walsh series and transforms: theory and applications

Walsh Series and Transforms

Mathematics and Its Applications (Soviet Series)

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

A. A. KIRILLOV, MGU, Moscow, U.S.S.R. Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R. N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R. S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R. M. C. POLYVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R. Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.

Volume 64

Walsh Series and Transforms Theory and Applications

by

B. Golubov Moscow Institute of Engineering, Moscow, l1.S.S.R.

A. EfImov Moscow Institute of Engineering, MOSCOlV, l1.S.S.R.

and

V. Skvortsov Moscow State University, Moscow, U.S.S.R .

.. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data

Golubov. B. I. (Borls Ivanovlch) [Rfădy 1 praobrazovlnl fă Uolshl. Engllshl Halsh ser les and transforms : theory and appllcatlons I by B.

Golubov. A. Eflmev. V. Skvortsov. p. c •. -- (Mathematlcs and lts appllcatlons. Sovlat sarlas

v. 64) Trans 1 at Ion of: R fădy 1 praobrazovan 1 fă Uo 1 sha. Inc 1 udes 1 ndax. ISBN 978-94-010-5452-2 ISBN 978-94-011-3288-6 (eBook) DOI 10.1007/978-94-011-3288-6 1. Halsh functlons. 2. Decempesltlon (Matha.atlcsl 1. Eflmov.

A. V. (Alaksandr Vasl1 'avlchl II. Skvorfiov. V. A. (Valantln Anatol 'evlch) III. T1tla. IV. SerlU, Mathantlcs and Its appllcatlons (Kluwer Academic Publlshers). SOvlBt sarlal : v. 64. CA404.5.06413 1991 515' . 243--dc20 90-26705

ISBN 978-94-010-5452-2

Prinred on acid-free pape.r

All Rights Reserved This English edition e 1991 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1991 Softcover reprint ofthe hardcover 1 st edition 1991

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

SERIES EDITOR'S PREFACE

'Et moi, ... , si j'avait su comment en revenir, je n'y se.rais point aile.'

Jules Verne

The series is divergent; therefore we may be able to do something with it.

O. Heaviside

One service mathematics has rendered the human race. It has put common sense back where it belongs, on !be topmost shelf next to the dusty canister labelled 'disc:arded nonsense'.

Eric T. Bell

Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences.

Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered computer science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote

"Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available.

If anything. the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on.

In addition, the applied scientist needs to cope increasingly with the nonlinear world and the

v

vi SERIES EDITOR'S PREFACE

extra mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines.

There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading -fortunately.

Thus the original scope of the series, which for various (sound) reasons now comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with:

- a central concept which plays an important role in several different mathematical and/or scientific specialization areas;

- new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have, and have had,

on the development of another.

Fourier series and the Fourier transform are of enormous importance in mathematics. They are based on the trigonometric orthogonal system of functions. However, this is but one orthonormal system and depending on the domain of interest other systems may be more useful, Le. better adapted to the phenomena being studied or modeled. Examples are the Haar system and various systems based on wavelets. Another most important example is the Walsh system which is based on rectangular waves rather than sinusoidal ones. These appear to be preferred in a number of cases, for instance in signal processing.

There is no doubt about the importance of Walsh-Fourier series and transforms in harmonic analysis, in signal processing, in probability theory, in image processing, etc. It is therefore slightly surprising that no systematic treatment of the topic appeared before. However, here is one by a group of authors which have contributed significantly to the field.

The shortest path between two truths in the

real domain passes through the complex domain.

J. Hadamard

La physique ne nous donne pas seulement

!'oceasion de resoudre des problemes ... elle

nous fail pressentir la solution.

H. Poincare

Bussum, December 1990

Never lend books, for no one ever returns

them; the only books I have in my library

arc books that other folk have lent me.

Anatole France

The function of an expert is not to be more

right than other people, but to be wrong for

more sophistica ted reasons.

David Butler

Michiel Hazewinkel

TABLE OF CONTENTS

Series Editor's Preface ..................................................................... v Preface ................................................................................... xi Foreword .............................................................................. . xiii

Chapter 1

WALSH FUNCTIONS AND THEIR GENERALIZATIONS

§1.1 The Walsh functions on the interval [0,1) ............................................... 1 §1.2 The Walsh system on the group ........................................................ 5 §1.3 Other definitions of the Walsh system. Its connection with the Haar system ........... 14 §1.4 Walsh series. The Dirichlet kernel ..................................................... IS §1.5 Multiplicative systems and their continual analogues .................................. 21

Chapter 2

WALSH-FOURIER SERIES BASIC PROPERTIES

§2.1 Elementary properties of Walsh-Fourier series. Formulae for partial sums ... · ........... 35 §2.2 The Lebesgue constants .............................................................. 40 §2.3 Moduli of continuity of functions and uniform convergence of Walsh-Fourier series ..... 43 §2.4 Other tests for uniform convergence ................................................... 47 §2.5 The localization principle. Tests for convergence of a Walsh-Fourier series at a point ... 50 §2.6 The Walsh system as a complete, closed system ....................................... 56 §2.7 Estimates of Walsh-Fourier coefficients. Absolute convergence of Walsh-Fourier series .. 59 §2.S Fourier series in multiplicative systems ................................................ 66

Chapter 3

GENERAL WALSH SERIES AND FOURIER-STIELTJES SERIES QUESTIONS ON UNIQUENESS OF REPRESENTATION OF

FUNCTIONS BY WALSH SERIES

§3.1 General Walsh series as a generalized Stieltjes series ................................... 71 §3.2 Uniqueness theorems for representation of functions by pointwise convergent

Walsh series .......................................................................... SO §3.3 A localization theorem for general Walsh series ........................................ 86 §3.4 Examples of null series in the Walsh system. The concept of U -sets and M -sets ........ S9

viii

Chapter 4

SUMMATION OF WALSH SERIES BY THE METHOD OF ARITHMETIC MEANS

TABLE OF CONTENTS

§4.1 Linear methods of summation. Regularity of the arithmetic means .................... 94 §4.2 The kernel for the method of arithmetic means for Walsh- Fourier series ............... 97 §4.3 Uniform (C, 1) summability of Walsh-Fourier series of continuous functions ............ 99 §4.4 (C,l) summability of Fourier-Stieltjes series .......................................... 103

Chapter 5

OPERATORS IN THE THEORY OF WALSH-FOURIER SERIES

§5.1 Some information from the theory of operators on spaces of measurable functions ..... 112 §5.2 The Hardy-Littlewood maximal operator corresponding to sequences of dyadic nets ... 117 §5.3 Partial sums of Walsh-Fourier series as operators ..................................... 119 §5.4 Convergence of Walsh-Fourier series in LP[O,l) ....................................... 124

Chapter 6

GENERALIZED MULTIPLICATIVE TRANSFORMS

§6.1 Existence and properties of generalized multiplicative transforms ..................... 127 §6.2 Representation of functions in Ll(O,oo) by their multiplicative transforms ............ 135 §6.3 Representation offunctions in LP(O, 00), 1 < p ~ 2, by their multiplicative transforms .147

Chapter 7

WALSH SERIES WITH MONOTONE DECREASING COEFFICIENTS

§7.1 Convergence and integrability ........................................................ 153 §7.2 Series with quasiconvex coefficients .................................................. 163 §7.3 Fourier series offunctions in LP ...................................................... 166

Chapter 8

LACUNARY SUBSYSTEMS OF THE WALSH SYSTEM

§8.1 The Rademacher system ............................................................. 173 §8.2 Other lacunary subsystems .......................................................... 177 §8.3 The Central Limit Theorem for lacunary Walsh series ................................ 185

TABLE OF CONTENTS

Chapter 9

DIVERGENT WALSH-FOURIER SERIES ALMOST EVERYWHERE CONVERGENCE OF WALSH-FOURIER SERIES OF L2 FUNCTIONS

ix

§9.1 Everywhere divergent Walsh-Fourier series ........................................... 194 §9.2 Almost everywhere convergence of Walsh-Fourier series of L2[O, I) functions .......... 198

Chapter 10

APPROXIMATIONS BY WALSH AND HAAR POLYNOMIALS

§10.1 Approximation in uniform norm .................................................... 213 §10.2 Approximation in the LP norm ..................................................... 219 §10.3 Connections between best approximations and integrability conditions ............... 230 §10A Connections between best approximations and integrability conditions

(continued) ........................................................................ 236 §10.5 Best approximations by means of multiplicative and step functions .................. 255

Chapter 11

APPLICATIONS OF MULTIPLICATIVE SERIES AND TRANSFORMS

TO DIGITAL INFORMATION PROCESSING

§11.1 Discrete multiplicative transforms .................................................. 260 § 11.2 Computation of the discrete multiplicative transform ................................ 270 §11.3 Applications of discrete multiplicative transforms to information compression ........ 281 §llA Peculiarities of processing two-dimensional numerical problems with discrete

multiplicative transforms ........................................................... 295 §1l.5 A description of classes of discrete transforms which allow fast algorithms ........... 298

Chapter 12

OTHER APPLICATIONS OF MULTIPLICATIVE FUNCTIONS AND TRANSFORMS

§12.1 Construction of digital filters based on multiplicative transforms .................... 310 §12.2 Multiplicative holographic transformations for image processing ..................... 313 §12.3 Solutions to certain optimization problems .......................................... 323

x TABLE OF CONTENTS

APPENDICES

Appendix 1 Abelian groups .............................................................. 341 Appendix 2 Metric spaces. Metric groups ................................................. 342 Appendix 3 Measure spaces .............................................................. 343 Appendix 4 Measurable functions. The Lebesgue integral ................................. 345 Appendix 5 Normed linear spaces. Hilbert spaces ......................................... 350 Commentary ............................................................................. 354 References ............................................................................... 359 Index .................................................................................... 365

PREFACE

The classical theory of Fourier series deals with the decomposition of functions into sinusoidal waves. Unlike these continuous waves, the Walsh functions are "rectangular waves". Such waves have been used frequently in the theory of signal transmission and it has turned out that in some cases these "waves" are preferred to the sinusoidal ones. In this book we give an introduction to the theory of decomposition of functions into Walsh series and into series with respect to the more general multiplicative systems. We also examine some applications of this theory.

The orthonormal system which is now called the Walsh system was introduced by the American mathematician J.L. Walsh in 1923. The development of the theory of Walsh series has been (and continues to be) strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results on Walsh series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of this can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of topological groups. This point of view leads in a natural way to a new domain of definition for the Walsh functions. As it is useful to consider the trigonometric functions "in complex form", i.e., defined on the circle group instead of the interval [0,21T), even so we shall see that it is convenient to define the Walsh system on a group which differs from the circle group in an essential way. This means that the Walsh system, and more generally each multiplicative system, provides an important model on which one can verify and illustrate many questions from abstract harmonic analysis. The Walsh system is also of great interest to anyone specializing in the theory of orthogonal systems since it is one of the simplest examples of a complete, bounded, orthonormal system. And, the so-called lacunary subsystems of the Walsh system play an essential role in probability theory.

In addition to progress made during the last 10-15 years in theoretical research on Walsh series, a number of works have been published which are concerned with applications of Walsh functions to scientific computing, coding theory, digital signal processing, e.t.c. Beginning in 1970 and repeating almost every year after that, there has been a conference in the United States of America which dealt with some aspect of applications of Walsh functions and has generated a collection of articles on the subject.

In 1969 and 1977, H. Harmuth published two monographs which were translated into Russian in 1977 and 1980 ([9), [10) 1). Although they contained much material on the applications of Walsh functions, the mathematical theory was almost completely missing. Thus the techniques implimented there were presented without an adequate theoretical foundation. Another monograph [27) was devoted to applied questions. But until now, no book has been published2 , either in the Soviet Union or abroad, which contains an account of the theoretical foundations of Walsh series and Walsh transforms accessible to a broad spectrum of specialists in applied mathematics.

One of the main goals of this book is to remove this flaw to some extent. This book is intended for a wide audience of engineers, technical specialists and graduate students preparing for a career in applied mathematics. In addition to these, this book may also be of interest to graduate students in any of the mathematical sciences, since it can be used as an introduction to further study of Fourier analysis on groups. This book will give the reader access to the literature on Walsh series (a fairly complete survey of this literature up to 1970 appeared in the article of Balasov and RubinsteIn [1), and surveys of more recent research in the article of Wade [4] and in the closing pages of the monograph [1]) and, for those who wish to continue to questions in a more abstract setting, it will give an introduction to monographs [1], [24], [26], and others. Acquaintance with the theory of

1 Numbers inside square brackets denote references which appear in the bibliography found at the end of this book. References to scholarly articles will contain the family name of the author and the number of the article as h appears in his Hst in the bibliography (for example, Efimov [1]). References without the author's name refer to monographs or textbooks which are listed separately in the bibliography (for example [1], [2]).

