a connection between completeness and uniqueness. If there are enough n ’s to guarantee that fz vanishes identically q

93

94

THE COMPLETENESS OF SETS OF COMPLEX EXPONENTIALS CH. 3

whenever fn  D 0 n D 1, 2, 3, . . ., i.e., if fn g is a set of uniqueness for fz, then the system of exponentials fein t g will be complete in the corresponding space. Therefore, every theorem on uniqueness generates a theorem on completeness, and conversely. In L 2 [, ], the situation is particularly simple. Here the system of exponentials fein t g is complete whenever fn g is a set of uniqueness for every function belonging to the Paley–Wiener space. Clearly, completeness in C[A, A] implies completeness in L p [A, A] for every value of p. It is often advantageous to permit repetitions among the n . Suppose, for example, that some n , call it , is repeated m times. In this case the understanding is that fz, be it of the form (1) or (2), shall have a zero of multiplicity m at z D . It follows that either t or the differential dωt will be orthogonal to each of the functions eit , teit , . . . , tm1 eit ,

3

and hence all of these functions will be available for the approximation. Conversely, the orthogonality of t or dωt to all functions of this form implies that  is a zero of multiplicity m of the corresponding entire function fz. Infinite repetitions among the n are excluded. For simplicity, we shall suppose in what follows that the n are distinct. The reader will have no difficulty in reformulating all completeness theorems to include functions of the form (3). We begin with a closer look at the trigonometric system.

1 THE TRIGONOMETRIC SYSTEM We have seen that the trigonometric system feint g1 1 is complete in every L space 1  p < 1 over an interval of length 2. In fact, if t is integrable on [, ] and   teint d t D 0 n D 0, š1, š2, . . ., p



then t must vanish almost everywhere. Once a collection of functions is known to be complete, the next concern should be for economy — are there any superfluous terms? Definition. A system of complex exponentials fein t g is said to be exact in L p [A, A] or in C[A, A] if it is complete, but fails to be complete on the removal of any one term. If the system becomes exact when N terms are

SEC. 1

95

THE TRIGONOMETRIC SYSTEM

removed, then we say it has excess N; if it becomes exact when N terms ei 1 t , . . . , ei N t are adjoined, then we say it has deficiency N. As we shall see (Theorem 7), the particular functions ei t added or removed are arbitrary; only their number is important. Proposition 1. The trigonometric system is exact in L p [, ] for 1  p < 1; in C[, ] it has deficiency 1. Proof. The first assertion is obvious — the trigonometric system is complete and its elements are mutually orthogonal. For the second assertion, argue as follows. It is clear that the trigonometric system is incomplete in C[, ] since the only continuous functions that can be uniformly approximated by trigonometric polynomials are those for which f D f. Let us adjoin the element ei t , where is not an integer. If f 2 C[, ], then for a suitable constant c the values of the function ft  cei t at t D  and t D  are the same. The new system is then complete, by virtue of the Weierstrass theorem on trigonometric approximation. Thus in C[, ] the trigonometric system has deficiency 1.  It is not difficult to show that if the system fein t g is complete in L 1 [A, A], then its deficiency in C[A, A] cannot exceed 1 (see Problem 2). Proposition 2. The trigonometric system is incomplete in L 1 over every interval of length greater than 2. Proof.

Indeed, given ε > 0, choose  1, t <  C ε, t D 0, jtj    ε, 1, t >   ε,

and put



fz D

Cε

ε

teizt d t.

A straightforward calculation shows that fz D

4i sin z sin εz, z

and hence fn D 0 for every integer n. Since ε was arbitrary, the result follows. 

96

THE COMPLETENESS OF SETS OF COMPLEX EXPONENTIALS CH. 3

Consider now the set of functions feint g1 nD1 obtained from the trigonometric system by removing roughly “half ” the terms. It seems plausible that the resulting set will be complete in L p over an interval of length  and over no larger interval. Surprisingly, this is not the case. The set feint g1 nD1 is complete in L p and in C over every interval of length less than 2. This is a consequence of the following theorem of Carleman [1922]. Theorem 1. Let fn g be a set of positive real numbers and suppose that for some positive number A,

A 1  1 lim sup > . 1 log R