2Translator's note: A comprehensive monograph on the theory of Walsh series and transforms was published in 1990 by Adam Hilger (Institute of Physics) Publishing, Ltd, Bristol and New York. It is "Walsh Series: An Introduction to Dyadic Harmonic Analysis" by F. Schipp, W.R. Wade, and P. Simon, with assistance from J. Pal.

xi

xii PREFACE

Walsh series is also useful for the study of general questions in the theory of orthogonal series. The first 10 chapters of this book deal with foundations of a theoretical nature, and Chapters 11

and 12 are connected with applications. Chapters 1 and 2 are fundamental for all that follows and by themselves are sufficient preparation for further study of both theoretical and applied material. Chapters 3-5 contain results concerning uniqueness of representation of functions by Walsh series, questions about summability and convergence in LP of Walsh-Fourier series. This material is not used directly in the final chapters and on first reading those primarily interested in applications may restrict themselves to a passing acquaintance with these chapters. On the other hand, the concepts considered in Chapter 6 concerning multiplicative transforms are widely used in the last two chapters.

Chapters 7 and 8, where we consider Walsh series with monotone coefficients and lacunary series, contain only elementary information about these important classes of Walsh series whose theory has extensive connections with other closely related areas, in particular, as was mentioned above, with the theory of probability.

Chapter 9 is devoted to the specific questions of convergence and divergence of Walsh-Fourier series and is intended primarily for mathematicians. Here we give Hunt's proof of the Walsh analogue of Carleson's Theorem, about convergence of Fourier series of functions in the class L2, which in its basic features coincides with the trigonometric proof. However, many of the technical details are simpler in the Walsh case and this allows the reader to grasp more easily the basic ideas of the proof and will prepare him for further study of the trigonometric case.

In Chapter 10 we consider the problem of approximation by Walsh polynomials and by polynomials in the multiplicative systems. This problem is fundamental for many applications of the Walsh system. Finally, in Chapters 11 and 12 we examine methods for applying the Walsh system and its generalizations to digital information processing, to construct special computational devices, to digital filtering, and to digital holograms.

In order to aid the reader who is only acquainted with an undergraduate curriculum in mathematics, we include at the end of this book several Appendices containing background information about more advanced mathematical material which is used in the body of this book, namely, information about group theory, measure theory, the Lebesgue integral, and functional analysis. These appendices are followed by a commentary which includes brief remarks of a historical nature and references for sources of material which appears in each chapter. In view of the fact, as was mentioned above, that the theory of Walsh series already has some excellent and comprehensive survey articles our commentary gives further information only about the latest developments in this area.

Chapters 1-5 (except §1.5, §2.5, and §2.7) and Chapter 9 were written by V.A. Skvorcov, Chapters 7,8,10 (except §10.5) and §2.7 were written by B.1. Golubov, Chapters 6, 11, 12, and §10.5 by A.V. Efimov, §1.5 was written jointly by A.V. Efimov and V.A. Skvorcov, and §2.5 was written jointly by B.1. Golubov and V.A. Skvorcov.

The authors hope that this book will draw attention to the applicability of our subject and at the same time precipitate further theoretical investigations of solutions to applied problems.

The authors convey sincere thanks to B.F. Gaposkin and A.1. RubinsteIn, who read a manuscript version of this book and gave several valuable remarks which helped improve the presentation of this material.

The Authors

FOREWORD The sections in each chapter have interior enumeration and theorems and formulae each have their

own enumeration in each section. For example, §1.5 is the fifth section in Chapter 1, 2.3.5 is the fifth theorem in §2.3, and (2.3.5) is the fifth formula in §2.3. Similarly, A5.2 is the second section of Appendix 5, and A5.2.3 is the third theorem in A5.2. The beginning of a proof will be marked by "PROOF.", and the end of a proof will be marked with a" I".

The sign "::" frequently denotes equal by definition. The Lebesgue measure of a set A will be denoted by mes A, but (and this is so especially when A

is an interval Ll) we shall also use the shorter notation IAI. We presume that the reader is familiar with the symbolism generally accepted from set theory,

including the symbol 0, which represents the empty set. As usual, (a, b) denotes the open interval from a to b, [a, b) the closed interval from a to b, and

[a, b) the half open interval from a to b which contains the point a but not the point b. Other notation will be introduced as needed.

xiii

Chapter 1


§1.1. The Walsh functions on the interval [0,1). Consider the function defined on the half open unit interval [0,1) by

ro(x) = { 1 -1

for x E [0,1/2) ,

for x E [1/2,1) .

Extend it to the real line by periodicity of period 1 and set rk(x) == ro(2kx) for k = 0,1, ... and real x. The functions rk(x) are called the Rademacher functions.

It is evident from this definition that

(1.1.1)

and

(1.1.2)

for all integers m, k 2:: o. It is also clear that each rk(x) has period 1/2k, is constant (with constant value +1 or -1) on the dyadic intervals [m/2 k+1, (m + 1)/2k+1), m = 0, ±1, ±2, ... , and although it has a jump discontinuity at each point of the type m/2k+l, it is always continuous from the right. A graph of rk(x) on [0,1) for k = 1 can be found in Fig. 1.

The Rademacher functions are sometimes defined by

where

sgn t = { ~ -1

for t > 0,

for t = 0,

for t < O.

This definition differs only slightly from the one above. Namely, these rk's are 0 at the jumps instead of being continuous from the right. We draw attention to this fact to emphasize that for us the Rademacher functions never assume the value o.

The Walsh system {w n ( x)} ~=o is obtained by taking all possible products of Rademacher functions. In connection with this we shall use the following enumeration of the Walsh system. (This enumeration is called the Paley enumeration; see

2 CHAP1ER I

Y, ~ I I I I I I I I I I I I

0 z

-1

Figure 1.

the commentary on Chapter 1). Set wo(X) == 1. To define wn(x) for n ;::: 1, represent the natural number n as a dyadic expansion, i.e., in the form

(1.1.3) k

n = Lc;2;, ;=0

where Ck = 1 and C; = 0 or 1 for i = 0,1, ... , k - 1. Such an n obviously satisfies 2k ::; n < 2k+l, where k = ken). Set

k k-l

(1.1.4) Wn(X) = II(r;(x)yi = rk(X) II(r;(xWi. ;=0 ;=0

In particular, each Walsh function wn(x), n ;::: 1, takes on only the values 1 and -1, and is continuous from the right.

The definitions we have given for the Rademacher functions and the Walsh functions make sense on the entire real line. However, frequently the Walsh functions are considered only on the interval [0, 1). This is the natural domain of definition, as will be seen in §1.2 where we shall give another definition of the Walsh functions.

Suppose 2k ::; n < 2k+l. Notice by (1.1.3) that n - 2k = E~':-~ c;2i. It follows from (1.1.4) that

(1.1.5)

In particular, rk(x) = W2k(X), We also notice that the following results are true.

1.1.1. The finite product of integer powers of Rademacher functions is a Walsh function w£(x). Moreover, if k is the maximal index of the Rademacher functions

WALSH FUNCTIONS AND THEIR GENERALIZATIONS 3

appearing in tbis product, tben i < 2k+1, and i = 0 if and only if eacb rj appears in tbis product witb an even power.

PROOF. Notice from definition that

ri(x) = { 1 ri(x)

when m is even,

when m is odd.

Thus a finite product of powers of Rademacher functions which contains s odd powers reduces to a product of the form

ri1(x). ri2(x)· ... · rj,(x),

for certain indices i1 < i2 < ... < is ~ k. It follows from (1.1.4) that this product is precisely the Walsh function we(x) where i = 2i1 + 2i2 + ... + 2i,. It follows that o < i < 2i,+1 ~ 2k+1. If a finite product has no odd powers, then it is everywhere equal to 1 and coincides with the Walsh function wo(x). This finishes the proof of 1.1.1. •

1.1.2. Tbe product of two Walsb functions wn(x) and wm(x) coincides identically witb a tbird Walsb function we(x), i.e., wn(x) . wm(x) = we(x). Moreover, if m ~ n < 2k+1, tben i < 2k+1, and i = 0 if and only if n = m.

PROOF. By (1.1.4) the product of wn(x) and wm(x) is a product of powers of Rademacher functions whose indices do not exceed k. Hence the result follows from 1.1.1. •

In §1.2 we shall be more precise about the relationship between the indices n, m and i which appear in Theorem 1.1.2.

It is evident from the definition that each Walsh function is constant on certain half-open dyadic intervals. We shall use a special notation for such intervals. Namely, we shall denote the (dyadic) interval3 of rank k ~ 0 by

(1.1.6) ~~) == [m/2k, (m + 1)/2k), 0 ~ m ~ 2k-1.

For convenience we also set ~~O) == [0,1). We shall call the collection of all intervals of rank k the dyadic (or binary) net of

rank k and denote it by Nk . We shall occasionally refer to the intervals ~~) as nodes of the net N k • Each of these nets provides a partition of the interval [0,1), namely

2k_1

[0,1) = U ~~). m=O

The interval of rank k which contains a point x will be denoted by ~~:~. Of course

for each point x the sequence ~~:~ is a nested set of intervals which shrinks to the point x.

Notice that any two (dyadic) intervals are either disjoint or subsets of one another.

4 CHAPTER 1

1.1.3. For each 0 ~ n < 2k+I , the Walsh function w n ( x) is constant on the intervals

~~+I), ° ~ m < 2k+I and takes on the value lor -1. Moreover, wn(x) = 1 for all E A (k+I)

x DO .

PROOF. If n < 2k+I then the Rademacher functions appearing in the product (1.1.4) (which defines wn(x)) have indices which do not exceed k. Thus it is clear

that wn(x) is constant on each ~~+I) and identically 1 on ~~k+1}. I Notice for 2k ~ n < 2k+l that the intervals ~~+I) are the largest dyadic intervals

on which the function w n ( x) is constant.

1.1.4. If2k ~ n < 2k+l for some k ;::: ° then

(1.1. 7)

for each m = 0,1, ... 2k - 1.

PROOF. Write w 1l (x) as a product by (1.1.5). Notice by (1.1.3) that the function

Wn_2k (x) is constant on the interval ~~) with value equal to 1 or -1. In view of (1.1.2) it follows that

This proves (1.1.7). I It is immediate from (1.1.7) that

(1.1.8) for n = 0,

for n ;::: 1.

1.1.5. The Walsh system satisfies the orthogonality condition

for n = m,

for n i= m,

i.e., the Walsh system forms an orthonormal system on [0,1).

PROOF. This fact follows immediately from 1.1.2 and (1.1.8). I By using periodic extensions of the Walsh functions to the whole real line, it is

easy to see that

(1.1.9)


Indeed, write k k

112m = 2m L ci2 i = L ci2i+ m

;=0 ;=0

and combine (1.1.4) with (1.1.1). An important consequence of (1.1.9) is that

(1.1.10)

for 11 = p,

for 11 -=J p.

§1.2. The Walsh system on the group. In this section we introduce another method of generating the Walsh system

which changes the domain of definition. It turns out that a better and more natural domain on which to define the Walsh

functions is given by the set of sequences whose entries are either ° or 1, namely, sequences of the form

(1.2.1) * x = {XO,Xl,X2, ... ,Xj, ... },

where Xj = ° or 1 for j = 0,1, .... Each such sequence gives rise to a series

00

(1.2.2) "x··Tj-l 6 J ,

j=O

which is the dyadic expansion of some point x in the interval [0,1]' namely,

00

(1.2.3) " 2-J·-1 X = 6Xj .

j=O

Notice that the correspondence between the set of sequences of the form (1.2.1) and the set of series of the form (1.2.2) is evidently 1-1 but the correspondence between series of the form (1.2.2) and points in the interval [0,1] which are represented by the formula (1.2.3) is not 1-1 because every dyadic rational has two expansions of the form (1.2.2) : one which is finite and one which is infinite (with Xj identically 1 for j large).

Because of this, the usual interval [0,1] is not a suitable model for a geometric interpretation of the set of sequences (1.2.1). In stead, we use the so-called modified

6 CHAPTER 1

interval [0,1]*, which consists of expressions of the form (1.2.2) but not their sums. One can think of the modified interval [0,1]* as the usual interval in which each interior dyadic rational x has been split into two points, a left point x - 0 which is the infinite dyadic expansion of x, and a right point x + 0 which is the finite expansion of x. The dyadic rationals 0 and 1 are not split because they each have only one expansion of the form (1.2.2), the expansion of 0 being the expansion in which every coefficient is zero.

We shall now show that the set of sequences (1.2.1) can be made into a commutative group. (For the definition of a group, see A1.l.) Define an algebraic operation EB, which we call addition, by the following process. The sum of two sequences

; = {Xj}~o and y = {Yj}~o is the sequence; = {Zj}~o given by

(1.2.4)

where

* * * { }= z=xEBy= XjEBYj j=O

for Xj + Yj = 0 or 2,

for x j + Y j = 1, i.e., the operation EB is coordinate addition of two sequences modulo 2.

It is obvious that this operation is associative and commutative. The inverse operation 8, defined by ; == ; 8 y if and only if; EB y = ;, evidently coincides with

* the operation EB. In particular, if 0 represents the sequence all of whose entries are * * * * * zero, then 0 is the zero element of the group and Y EB Y = 0 for all sequences y.

The commutative group whose elements are sequences of the form (1.2.1) and whose addition is the operation EB defined by the formula (1.2.4) will be denoted by G.

Since there is a 1-1 correspondence between the sequences (1.2.1) and the modified interval [0, 1] *, the operation EB can be carried over in a natural way to [0,1] * , making it a group as well.

The group structure of G gives a convenient domain of definition for the Walsh functions, whose properties are intimately connected with this structure.

We shall use {~n(;)}~=o to denote the Walsh system whose domain is the group G or the modified interval [0,1]*. This system is defined in the following way. Let

n be a natural number whose dyadic expansion has the form (1.1.3) and let; be an element of the group G of the form (1.2.1). Set

(1.2.5) * * ~k ~.x. wn(x) = (-1)6i=0 • '.

For n = 2i we see that ~2i(;) = (-lyi. Naturally, this subsequence of the Walsh system will be called the Rademacher system on the group G and will be denoted by {;i(;)}. Using this notation we see that (1.2.5) is analogous to (1.1.4), namely,

k k

(1.2.6) ~n(;) = II((-I)Xi)"i = II(;i(;))"i. i=O ;==0

It is clear from definition (1.2.5) that for each n < 2k+I, the function ~n(~) is

constant with value 1 or -Ion sets of the form {~ : xi = x~, j = 0,1, ... , k}, for each choice of fixed coordinates {xg, x~ , ... , xV. These sets can be indexed by integers 0 ~ m < 2k+l in the following way. For each such m let the x~'s be determined by

(1.2.7)

and set

(1.2.8)

k

m = Lx~2k-i, j=O

~(k+l)_{*. ,_ 0 '-01 k} U m = x. X 1 - xi' J - , , ... , .

We shall presently see that this notation consistent with the notation introduced in §1.1.

Consider the transformation A : G --+ [0, I] which takes each sequence;; of the

form (1.2.1) to the number x = A(;;) which satisfies (1.2.3). Thus;; is the sequence

of coefficients of the dyadic expansion of A(;;). The transformation A is well-defined but, as we remarked above, is not 1-1. We shall examine what happens to the sets (1.2.8) under the transformation A, i.e., look at the set of points x of the form (1.2.3) which are images under A of sequences from the set (1.2.8). Clearly, for such points x,

But (1.2.7) implies

i.e., each such point satisfies the inequality

It follows that each element of the set ';'~+1) is a sequence of coefficients of dyadic expansions of points from [m/2 k+I, (m + 1)/2k+I], the closure ofthe interval .6.~+I). This shows that the notation introduced in (1.2.8) is consistent with the

notation introduced in (1.1.16). Moreover, one can interpret ';'~+I) geometrically as the subset of the modified interval [0,1]* which differs from the usual interval

8 CHAPTER 1

[m/2k+I, (m+ 1)/2k+lj in that its interior dyadic rationals have been split into two points and its endpoints are understood to be (m/2k+l )+0 and ((m+ 1)/2k+l )-0.

Notice that as interpreted the intervals ;~~i) and ;~+1) have no common points, since the right endpoint of the first interval is (m/2k+ 1 ) - 0 and the left endpoint of the second interval is (m/2k+l) + o.

It is easy to verify that the sets ;~k), k = 0,1, ... , (;~o) := G), are subgroups of

G and for each fixed k, that the sets ;~), m = 0,1, ... ,2k - 1, exhaust the cosets * (k) (see A1.2) of the subgroup 6 0 in the group G.

The following identity is important to the theory of Walsh series1 :

(1.2.9) n = 0, 1, ....

It reveals a connection between the properties of the Walsh functions as defined on the group G, and the group structure of G. Its proof follows easily from the definitions of the operation EB and the Walsh functions. Indeed, it is clear from (1.2.4) that the usual sum Xi + Yi and the sum Xi EB Yi are either both even or both

odd. This same observation holds for the sums 2:7=0 Ci(Xi +Yi) and 2:7=0 Ci(Xi EBYi),

where Ci are defined by the decomposition (1.1.3). It remains to apply (1.2.5). In order to formulate the next property of the Walsh functions, we introduce an

operation on the non-negative integers similar to the operation EB on the group G. For this, represent each non-negative integer n as an element of the group by setting

where the coefficients ej are determined by the equation (1.1.3). Let Go represent the collection of sequences in G which contain only finitely many non-zero entries, i.e., sequences which from some point on are identically zero. Clearly, the map n -+ ~ is a 1-1 transformation from the collection of non-negative integers onto

the subgroup Go. Let 1~ be the element of the subgroup Go which corresponds to

m. Then 1; EB ~ belongs to Go and, under this transformation, has a non-negative integer preimage which we shall denote by nEB m. Thus to obtain the dyadic sum of two numbers nand m, take their dyadic expansions and add the dyadic coefficients coordinatewise modulo 2.

As an analogue of (1.2.9) we obtain

(1.2.10) X E G.

By using the Walsh functions as defined on the group G we can give a new definition of the Walsh functions on the unit interval which is equivalent to that

1 Identity (l.2.9) shows that ~n(;) are the characters of the group G.


given in §1.1. For this we look at the connection between the half-open interval [0,1) and the group G in more detail. We mentioned above that the transformation from G to [0,1) fails to be 1-1 only at dyadic rational points, because each dyadic rational in (0,1) has two dyadic expansions. If we agree to use only the finite expansion of these dyadic rationals, then each point x E [0,1) uniquely determines the sequence (1.2.1). Thus we can define a transformation g : [0,1) -> G by the formula

(1.2.11)

where x j is determined by the formula (1.2.3), with the convention about dyadic rationals which we just agreed upon. Let G' be that subset of G obtained by removing all sequences which are identically 1 from some point on. It is clear that g(x) -=I- g(y) for x -=I- y, so the transformation g is a 1-1 map from the half open interval [0,1) onto G'. Hence we can define the inverse transformation g-l on G' which apparently coincides with the restriction to G' of the map A defined above.

Earlier, when we showed that the transformation A takes sets of the form ~~+1) to intervals, we in fact showed that the finite dyadic expansion of the point m/2(k+1)

has the form 2:7=0 x~2-j-1, where the coefficients xJ are defined by (1.2.7). It is easy to check that the dyadic expansions of the interior points of the interval [m/2k+1, (m + 1 )/2k+1) = ,0.~+1) has coefficients x j which coincide with x~ for

0:::::; j :::::; k. Consequently, the transformation g takes the interval ,0.~+1) to a subset

of ~~+l). In fact, the image of ,0.~+1) under g is precisely the set ~~+l) n G'. It is now possible to give the following definition of the Walsh functions on [0,1):

(1.2.12) n = 0,1, ... ,

where {ci} is determined by (1.1.3) and {x;} is the sequence of coefficients from the dyadic expansion of x with the convention that the finite expansion is used when x

is a dyadic rational. We shall show that this definition is equivalent to the definition given by (1.1.4).

For this, notice first that by (1.1.4) and (1.2.6) it is enough to prove

(1.2.13)

for i = 0,1, .... To prove this last identity, notice that the coordinate Xi of the sequence g( x) (see

(1.2.11») is fixed as x ranges over an interval of the form [m/2 i+1, (m + 1)/2i+1 ),

since as was noticed above, g takes such an interval to a subset of ~~+1). Moreover, the value of Xi (see (1.2.8)) coincides with x? from the expansion of m in (1.2.7), where k has been replaced by i. Since the coefficients x? alternate between ° and 1 as m ranges from ° to 2i+1 - 1, it follows that the values of (-lyi alternate

\0 CHAPTER 1

between 1 and -1 as x moves through the intervals of rank i. Hence the description above of ri coincides with the definition given in §1.1. This completes the proof of identities (1.2.13) and (1.2.12). In particular, it is natural to restrict the domain of the Walsh functions of a real variable to the half open interval [0,1).

By using the transformation 9 one can transfer the group operation EEl from G or from the modified interval [0, 1)* to the usual unit interval in the following way. First recall that the map 9 is only defined on the subgroup G', which consists of sequences which are not identically 1 from some point on. Set

(1.2.14) x EEl y = g-l(g(X) EEl g(y)), g(x) EEl g(y) E G'.

Thus to find the sum x EEl y it is necessary to look at the sequences ; and y of the form (1.2.1), obtained from the dyadic expansions (1.2.3) of the points x and y (where the finite expansion has been used when a dyadic rational is involved). Add them to obtain; EEl y and if this sequence does not terminate in 1 's, use the transformation g-1 to take; EEl y to a point in [0,1). This point is the value of x EEl y.

It is evident that the sum x EEl y is not defined for all pairs x, y E [0,1). This is done so that the operation EEl on [0,1) will preserve many of the important properties that are enjoyed by the group operation. For the theory of Walsh series the

most significant of these properties is that the shift operation G -+ G EEl Y is a 1-1 transformation from the group G onto itself, and that for each fixed k it induces

a permutation of the finite collection of cosets b.jkl , ° ::; j ::; 2k - 1. (See A1.3). Examining the corresponding shift operation on [0,1) (with respect to EEl as defined above), we shall see that if we neglect a countable subset then it too is a 1-1 transformation. Before we formulate the corresponding properties concerning cosets, we shall make several preliminary remarks.

Recall that the subset G \ G' of the group G consists of all sequences of the form (1.2.1) whose entries are identically 1 from some point on. Hence G\ G' is countable.

If for some fixed y we denote by (G \ G') EEl Y the collection of all sums of the form ; EEl y, where; E G \ G' , then it is also clear that (G \ G') EEl y is countable for each y E G.

Define a set Iy == )..( G' EEl g(y)), where).. is the transformation defined above which takes the group G onto the unit interval [0,1]. Since [0,1) \Iy C )..( G\ (G' EElg(y))) = )"((G \ G') EEl g(y)) and this last set is the image under).. of a countable set, we see that the set [0,1) \ Iy is also countable.

We are now prepared to formulate and prove the following result.

1.2.1. For each fixed y the sum x EEl y is defined for all x E Iy == )..(G' EEl g(y)), i.e., at all but count ably many points in [0,1). The transformation x -+ x EEl y is a 1-1 transformation of the set Iy onto itself. Moreover,

WALSH FUNCTIONS AND TIlEIR GENERALIZATIONS 11

for each dyadic interval ll)k), where the integer iI depends on y and satisfies il = i if and only if y E ll~k). If y is a dyadic rational then Iy = [0,1). In particular, the shift operation x --+ x EB y maps [0, 1) onto itself when y is a dyadic rational.

PROOF. Recall that the transformation g takes [0,1) onto G' and that on the set G', the transformation oX coincides with g-l. Consequently, the transformation oX = g-l is I-Ion the set (G' 67g(y» n G' and we have both g(Iy) = (G' 67g(y» n G'

and g(ll)k) nly) = b.)k) neG' EB g(y» nG'. Furthermore, it is clear that

(G' EB g(y» nG') EB g(y) = G' neG' EB g(y».

It follows from (1.2.14) that the sum x EB y is defined for all x E Iy and that the map x --+ x 67 y is a 1-1 transformation from Iy onto itself.

It is well known (see A1.3) that on any group a shift preserves cosets. Thus a

shift by g(y) takes a coset b.)k) of the group G to another one, say b.)~). Moreover, it is clear that

(b.)k) neG' EB g(y» n G') EB g(y) = b.)~) n G' neG' EB g(y».

This verifies the statements concerning the intervals ll)k).

It is evident that b.)k) EB g(y) = b.)k) if and only if g(y) E b.~k), i.e., y E ll~k). Moreover, for each dyadic rational y the sequence g(y) is finite so G' EB g(y) = G'. These observations complete the proof of 1.2.1. I

It is now possible to obtain the following analogue of (1.2.9) for the Walsh functions wn(x):

1.2.2. Let x, y E [0,1). If the sum x EB y is defined then

(1.2.15)

for each n = 0,1, .... Thus for a fixed y identity (1.2.15) holds for all but countably many points x E [0,1), and when y is a dyadic rational it holds for all x E [0,1).

PROOF. Fix a natural number n. In view of 1.2.1 we need only prove (1.2.15) in the case when the sum x 67 y is defined. Suppose, then, that g( x) EB g(y) E G'. Hence g-l(g(X) EB g(y)) exists and we may apply definitions (1.2.14), (1.2.12), identity (1.2.9), and again definition (1.2.12). We obtain

Wn(x EB y) = Wn(g-l(g(X) EB g(y))) = t;;n(gg-l(g(x) EB g(y)))

= t;;n(g(X) EB g(y» = t;;n(g(X))1tn(g(y)) = wn(x)wn(y). I

12 CHAPTER I

It is now easy to see that

(1.2.16)

Indeed, write 11 = 2k + 2:~':~ ci2i and notice by definition (1.2.12) that

Since (1.2.15) implies

it follows that (1.2.16) holds. An immediate consequence of (1.2.12) is that an analogue of (1.2.10) holds with

out the kind of restriction on the domain which appeared in the statement of 1.2.2. Namely, the following identity is true:

(1.2.17) x E [0,1).

This identity is a more precise version of 1.1.2. In connection with this remark notice that if In ::; 11 < 2k+ I then 11 EEl m < 2k+1 and 11 EEl m = ° if and only if 11 = nt.

1.2.3. Let; and y be elements of the group G and A be the transformation defined above whicil takes G onto [0,1]. Then

(1.2.18)

PROOF. Let; = {x;}~o and y = {y;}~o' If A(Y) = A(;) the inequality is

obvious. By symmetry we may suppose that A(Y) > A(;). Consider the sequence

~ = {z;} ~o defined by

{ y'

Zi = 1: ifxi=O,

if Xi = 1.

Since for each i we have Zi 2: Yi, it is clear that

00 00

A(;) = L ziTi - 1 2: LYiTi-1 = A(Y). i=O i=O

Furthermore, Zi 2: Xi and Zi - Xi = Zi EEl Xi ::; Yi EEl Xi for each i. It follows that A(~) - A(;) = 2::0 Zi2-i-l - 2::0 Xi2-i-1 ::; 2::o(Yi EEl xi)2- i- 1 = A(y EEl;). But


A(Y) - A(~) ~ A(;) - A(~). Consequently, A(Y) - A(~) ~ A(Y E9~) which verifies (1.2.18) in the case under consideration .•

The quantity Pa(;;, y) == A(;; EBy) = A(;; 8 y) plays the role of a distance2 between

two elements;; and y in C. Similarly, a new concept of distance between two points x and y in the unit interval [0,1) can be defined by

(1.2.19) p*(x, y) == A(g(X) EB g(y)).

If x and y are points for which the sum (1.2.14) is defined, then p*(x, y) = x E9 y. Since x = A(g(X)) and y = A(g(y)) always hold, it follows directly from (1.2.18)

that

(1.2.20) Iy - xl ~ p*(x, y) x, Y E [0,1).

This inequality reveals the connection between p* and the usual metric on the real line.

From definition (1.2.19) it is clear that the distance p* is invariant under translation, namely, if x EB z and y ttl z are defined then

(1.2.21 ) p*(x, y) = p*(x ttl z, y ttl z).

1.2.4. a) Ifx,y E f::.Jkl , then p*(x,y) ~ 1/2k.

b) If p*( x, y) ~ 1/2k then there exists a dyadic interval of the form f::.Jk- Il which contains both x and y.

PROOF. a) Since we have agreed to use the finite dyadic expansion for each

dyadic rational, it is clear that if x, y E f::.;kl then the dyadic coefficients of x and y satisfy Xi = Yi for i = O,I, ... ,k -1. Thus the sum g(x) ttlg(y) has the form (0,0, ... ,0, Zk, Zk+1, ... ) which implies

00 z' 1 p*(x, y) = A(g(X) ttl g(y)) = L 2j~1 ~ 2k '

j=k

b) Suppose to the contrary that x and y belong to different dyadic intervals of rank k - 1. Then we can choose an integer ° ~ i ~ k - 2 such that the coefficients of 2-i - 1 in the dyadic expansions of x and yare different. Consequently, the i-th element of the sequence g( x) ttl g( y) equals 1. This means that the dyadic expansion of the number A(g( x) ttl g(y)) contains the term 2- i - 1 • Since i ~ k - 2 it follows that p*(x,y) 2 2-k+ 1 > 2- k • This establishes b) .•

2That the function A(~ e y) satisfies the triangle inequality follows directly from (1.2.18). After recognizing this, it is easy to verify that this function satisfies all the usual properties of a metric.

14 CHAPTER 1

1.2.5. If Yn -+ Y in the usual sense and Y is a dyadic irrational then P*(Yn' y) -+ 0.

PROOF. For any k choose a natural number no(k) such that y, Yn E t1jk) for all n 2': no and some integer j. It remains to apply 1.2.4 a) .•

It is not difficult to verify that G (respectively, [0,1)) is a metric space under the distance Po (respectively, pO) (see A2.1 and A2.2).

It is important to notice that on these metric spaces the Walsh functions are continuous. This is connected with the fact that each Walsh function has a jump discontinuity, in the classical sense, only at dyadic rationals. And, on the modified interval each dyadic rational is not only split into two pieces but these pieces are sufficiently far from each other as measured by the distance Po; we leave it to the reader to verify this fact and provide the actual calculations. An similar situation prevails on the unit interval [0,1) for the distance p*. In this case, the dyadic rationals are not split in two, but a pair of disjoint dyadic intervals are a positive distance from each other.

§1.3. Other definitions of the Walsh system. Its connection with the Haar systelll.

For each ° :::; n < 2k+1 and ° :::; m < 2k+1 denote by w~~~1) the constant value

which the Walsh function wn(x) takes on the dyadic interval ~~+I) (and which the

Walsh function ~n(~) takes on the set ~~+I»). We shall examine the 2k+1 x 2k+1

matrix (w~~~I)). Our interest in this matrix stems from the fact that it completely determines the first 2k+1 Walsh functions.

1.3.1. The matrix (w~~;!; I») is symmetric and orthogonal.

PROOF. Since the elements of this matrix depend on k but k is fixed, we shall drop the superscript (k + 1) from the notation in this proof.

We first verify that Wn,m = wm,n' Write n in the form (1.1.3) and write m in the

form (1.2.7). Recall that the numbers x~, j = 0,1, ... , k, determine the set ~~+I) (see (1.2.8)). We obtain from definition (1.2.12) that

Reverse the roles of nand m. By (1.1.3), (1.2.7), and (1.2.8) we see that the set

~~k+I) is determined by the numbers Xi = Ck-i for i = 0,1, ... ,k. Moreover, for each i the coefficient of 2i in the dyadic expansion of m of the form (1.1.3) is given by xLi' Therefore, we have by formula (1.2.12) that

",k 0 wm,n = (-l)L..ii=o Xk_i€k-i.

This coincides with the formula for wn,m given above. In particular, the matrix

(w~~~I») is symmetric.


Orthogonality of the matrix follows easily from orthogonality of the Walsh system {wn(x)}. Indeed, by 1.1.3 we have

Thus it follows from 1.1.5 that the matrix (w~~~1)) is orthogonal. • Theorem 1.3.1 allows us to prove the following useful result:

1.3.2. Any function P( x), which is constant on all dyadic intervals of the form ll.~), 0 ~ m ~ 2k - 1, can be represented in the form

2k_I

P(x) = L aiwi(x), i=O

i.e., P(x) is a Walsh polynomial whose non-zero coefficients have indices no greater than 2k - 1. Moreover, this representation of P( x) is unique.

PROOF. Use the notation w~~~1) introduced above and let Pm denote the constant

value which P(x) assumes on the dyadic interval ll.~). We obtain the following system of 2k equations in the unknowns ai, 0 ~ i ~ 2k - 1:

2k_I

'" (k) L....J aiwi,m = Pm, i=O

By Theorem 1.3.1, the determinant of this system is non-zero. Hence the function P( x) can be represented as promised, and this representation is unique among the Walsh polynomials of order no greater than 2k - 1.

Since by hypothesis the function P( x) is also constant on any dyadic interval ll. (I) of rank £ > k, the remarks above remain true if £ is substituted for k. Hence the representation of P( x) is unique among the class of Walsh polynomials of order no greater than 2c - 1. But this class contains the class of Walsh polynomials of order no greater than 2k -1. Since the representations are unique in both classes, it follows that there is but one representation. In particular, the representation must be unique in the class of all Walsh polynomials .•

We shall now show that the matrix (w~:~1)) can be constructed from the matrix

(w~~~). First, we establish the relationships

(1.3.1) { (k+I) (k+I) (k)

w 2n ,m = w 2n+ I ,m = wn,m,

w(k+I) _ _ W(k+I) _ w(k) 2n,2k+m - 2n+I,2k+m - n,m

16 CHAPTER I

for 0 ~ n ~ 2k - 1 and 0 ~ m ~ 2k - 1. If n = E~~; 10;2;, then 2n = E~=l ci-12;

and 2n + 1 = 2° + E~=l ci-12;. Furthermore, if m is the index of the interval .6.~), written in the form (1.2.7) with k - 1 in place of k, then converting to intervals of rank k + 1 we see that the expansions of m and 2k + m can be written in the form m = E~=l xL12k-i, 2k + m = 2k + E7=1 x?_12k-;. Applying formulae (1.2.8), (1.2.5), and (1.2.12) we obtain

(k) ",k 0 ",k-l 0 W2 ·+1 = (-I)L..Ji=l ~i-lXi_l = (-I)L..Ji=o EiX. = w(k) ,

n,ffi n,m

and

This proves (1.3.1). Since the matrices (w~~?n) and (w~~;!;l») are symmetric, (1.3.1) can be rewritten

in the form

(1.3.1')

for 0 ~ n ~ 2k - 1, and 0 ~ m ~ 2k - 1. The equations in (1.3.1) can be used to generate the matrices (w~~?n) recursively.

To obtain the matrix (w~~;!;l)) write each row of (w~~?n) twice, with the new copies under the old ones. This makes an intermediate 2k+1 X 2k matrix. To fill in the rest of the columns of (w~~;!;1)), write a copy of each row of the intermediate matrix to the right of its row but multiply each element of the copied even rows by -1. To ill ustrate this process, we write the matrices of order 2 x 2, 22 X 22, 24 X 24:

1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1

-\), ( 1 1 1 _11 )

1 1 -1 -1 1 1 -1 -1

( 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 ' 1 -1 1 -1 1 -1 1 -1

1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1

Since these matrices are symmetric, the doubling process works equally well with columns. Thus it is easy to see that the values of the first 2k Walsh functions, as


prescribed by the matrices (W~~~), (w~~;!;1)), and all subsequent ones, are consistent with one another. Indeed, the first identity of (1.3.1') shows exactly that for 0 ~ n ~ 2k -1 the values of the function w n ( x) generated by matrix of order 2k+l X 2k+l

on each of the intervals ~~~+I) and ~~:.+~~ which make up the interval ~~) coincide

with the values of this function on the interval ~~) as prescribed by the matrix of order 2k X 2k.

Thus it is clear that the collection of Walsh functions is uniquely determined

by the sequence of matrices (w~~~)~o, where (w~~6) = (1). On the other hand equations (1.3.1) together with the agreement that the matrix of order 1 X 1 has the form (W~~6) = (1), completely determine the matrices (w~~~), k = 0,1, .... This gives a new, equivalent definition of the Walsh system, not only on the interval [0,1), but also on the group G.

Finally, the matrices (w~~~) allow us to establish a connection between the Walsh system and the Haar system, another system used widely in the theory of orthogonal expansions of functions. This will give us one more equivalent definition of the Walsh system on [0,1), this time in terms of Haar functions.

The Haar system {hn(x)}~=o is defined on the interval [0,1) in the following way. For all x E [0,1) set ho(x) = 1. Write each integer n ~ 1 as n = 2k + m, for some integers k ~ 0, 0 ~ m ~ 2k - 1, and set

(1.3.2)

£ A (HI) or x E U2m ,

£ A (k+l) or x E u2m+I'

for x E [0,1) \ ~~) .

(We notice that according to this definition, the Haar functions are continuous from the right at each point of discontinuity; this convention differs from that frequently found in the literature, where the Haar functions are defined at jump discontinuities to be the average of their left and right limits.)

Clearly, the Haar function with index n = 2k + m, 0 ~ m ~ 2k - 1, is non-zero only on the dyadic interval ~~), m = 0,1, ... ,2k - 1.

By using the matrices (w~~~), we shall now show that the Walsh functions w;(x), 2k ~ i ~ 2k+l -1, can be written as a linear combination of Haar functions with indices in the same range, namely, that

(1.3.3)

It is easy to see that the sum on the right side of (1.3.3) is constant on the interval (HI) . (k) • (HI). (k)

~2m wIth value Wn,m, and constant on the mterval ~2m+1 wIth value -wn,m.

Hence the values of W2k+n(X) for 0 ~ n ~ 2k - 1, as calculated by the formula (1.3.3), coincide with the true values that these functions must satisfy as described

18 CHAPTER I

in the second formula of (1.3.1 '). Thus definition (1.3.3) is equivalent to the one

given above using the matrices (w~~~). This establishes (1.3.3). Since it holds for any k ~ 0, it defines each Walsh function Wi(X) for i ~ 1.

Equations (1.3.3) can be interpreted in the following way. The linear transfor

mation induced by the matrix (w~~~) takes the vector (h2k, h 2k+1, .. . , h 2k+l+I) to

the vector (W2k,W2k+I, ... ,W2k+1+1). Since the matrix (w~~~) is symmetric and orthogonal hence its own inverse, we see that (1.3.3) has the following analogue which defines Haar functions in terms of Walsh functions:

(1.3.4)

Defining the Walsh system by using (1.3.3) is most useful in those cases when a certain property of Haar series corresponds to a similar one for Walsh series.

§1.4. Walsh series. The Dirichlet kernel. By a Walsh series we shall mean a series of the form

(1.4.1) 00

2: aiw i(X), i=O

where the coefficients ai are, by convention, real. We shall isolate several properties enjoyed by the partial sums

n-I

( 1.4.2) Sn(x) = 2: aiwi(x) i=O

of (1.4.1).

1.4.1. For each n, 1 ~ n ~ 2HI, the sum Sn( x) has a constant value on the intervals ~~+I), 0 ~ m < 2HI.

PROOF. This observation follows directly from 1.1.3 .• For each positive integer n ~ 2k, the constant value which a partial sum Sn(x)

assumes on an interval ~~) will be denoted by s~~:n.

1.4.2. For each interval ~~) = ~~~+I) U ~~~+~~ and each integer n satisfying 2k < n ~ 2HI , the equations

(1.4.3) (HI) (HI) (k) Sn 2m + Sn 2m+I = 2S2k , , , ,m

and

(1.4.4)

hold.

PROOF. Write the sum Sn( x) in the form

n-l

Sn(X) = S2k(X) + L ajWj(x). j=2 k

Thus by (1.2.16) we have

(1.4.5) Sn (x EB 2k: 1 ) - S2 k (x EB 2k: 1 ) = -Sn(x) + S2k(X).

Using the definition of the operation EB it is not difficult to see that if x belongs to one of the intervals .6.~~1) or .6.~~~I, then x EB 1/2k+1 belongs to the other one. Moreover, both these points belong to the same interval .6.~). Consequently, S2k(X EB 1/2k+l) = S2k(X) and we obtain from (1.4.5) that

(1.4.6) Sn(x) + Sn (X EB 2k: 1 ) = 2S2k(X).

In view of the notation introduced above, this identity verifies (1.4.3). We obtain (1.4.4) multiplying (1.4.3) by 1.6.~~1 and recalling that 1.6.~)1 = 21.6.~~1 .•

Apply (1.4.4) for n = 2k+l and iterate. We obtain

(1.4.7) £ ~ O.

The partial sums of the series L:i Wj(x) play such an important role in the theory of Walsh-Fourier series that they receive a special notation:

n-l

(1.4.8) Dn(x) = L Wj(x). ;=0

Analogous to the trigonometric case, we shall refer to these partial sums as the Dirichlet kernels for the Walsh system.

The Dirichlet kernels satisfy the following properties.

1.4.3. For 1:::; n:::; 2k+l the kernel Dn(x) is constant on each interval.6.~+l), and

(1.4.9) for x E .6.~k+1).

PROOF. The first part of this result is a special case of 1.4.1. Identity (1.4.9) follows directly from the fact that for x E .6.~k+l) and 0 :::; i < 2k+l, each Walsh function Wi( x) is identically 1 (see 1.1.3). •

20 CHAPTER 1

Relationship (Ll.8) implies

(1.4.10) 11 Dn(x)dx = 1, n = 1,2, ....

Let n be an integer written in the form n = 2k + m where 1 ::; m ::; 2k. Then

2k+m-I

Dn(x) = D 2k(X) + E w;(x). ;=2>

Applying (1.1.5) to each of the functions w;(x) for 2k ::; i < n -1 < 2k+I, we obtain

2k+m-I m-I

Dn(x) = D 2>(x) + rk(x) E W;_2 k(X) = D 2>(x) + rk(x) E w.(x) . • =0

Consequently, we have verified the formula

(1.4.11)

for all n = 2k + m, 1::; m ::::; 2k. Substituting m = 2k, we see that

(1.4.12)

We shall now establish the identity

(1.4.13) for x E 6(k) o ,

(k) for x E [0, 1) \ 6 0 •

The proof is by induction on k. For k = 0 it is obvious. Suppose the formula holds for some k ~ O. To obtain it for k + 1, combine (1.4.12) with the facts that

(k) (k+I) U (k+I) 6 0 = 6 0 6 1 and the Rademacher functions were defined so that

for x E 6 (k+I) o ,

for x E 6~k+I).

It follows that D2>+1(X) = 2D2k(X) = 2k+I for x E 6~k+I) and D2k+l(X) = 0 for

x E 6~k+I). This verifies (1.4.13) for k+1. Hence (1.4.13) holds for all non-negative integers k.

The inequality

(1.4.14) A (;) • 1 2 1 ? x E UI , Z = , , ... , n = , ~, ...

will play an important role for us. To prove this inequality, fix i ~ 1 and proceed by induction on k, where 2k < n :5 2k+1. Notice that the inequality is obvious for n = 1 = 2°. Suppose that the inequality holds for n :5 2k and let 2k < n :5 2k+1. Let m = n - 2k and notice that (1.4.11) holds.

We consider three cases: i :5 k, i = k + 1, and i > k + 1. If i :5 k then ~~i) C [0,1) \ ~~i) C [0,1) \ ~~k). Hence by (1.4.13), D2k (x) = a for

x E ~~i). In view of (1.4.11), it follows that Dn(x) = rk(x)Dm(x) for such points x. In particular,

(1.4.15) x E ~~i)

Since m = n - 2k :5 2k, inequality (1.4.14) holds for Dm(x) by the inductive hypothesis. Hence (1.4.15) implies (1.4.14) for Dn(x) when i :5 k.

Passing to the case i = k + 1, notice that r k (x) = -1 for x E ~~k+ 1). Recall also that Dm(x) = m for x E ~~k+1) C ~~k) (see (1.4.9». Thus it follows from (1.4.11)

and (1.4.13) that Dn(x) = 2k - m for x E ~~k+1). In particular, we have proved (1.4.14) in the case i = k + 1.

It remains to examine the case i > k + 1. In this situation ~~;) C ~~k+1) and therefore by (1.4.9), Dn(x) = n :5 21<+1 :5 2i - 1 • Thus inequality (1.4.14) holds in this case as well.

This completes the inductive step from k to k + 1. Thus the proof of (1.4.14) is finished.

It is not difficult to see that inequality (1.4.14) implies

(1.4.16) IDn(x)1 < l/x, x E (0,1), n = 1,2, ....

Indeed, if x f a then choose an i such that x E ~~i), i.e., 2- i :5 x < 2- i+1 and apply (1.4.14) to obtain

§1.5. Multiplicative systems and their continual analogues. The Walsh system is a special case of a more general class of function systems,

the so-called multiplicative systems. We shall define these systems here by means of a direct generalization of the definitions in §1.2. As is the case for the Walsh system, these systems can be defined on the interval [0,1), extended to the whole real line by periodicity of period one, or defined on some compact group similar to the group G.

We begin with a description of a class of groups of interest to us. Let

(1.5.1) p == {P1,P2,. ··,Pi,···}, Pi ~ 2, j ~ 1

22 CHAPTER 1

be a fixed sequence of natural numbers. Using P we define a set of sequences of integers of the form

(1.5.2) ° ~ Xj ~ pj - 1, j ~ 1.

This set becomes a group, which we shall denote by G(P), if we use the binary operation

(1.5.3) • • _ { }OO x Ef) Y = Xj Ef) Yj j=1' Xj Ef) Yj = Xj + Yj (mod Pj).

In the special case when the elements Pj of the sequence P are identically 2, then the group G(P) coincides with the group G introduced in §1.2. Notice, however, that in contrast to the dyadic case, the operation 8 ( which is the inverse of Ef)

on G(P)) is different from Ef). In fact, this operation can be defined as coordinate subtraction modulo Pj, i.e.,

Xj8Yj= 1 l' {X' - y'

Pj+Xj-Yj,

Xj ~ Yj,

Xj < Yj.

We shall define a multiplicative system on the group G(P) which will be indexed by the non-negative integers n. To do this we use the sequence P to write each non- negative integer n in P-adic form. This is a direct generalization of the dyadic expansion (1.1.3). First, set

(1.5.4) j

rno = 1, rnj = IIps,

s=1

where Pa are the members of the sequence (1.5.1). Next, write each n in the form

(1.5.5) k

n = L 0ij1nj_1,

j=l

° ~ Oij ~ Pj -1, j = 1,2, ... ,k.

This will be called the P-adic expansion of n. As in the dyadic case, each number n corresponds to an element;' of the group

G(P), namely, if n has P-adic expansion (1.5.5) then

* n = {0iI,0i2,oo.,0ik,O,O,oo.}.

This element is a finite sequence. As in §1.2, the map n -+ ;, allows us to transfer the group operation Ef) from the group to the set of non-negative integers.


For each integer n with P-adic expansion (1.5.5) and for sequence ~ of the form (1.5.2), define the n-th function of the system {Xn(~)}~o by3

(1.5.6)

It is easy to see that these functions satisfy the equations4

(1.5.7)

and

(1.5.8)

{ Xn(~ E9 y) = Xn(~)Xn(Y),

Xn(i e y) = Xn(i)Xn(Y),

{ XnE!lm(~) = Xn(;)Xm(;),

Xnem(;) = Xn(X)Xm(X),

analogous to equations (1.2.9) and (1.2.10) for the Walsh system. Notice that the system of functions (1.5.6) becomes the Walsh system {~;} in the particular case when pj = 2 for all j.

As it was for the Walsh system, the domain of definition for the functions (1.5.6) can be transformed in a 1-1 fashion to a "modified" interval [0, l]p, or, if we relax the 1-1 condition on some countable set, can be transformed to the unit interval [0,1). The details are as follows. Notice that each sequence ~ of the form (1.5.2) from the group G(P) corresponds to a series

00

(1.5.9) 2:)xj/mj), j=l

where the mj's are defined by equation (1.5.4). This series evidently converges and is the P-adic expansion of some number x which equals the sum of this series. The transformation Ap : x -+ x = ,£j=.l(Xj/mj) takes the group G(P) onto the interval [0,1]. It is not 1-1 since each P-adic rational has two expansions, a finite one and an infinite one. If we consider these two expansions as different points then, as in the dyadic case in §1.2, we obtain a "modified" interval [0, l]p, which gives a geometric interpretation of the group G(P).

Corresponding to the system {Xn} there is a multiplicative system of functions on the interval [0,1) defined analogously with the definition (1.2.12) which uses

3The systems described here, which in the literature are usually called Price systems (see [1), p. 68), do not exhaust the entire class of multiplicative systems.

4Hence the system {Xn(;)}g" is the character system for the group G(P).

24 CHAPTER 1

the transfonnation gp : x -7 ~. This transformation is defined so that each point x E [0,1) whose expansion is given by (1.5.9) corresponds to the sequence of the form (1.5.2) whose entries are the P-adic coefficients of x. As before, we adhere to the convention that the expansion used for each P-adic rational is the finite one. Thus for each n of the form (1.5.5) we have

(1.5.10) Xn(x) = Xn(gp(x)) = exp (27ri f QjX j )

j=l Pl

Let x be any real number. Notice that the P-adic coefficients x j of the expansion (1.5.9) of x can be computed by the formula

j 2': 1,

where for each real number a, [a] represents the greatest integer in a. Moreover, for P-adic rational points this formula gives the coefficients of the finite expansion. Similarly, the coefficients Qj of the expansion (1.5.5) of a natural number n can be computed by the formula

It is clear that the functions

(1.5.11) X m _ 1 (x) = exp (27ri x j) , J pj

j = 1,2, ... ,

playa role here analogous to that played by the Rademacher functions in the definition of the \i\Talsh system. In particular, we can write each Xn(x) in the form

(1.5.12) k

Xn(x) = IT (Xmi-l (x))"i . j=l

It is easy to check for n < m k that the function X n (x) is constant on intervals of the form

(1.5.13)

These intervals are P-adic analogues of the dyadic intervals ~~k). On the group G(P) or on the modified interval [0, l]p they correspond to the sets

*(k) _ *. . _ 0 ._ or -{x.x1 -xj,)-1,2, ... ,k},

where r = E~=l X~mk-j. Notice once and for all that

(1.5.14)

and that

(1.5.15) Xk = s (mod Pk),

As in §1.2 we can transfer the operation EB from the group G(P) to the unit interval [0,1). Moreover, for each fixed y the identity

(1.5.16)

holds for all but count ably many points x in [0,1). (The proof of this fact is similar to that of 1.2.2.)

We shall make several more observations about the system {Xn(x)}. By using formula (1.5.11) and summing the resulting geometric series, it is not

difficult to verify

(1.5.17) (k) for x E li rpk , r = 0,1, ... , mk-l - 1, (k-l) \ (k) for x E lir lirpk •

The system {Xn(X)} is orthonormal. This will follow from (1.5.8), (1.5.10) when we establish

(1.5.18) r Xn(x) dx = 0, J6(k)

"

To prove (1.5.18) notice for j < k that the function Xmj_l(X) is constant on 6~k-l) and has modulus 1 there. Thus by (1.5.12) we have

Since Xmk_l(X) is constant on the intervals 6i k ), we see by (1.5.14), (1.5.11), and (1.5.15) that this last integral reduces to the sum

26 CHAPTER I

This proves (1.5.18) and shows that the system {Xn(x)} is orthonormal. The Dirichlet kernels for the system {x n (x)} will share the same notation we

used for the Walsh system:

n-l

(1.5.19) Dn(x) = L Xs(x). s=O

(No confusion will arise from this choice of notation. It will be clear from the context which system we are talking about.)

Analogous to (1.4.11), one obtains immediately from definitions (1.5.10) and (1.5.12) that

for n = D:kmk-l + r, 1:::; r:::; mk-l. Here, as before, we have set Do(x) == O. Since

it follows that

(1.5.20)

"'k- 1

D"'kmk_I(X) = L (Xmk_l(x»qDmk_l(x), q=O

"'k -1

Dn(x) = Dmk _1 (x) L (Xmk_1 (x»q + (Xmk_1 (X»)"'k Dr(x) q=O

for n = D:kmk-l + rand 1 :::; r :::; mk-l. In particular, by setting D:k = Pk - 1 and r = mk-l we obtain

Pk- 1

Dmk(X) = D mk _1 (x) L (Xmk_tCx»q. q=O

We have therefore by induction and (1.5.17) that

(1.5.21 ) for x E b(k) o ,

(k) forxE[O,l)\bo ·

We shall now pass to the construction of a continual analogue of multiplicative systems, that is a system whose index set is a continuum.

Notice that the system of functions (1.5.6) can be viewed as a single function of

two variables, namely, X(~,;;,) == Xn(~). From this point of view, the role of both variables in definition (1.5.6) is similar, and the only difference between them is that the second variable does not take values from the whole group G(P), but only from the countable subgroup consisting of finite sequences.


We are interested in a generalization of the function X(~,~) in which the second variable ranges over a continuwn of points instead of the discrete set of points in G(P) consisting of finite sequences. This continuum of points must itself also be a group.

We begin by constructing a class of "continuum" groups whose cartesian products will form the region of definition of the function x. This function can be viewed as an extension of x(~,~) and thus a generalization of multiplicative systems.

Let P be an arbitrary doubly infinite sequence of natural numbers of the form

(1.5.22) P == { ... ,P-j,··· ,P-l,Pl,P2, ... ,Ph···},

where pj ~ 2 for j = ±1, ±2, .... (For convenience we have omitted the index 0 here.) Define a group G(P) as the set of sequences of the form

(1.5.23) * x = { ... ,x_j, ... ,X-l,Xl,X2, ... ,Xj, ... },

where 0 :5 Xi :5 Pi - 1 for j = ±1, ±2, ... and x-i = 0 for j > k(~) ~ 1. Define a group operation EB on G(P) by using (1.5.3) with one difference: the index j takes on all integer values except zero. Clearly, the group G(P) consists of sequences which are only infinite to the right.

Let pI represent the reverse sequence of P, i.e.,

(1.5.24) '1"11 _ {' I I I I } r = ... ,P-j, ... ,P-l,Pl,P2, ... ,Pj, ...

where

(1.5.25) I Pj = P-j, j = ±1,±2, ....

Thus the group G(PI ) consists of sequences of the form

(1.5.26)

where 0 :5 xj :5 P_j - 1 for j = ±1, ±2, ... and x' _j = 0 for j > k(~I) ~ 1.

Define a functionS of two variables x( ~';') for each (;, ;1) E G(P) X G(PI ) by

(1.5.27) *(* *1) 2. ~ XjX -j ~ X_jX j { (k(;') I k(;) I ) }

X x, x = exp 7rZ L..J --- + L..J --- . j=l Pi j=l P-j

5 X(;, ;') as a function of the first variable is a character of the locally compact group G(1').

28 CHAPTER 1

Let

j j

(1.5.28) ma == 1, mj == IIps, m_j == ITp-s, j = 1,2, .... s=l s=l

In view of (1.5.25) we have

j j

m~ = ma = 1, mj == IIp~ = m_j, m'-j == IIp'-s = mj s=l s=l

for j = 1,2, .... We shall describe transformations which take the groups G(P) and G(PI ) into

the set of real numbers. In contrast to the groups G(P) which were by (1.5.9) identified with the unit interval, these groups G(P) and G(PI ) will be identified with the positive real axis. To accomplish this, correspond each sequence (1.5.23) to a series of the form

(1.5.29)

and each sequence (1.5.26) to a series of the form

(1.5.30)

The transfonnation Ap which takes an element ~ E G(P) to the sum of the series

(1.5.29) (respectively, the transfonnation Api which takes an element ;1 E G(PI) to the sum of the series (1.5.30)) is a map from the group G(P) (respectively, G(PI ))

onto the positive real axis [0,00). Moreover, these transformations fail to be 1-1 only at P-adic rationals (respectively, P'-adic rationals). If, in analogy with the modified interval [0,1 )p, we form the modified rays [0,00);' and [0,00);', in which each P- adic rational (respectively, P'-adic rational) has been split into two points, then the groups can be mapped in a 1-1 fashion onto these modified rays. Thus we obtain a suitable geometric model for the groups G(P) and G(PI ).

The usual ray [0,00) can be mapped in a 1-1 fashion into the groups G(P) and G(PI ) by transformations gp and gp,. These transformations are defined in the following way. Let gp : x -. {Xj}±j=l and gpl : x' -. {xj}±j=l' where Xj and xj are the coefficients of the corresponding expansions: (1.5.29) for x, and (1.5.30) for x'. Here, as before, we agree to take the finite expansion when x is a P-adic rational or x' is a P'-adic rational.


These coefficients are determined by the equations

(1.5.31 ) { XJ,. = [xmj] (mod Pj), X_j = [x/m1_j]

Xj = [xm_j] (mod P_j), X'-j = [x'/mj_1]

(mod P_j),

(mod Pj)

for j = 1,2, ... , where [a] represents the greatest integer in a. Thus,

(1.5.32) {X = 2:7~x{ X-jm1_j + 2:j:1 xj/mj == [x] + {x},

, ",k(x' ), ",=, / - ['] {'} x = L.Jj=1 X_jmj_1 + L.Jj=1 Xj m_j = x + x .

29

(Here, {a} represents the fractional part of a number a.) Using this notation, we define a function on [0,00) X [0,00) by

(1.5.33) X(x, x') = X(gp(x),gPI(x')) = exp {27ri (kf XjX'-j + ~ Xj:_j) } . j=1 PJ j=1 P J

We shall make a number of observations about the function X(x, x') and the corresponding function xC:':':, ;').

1.5.1. X(O, x') = X(x,O) = Ix(x, x')1 = 1. Similarly, if Op is the zero element of the group G(P) and Opl is the zero element of the group G(P') then X(Op, :':':') =

X(X,Opl) = Ix(;,;')1 = 1.

1.5.2. Let P be a sequence of the form (1.5.22) and consider the sequences P = {Pj}~1 and P' = {Pj}~1' where pj = P_j. Ifx' = n for some non-negative integer n then X(x, n) = Xn({ x} )(P), where (Xn(x ))(P) is the multiplicative system of type (1.5.10) determined by the sequence P. If x = n for some non-negative integer n then x(n, x') = Xn( {x'} )(P/), where (Xn(x ))(P/) is the multiplicative system of type (1.5.10) determined by the sequence P'.

(Similar properties hold also for the function X(;,;').) PROOF. These facts follow directly from the definitions, since if the fractional

part of one of the variables in (1.5.33) is zero then the value of the function does not depend on the integer part of the other variable .•

1.5.3. Using the notation introduced in 1.5.2,

X(x, x') = X( {x}, [x'])· X([x], {x'}) = X[xll( {x })(P) . X[xl( {x'} )(P/).

(This identity shows that X(x,x') is the cross product system of Xn({x'})pl and Xm({x})p. (See Vilenkin, Zotikov [1].)

PROOF. To verify this identity it is enough to write (1.5.33) in the form

x(x, x') = exp (27ri ~ XjX'-j) . exp (27ri ~ XjX~j) j=1 PJ j=1 p-J

and combine (1.5.32) with 1.5.2. I

30 CHAPTER 1

1.5.4. Let k ~ 1. For each fixed x, < mk (where mk is defined by (1.5.28)), the function x(x, x') is constant in x on intervals of the form (1.5.13), r = 0,1, .... For each fixed x < m-b the function X(X,X') is constant in x' on intervals of the form

r = 0,1, ....

PROOF. Fix x' and observe that the second factor in the statement of 1.5.3 is constant in x on the interval [[x], [x] + 1). Similarly, the first factor is by 1.5.2

constant on each 8~k). This proves the first half of 1.5.4. The second half is proved in a similar way .•

1.5.5. The function x(~, ~I) is multiplicative in each of its variables, i.e.,

and x(~, Y)x(~,;) = X(~, Y 8 ;), x(~, y)x(;, y) = X(~ 8;, y).

PROOF. This result follows directly from definition (1.5.27) .• Notice that these identities generalize the identities (1.5.7) and (1.5.8). An analogue of 1.5.5 holds for the function X(x, x') with the same reservations

concerning the domain that were imposed for equation (1.5.16). Specifically, for each fixed x and y

(1.5.34) x(x, Y)X(z, y) = X(x EB z, V), X(x, Y)X(z, y) = X(x 8 z, y)

hold for all but count ably many z in [0,00). For certain applications involving the multiplicative function X(x, x'), it is im

portant to impose an additional symmetry assumption on the sequence (1.5.22), namely that

(1.5.35) P-j = Pi> j = 1,2, ....

In this case the group G(P') coincides with the group G(P), and clearly both groups are completely determined by the sequence P = (PI ,P2,'" ,Pj,"')' In addition to

properties 1.5.1 -1.5.5, the corresponding functions x( ~)I) and X(x, x') also satisfy the following symmetry conditions:

(1.5.36) *(* *1) *(*, *) Xx,x =xx,x, x(x,x' ) = X(x',x)

As an analogue of Dirichlet kernels for the multiplicative function X(x, x'), we introduce the kernel

(1.5.37) X'

DXI(X)=D(x,x' )= 1 x(x,t)dt.

(The sign v appears above the variable over which we integrate.) The connection between the Dirichlet kernel D( x, x') and the kernel Dn (x )(P)

defined in (1.5.19) is given by the following formula:

(1.5.38) for 0 ::; x < 1,

for x ~ 1.

We shall prove this formula. For 0 ::; x < 1 we use 1.5.3 and 1.5.1 to obtain

x'

D(x,x') = 1 X[t]({X})(P)X[x]({t})(p/)dt

1[,,/] j[X/]+{X/}

= X[t](X)(p) dt + X[t](X)(P) dt o [x']

[x/]-1 ( n+1 )

= ~ 1 Xn(X)(p) dt + {x'}x[X/](X)(P)

= D[x/] (x )(P) + {x' }x[X/] (x )(P)·

On the other hand, let x ~ 1. Since

I n +1

n Xk( {t} )(P/) dt = 0

for k = 1,2, ... , n = 0,1,2, ... , we have

[X']

D(x,x') = 1 X[t]({X})(P)X[x]({t})(p/)dt

j[X/]+{X / }

+ X[t]( {x })(P)X[x]( {t})(P/) dt [x']

[x/]-l ( n+1 )

= ~ 1 Xn({X})(P)X[x]({t})(p/)dt

j [X/]+{X / } + X[x/]({X})(P)X[x]({t})(p/)dt

[x']

j[X/]+{X /}

=X[X/]({X})(P) X[x]({t})(p/)dt [x']

l {X / }

=X[X/]({X})(P) 0 X[x](t)(p/)dt.

This proves formula (1.5.38).

32 CHAPTER 1

If we interchange the roles of x and x' we obtain another generalization of the Dirichlet kernels for the function X(x, x'):

(1.5.39) DX(X') == D(x,x' ) == l x X(t,x')dt,

Here x plays the role of an index and x' that of a variable. For this kernel, the formula analogous to (1.5.38) turns out to be:

for 0 ~ x' < 1,

for x' ~ 1,

where D[xJ(x')(PI) is the kernel of the form (1.5.19) for the system {Xn(x')}(PI). We close this chapter with the following theorem.

1.5.6. For each k ~ 0 the systems

and

{ Vk(X) == ~X (x, _v )}OO ym_k In_k v=o

are uniformly bounded and orthonormal on the respective intervals [0, mk) and [0, m-k).

PROOF. We shall verify the result for the first system. The proof for the second system is similar.

Clearly,

rmk X (~,y) X (~,y) dy = rmk

1 X (~,y) 12 dy = mk Jo mk mk Jo mk

for each v. If v =I- J-l then by the second equation in (1.5.34) and identities (1.5.37) and (1.5.38) we have

rmk X (~,y) X (~,y) dy = l mk X (~8 ~,y) dy Jo mk mk 0 mk mk

=D(~8~,mk) mk mk

= { Dmk (v/mk 8 J-l/mk\p) for 0 ~ v/mk 8 J-l/mk < 1,

o for v/mk 8 J-l/mk ~ 1.


But v =f. p, so v/mk 8 p,/mk ~ 1/mk. In particular, v/mk 8lt/mk rt c5i. Thus by (1.5.21) Dmk (v/mk 8 p,/mk) = O. Consequently,

Since the uniform boundedness of this system is obvious, the proof of this theorem is complete. I

Chapter 2

WALSH-FOURIER SERIES. BASIC PROPERTIES.

Within the collection of all Walsh series, Walsh-Fourier series playa crucial role. These are the series of the form (1.4.1) whose coefficients are given by the formula

aj = 11 f(t)Wj(t) dt,

for some integrable function f. From this definition it is clear that the concept. of a Fourier series is intimately connected with the theory of measures and integrals.

For the case considered in §1.2, namely the system {~n(;)} defined on the group G, the formula for the Fourier coefficients is similar:

{ * * * aj = laf(t)wi(t)djJo.

Here the It represents Haar measure for the group G (sec A3.5) and the integral is meant in the sense of Lebesgue. This integral satisfies a fundamental property which is extremely important for the theory of Fourier series, namely it is translation invariant, i.e.,

for any integrable function f and any element; E G. This property, on which many results of Fourier series are based, once again emphasizes that the most natural domain on which to define the Walsh functions is the group G. The theory of Walsh- Fourier series is more elegant on the group G than on the real line or the unit interval because results can be formulated on the group with less restrictions than are necessary to formulate the same results on the unit interval. This is due in part to the fact that the unit interval is not a group under the operation (fl. But, the theory in the group setting requires mastery of certain techniques including the concept of Haar measure, and other closely related ideas. In connection with this, the language of the theory of Walsh-Fourier series on the group may seem at first unnecessarily abstract and perhaps unusual for the uninitiated reader who is interested most of all in the applications.

For this reason we have decided to set forth here the foundations of the theory of Walsh series on the more intuitive version of the Walsh system which is defined on the unit interval [0,1). The reader who has mastered the concepts mentioned above is urged while studying these results about series in the system {wn(x)} to

34

WALSH-FOURIER SERIES: BASIC PROPERTIES 35

constantly have in view the parallel results about series in the system {~n (;)}. In this light, he should interpret wn(x) as ~n(;)' the dyadic intervals 6.;k) as the

equivalence classes b.Jk) of the group G, and all integrals as integrals with respect to Haar measure on G.

In those few cases when translation to the language of the group is not merely mechanical, it will be made sufficiently clear.

§2.1. Elementary properties of Walsh-Fourier series. Formulae for partial sums.

As we mentioned above, the Walsh-Fourier series of a function I, Lebesgue integrable on [0,1), is a series of the form (1.4.1) whose coefficients are given by

(2.1.1) ai = 11 I(t)w;(t) dt.

We notice at once that every uniformly convergent Walsh series must be a WalshFourier series. This follows from the fact that a uniformly convergent series can be integrated term by term and the fact that the Walsh system is orthononnal (see Theorem 1.1.5). In particular, any polynomial can be viewed as the Fourier series of its sum.

We shall presently see that on the unit interval the Lebesgue integral is translation invariant with respect to the operation EB, i.e.,

(2.1.2) l 1I(t EB X)dt= l 1I(t)dt.

This property plays an important role in the study of Fourier series. (We note for each fixed x that the integrand on the left side of (2.1.2) fails to be defined at count ably many points t (see §1.2). However, countable sets are of Lebesgue measure zero and the Lebesgue integral cannot distinguish between functions which differ only on a set of measure zero.)

To prove (2.1.2) we begin by showing that Lebesgue measure is translation invariant under the operation EB, i.e., that for any Lebesgue measurable set E C [0,1) and any x E [0,1) the set E EB x == {t EB x : tEE} is measurable and

(2.1.3) mes(E EB x) = mes(E).

(In the group case this property is built into the construction of Haar measure.) Recall from Theorem 1.2.1 that for each positive integer k, the map 6. (k) _ 6. (k) EBx is a permutation on the collection of dyadic intervals of rank k. Thus if E is a union of non-overlapping dyadic intervals then E EB x is also a union of non-overlapping dyadic intervals (except countably many points) and thus its measure does not change. Since any open set can be written as a countable union of dyadic intervals,

36 CHAPTER 2

(2.1.3) holds for any open subset of [0,1). In view of the definition of measurable sets (see A3.2.3) it follows that (2.1.3) holds for all measurable sets E.

It is now clear that (2.1.2) holds for characteristic functions of any measurable set E C [0, 1) since

11 XE(t) dt = mes(E).

Hence (2.1.2) holds for all simple functions, i.e., for functions of the form L:~1 C;;'(Ej

where Ei are non- overlapping and measurable. Finally, since the Lebesgue integral is defined in terms of limits of integrals of simple functions (see A4.2), we conclude that (2.1.2) holds for any integrable function.

Defore we isolate several fundamental properties enjoyed by 'Walsh- Fourier series, we introduce additional notation.

Analogous to the trigonometric case, the Fourier coefficients (2.1.1) will frequent.ly be denoted by fci). Thus we shall write the Walsh-Fourier series of a function f in the form

00

(2.1.4) L fci)w;(x), i=O

where

(2.1.5) fci) = 11 f(t)Wi(t) dt.

We shall use the symbol ito represent the sequence {i(i)}~o of Fourier coefficient.s of a function f, i.e., the map i -+ fci) which takes the set of non-negative integers into the set of real numbers. We shall also consider i as the ima.ge of a function f under the map f -+ i defined on the set of functions integrable on [0,1). We shall call this transformation the Walsh-Fourier transformation for the interval [0,1). Frequently the transformation f -+ i will be denoted simply by the symbol 1.

We point out several properties which this transformation satisfies.

2.1.1. The transformation i is linear, i.e., if f, g are integrable functions and (\', (3 are real numbers then

(af + (3g) = af + (3g.

PROOF. This property follows immediately from definition (2.1.5) .•

2.1.2. If faCt) == J(t EB a) then


PROOF. Since the integral is translation invariant under EEl and since (tEEla) EEl a = t, we see by identity (1.2.15) that

L(i) = 11 J(t EEl a)w;(t) dt

= 11 J(t)Wi(t EEl a)dt

= Wi(a) 11 f(t)Wi(t) dt

= wi(a)1ci). •

Proposition 2.1.1 shows that under the Walsh-Fourier transformation, the sum of two functions corresponds to the sum of their Fourier coefficients. It is natural to ask whether there is an operation of two functions which corresponds to the product of their Fourier coefficients. In other words, given two integrable functions J and 9

can we find a third function ¢ such that ~(i) = i(i) . g(i) for all i = 0,1,2, .... By looking at polynomials, it is easy to see that such a function ¢ will not be obtained by multiplying the functions J and g. However, such a function can be obtained from a generalized product * which is called the convolution. The cOl1Yolution of two functions J and 9 is defined by the identity

(2.1.6) ¢;(x) = (f * g)(x) == 11 J(t ffi x)g(t) dt.

It is not at all obvious that the integral in (2.1.6) exists and defines an integrable function. We shall show that this function is defined at least for almost every x in [0,1) (i.e., for all x except those in some set of measure zero), is Lebesgue integrable on [0,1), and satisfies the inequality

(2.1.7)

If we denote the norm of the space L([O, 1)) of integrable functions by II . IiI (see A5.2), then (2.1.7) can be written in the form

Since

(2.1.8) 11 l(f * g)(x)1 dx ::; 11 11 IJ(t ffi x)g(t)1 dtdx = 11 I(IJI * Igl)(x)1 d.x,

38 CHAPTER 2

and since each integrable function is almost everywhere finite, it suffices to show that the integral Iol(IJI * Igl)(x) dx is finite. But by Fubini's Theorem (see A4.4.4) translation invariance of the Lebesgue integral we see that

11 11 IJ(t ffi x )llg(t)1 dt dx = 11 11 IJ(t ffi x )llg(t)1 dx dt

= 11Ig(t)1 (11IJ(tffix)ldx) dt

= 11 Ig(t)1 (11 IJ(x)1 dX) dt

= 11 IJ(x)1 dx 11 Ig(t)1 dt.

Hence the integral in question is finite. Moreover, if we combine this inequality with (2.1.8) we conclude that (2.1.7) holds.

We shall now prove that if <1>( x) = (f * g)( x) then

(2.1.9) ~(i) = [(i)g(i), i = 0,1,2, ....

This identity can be obtained by several applications of Fubini's Theorem (see A4.4.4). To apply Fubini's Theorem it is necessary to show, as we did above, that the integrand is absolutely integrable on the square [0,1) x [0,1). Thus by Fubini's Theorem, (1.2.15), and translation invariance of the integral we obtain

~(i) = 11 (f * g)(x)w;(x) dx

= 11 w;(x) 11 J(t ffi x)g(t) dt dx

= 1111 Wj(x)J(t ffi x)g(t) dx dt

= 11 (11 J(tffiX)W;(tffiX)9(t)w;(t)dx) dt

= 11 g(t)Wj(t) (11 J(tffiX)Wj(tffiX)dX) dt

= !(i) 11 g(t)Wj(t)dt

= !(i)g(i).

We shall denote the partial sums of the Walsh-Fourier series of a function J by Sn(x,f) or more briefly by Sn(x). We shall find an integral expression for these

partial sums. Indeed, substitute definition (2.1.5) into the expression for the partial sums of a Walsh-Fourier series (2.1.4). By (1.2.15) we obtain

By the definition of the Dirichlet kernels (see (1.4.8» and translation invariance of the integral, we arrive at the formula

(2.1.10)

for n = 1,2, .... One can also arrive at this formula by starting with the definition of convolution and applying (2.1.9).

The partial sums of Walsh-Fourier series of order 2k are of fundamental importance. For such partial sums it is clear by (2.1.10) and (1.4.13) that

for any k 2': 0 and x E [0,1). If xED.. jkl, t E D..~kl, and x EEl t is defined, then by

Theorem 1.2.1 we have x EEl t E D..;kl (the countable set of values t where the sum x EEl t is not defined does not affect the integration above.) We conclude that

(2.1.11)

It is useful to notice that identity (2.1.11) is a characterization of Walsh-Fourier series. Indeed, if the partial sums S2k of a series (1.4.1) satisfies

(2.1.12) S2k(X) = ~(kl r f(t)dt lD..j I JOlJk )

x E D..jkl, j = 0,1,2, ... ,2k - 1,

for some f E L([O,l» and aU k 2': 0, then the series (1.4.1) is the Fourier series of the function f.

To prove this notice first that each coefficient ai of the series (1.4.1) can be viewed as a Fourier coefficient of any partial sum of this series Sn(x) for n > i. Next, fix i and choose k such that i < 2k. It follows that

(2.1.13)

40 CHAPTER 2

By 1.1.3 the function Wi(X) is constant on each interval 6.;k) and we denote this

constant value by Wi,j. Moreover, the sum S2'(X) is also constant on each 6.;k) and we denote this constant value by 82',j. Consequently, we have by (2.1.13) and (2.1.12) that

2'-1

ai = L r S2k(t)Wj(t) dt j=O J fl.]')

2' -1

= L Wj,j 8 2.,jl6.;k)1 j=O

2' -1

L Wi,j 1 (k) J(t) dt j=O fl.;

2' -1

~ i].) J(t)w;(t) dt = f(i).

We have proved the following theorem.

2.1.3. A ·Walsh series (1.4.1) is the Walsh-Fourier series of some function J, integrable on [0,1), if and only if all partial sums of this series of oreIer 2k satisfy identity (2.1.12).

§2.2. The Lebesgue constants. The Lebesgue constants for the Walsh system are defined by

(2.2.1 )

for n = 1,2, .... As we shall see below, these constants play an essential role in the study of questions concerning convergence of Walsh-Fourier series.

Notice by (1.4.13) that

(2.2.2)

Thus some subsequence of the Lebesgue constants is bounded. We shall show that there are other subsequences which grow without bound. Neyer the less, the following shows that logz n is an upper bound for the growth of any subsequence of Lebesgue constants.

2.2.1. The Lebesgue constants satisfy tile inequality

(2.2.3)

PROOF. For the proof suppose that 2k ::; n ::; 2k+l for some k ~ 2. Recall that

[a, 1) = .6.~k)U (Uf=I.6.~i») and 1.6.~i)1 = 2-i. Hence by estimate (1.4.14) and identity

(1.4.11) we have

k" " k < 1 + "2,-1. T' = 1 +-. - L...J 2

i=1

Since 2k ::; n implies k ::; log2 n, it follows that Ln ::; 1 + (log2 n) /2. In particular, (2.2.3) holds for n ~ 4 .•

The following proposition shows that the order of the upper estimate in (2.2.3) is exact, i.e., for some subsequence of natural numbers {nd the sequence Lnk grows like log2 nk. Moreover, we shall see that this subsequence can be chosen so that 2k ::; nk < 2k+l for each k ~ a. In fact, we shall show that one such subsequence is given by

8 • (2.2.4) " 2" n2. = L...J2 ., _" 2;+1 n2.+1 - L...J 2 , s = a, 1,2, ....

;=0 i=O

2.2.2. If na.tural numbers nk are defined by (2.2.4) then

(2.2.5) k = a, 1,2, ....

PROOF. Since

228+2 - 1 4 n2. = < _22•

3 3'

we have

(2.2.6) k = a, 1,2, ....

Consequently,

(2.2.7) tE[a,I), k=a,I,2, ....

To prove (2.2.5) we show first that

(2.2.8) k=a,I,2, ....

42 CHAPTER 2

We prove (2.2.8) for the case k = 28 by induction on 8, i.e., we shall show that

(2.2.9) 11 1 2_2._2IDn2,(t)1 dt ~ 2(8 + 1), 8 = 0,1,2, ....

For 8 = 0 (2.2.9) is obvious, since no = 1 and

11 3 1 ID1(t)1 dt = - > -.

1/4 4 2

For the general inductive step, suppose that

(2.2.10) 11 1 IDn2(,_I)(t)1 dt ~ -28

2- 2•

for some integer 8 ~ 1. By (1.4.13) we know that ID22.(t)1 = 228 for t E ~~28). On the other hand, (2.2.7) implies IDn2(._I)(t)1 < 228 /3 for all t E [0,1). Thus

(2.2.11) ID22.(t)I-IDn2('_I)(t)1 > ~22" t E ~~2'). But (2.2.4) implies

(2.2.12) n2. = 228 + n2{.-l).

Hence applying (1.4.11) in conjunction with (2.2.11) we obtain

IDn2.(t)1 ~ ID22.(t)I-IDn2('_I)(t)1 > ~22" t E ~~28). Bearing in mind that (2-28-2,2-28) C ~~28), we come to the inequality

2- 2•

(2.2.13) 1 IDn2• (t)1 dt > 4~2-28 . -32 .228 = -21 . 2- 2.-2

Another application of (1.4.11) in conjunction with (1.4.13) and (2.2.10) yields

(2.2.14) 11 11 1 IDn2.(t)1 dt = IDn2(._I)(t)1 dt > -28.

2- 2 • 2- 2•

Combining inequalities (2.2.13) and (2.2.14), we obtain (2.2.9) For the case k = 28 + 1 (8 = 0,1, ... ) we must show

1~2'_3IDn2'+1 (t)1 dt ~ ~ (8 + ~ + 1) . This inequality holds for 8 = 0 since in this case n28+1 = n1 = 2 and

11 ID2(t)1 dt = 2· ~ = ~. 1/8 8 4

The rest of the proof is similar to the case when 8 is even. Thus (2.2.8) is established. Notice by (2.2.6) that log2 nk < k+1. Thus we conclude by (2.2.8) that inequality

(2.2.5) holds as required. I

§2.3 Moduli of continuity of functions and uniform convergence of WalshFourier series.

To study the question of uniform convergence of a Walsh-Fourier series of some function f it is natural to estimate the difference Sn(x,f) - f(x). According to (2.1.10) and (1.4.10), this difference can be written in the form

(2.3.1) Sn( x, f) - f(x) = 11 (J(x (l3 t) - f(x ))Dn(t) dt.

By combining (2.1.11) with the obvious identity f(x) = (l/l.6.jk) I) J~(k) fex) dt, J

we see that for partial sums of order 2k, formula (2.3.1) has the form

(2.3.2) S2k(X, f) - f(x) = ~(k) r (J(t) - f(x)) dt l.6.j I J~?)

This leads us to the following theorem:

2.3.1. If f is a function continuous on the interval [0,1] then the subsequence of partial sums {S2k(X, f)} of the Walsh-Fourier series of f converges to f uniformly on [0,1).

PROOF. Let E > O. Since f is uniformly continuous on [0,1], choose 0 > 0 such that If(t) - f(x)1 :::; f for It - xl < o. Choose ko so large that Tko < o. Notice

that if points t and x belong to the same .6.Jk) for some k ~ ko then It - xl < o. Consequently, it follows from (2.3.2) that

k ~ ko .•

We can strengthen this theorem by determining how the rate of the approximation to the function f by the partial sums S2k (J) depends on the smoothness of the function f as measured by the modulus of continuity of the function f, i.e., the quantity

(2.3.3) w(o, f) = sup If(t) - f(x)l· It-xl9 t,xE[O,1)

It is apparent from (2.3.2) that

(2.3.4) IS2k (X, f) - f(x)1 :::; w (2\ ' f)

for all x E [0,1). This formula allows us to obtain the necessary estimates for various classes of functions of a given smoothness, for example the Lipschitz classes. Recall that a function f belongs to the Lipschitz class of order Q for some Q > 0, if w(o,!):::; COOl, where the constant C does not depend on o. Clearly, (2.3.4) implies the following proposition:

44 CHAPTER 2

2.3.2. If f belongs to the Lipschitz class of order a, then

The definition (2.3.3) of the modulus of continuity uses It - xl to measure the distance between x and t. In view of the importance of the operation EB for the theory of Walsh series, it is natural to look at another modulus of continuity. This one is defined using the metric p*(x, t) (see idC'lltity (1.2.19» in the following way:

(2.3.5) (;;(8,f) = sup If(t) - f(x)l· p' (t,x)<6 t,xE[O,l)

Since It - xl ~ pOet, x) (see (1.2.20», the usual modulus of continuity and the onc just defined are related by the relationship

(2.3.6) (;;( 8, f) ~ w( 8, f).

As we mentioned in §1.2, the metric p*(x, t) gives rise to a new concept of cont.inuityon [0,1). vVe shall define, for example, a g('neralization of llniform ("Ont.inllit.y. We shall say that a funct.ion f is uniformly p* -continuous on [0,1) if for cvcry E: > 0 there is a 8> ° such that If(t) - f(x)1 < E: for all p*(t,x) < 8. We leave it to the reader to verify that although a Walsh function may be discontinuolls in thc Ilsllal sense, every function from the Walsh system is llniformly p* -continllous on [0,1).

It is not difficult to velify that if x and t E 6. j k) then p*( t, x) < 1/2k. Hellce it. follows from (2.3.2) that estimate (2.3.4) can be strengthened as follows:

(2.3.7) XE[O,l).

In the same way we can strengthen Theorem 2.3.1 by replacing the condit.ion about continuity of f on [0,1] by the condition that f is uniformly p*-continllous.

The concept of a modulus of continuity and the estimates we obtained above arc easily carried over to the group G using the system {l~d and the distance pc(;,i)) which was introduced in §1.2. We shall not dwell on this in any more detail.

The fact that the partial sums of the form S2k (x, f) converge uniformly whell f is continuous in some sense is closely connected to the fact. that the Lebesguc constants of order 2k are uniformly bounded (see (2.2.2». This can be seen by a closer look at the proof of Theorem 2.3.1 in conjunction with (2.3.2).

We shall now obtain a general estimate which demonstrates the role that the Lebesgue constants play in questions concerning convergence of vValsh-Fourier senes.

WALSH-FOURIER SERIES: BASIC PROPERTIES

2.3.3. If f is integrable on [0,1) and n is any positive integer then

(2.3.8)

where k is determined by the relationship n = 2k + m for m :::; 2k.

PROOF. By (2.3.1) and (1.4.11) we have

(2.3.9)

Sn(x,f) - f(x) = 11 (f(x fIJt) - f(x))D 2 k(t)dt

+ 11 (f(xillt)-f(x))T'k(t)Dm(t)dt

== J1 + J2 ·

45

To estimate the first integral repeat the steps which lead to the inequality (2.3.7). We obtain

(2.3.10) * k Ihl:::; w(1/2 ,f).

The second integral can be written in the form

(2.3.11)

2k_l 2k_l

h = L 1 (k) (f(x ill t) - f(x)) T'k(t)Dm(t) dt == L JJil. j=O ~j j=O

(k) (HI) (HI) Recall that Dm(t) is constant on each.6 j = .62j U .62j+1 and that T'k(t) = 1 r A (k+l) d () r A (k+l) S A (HI) l' lOr t E D2j an T'k t = -1 lOr t E D2j+l' ince t E D2j imp les

t fT\ 1/2H1 E .6 (HI) \II 21+ 1 ,

it follows that

(2.3.12)

JJil = 1 (f(x ill t) - f(x)) Dm(t) dt ~~~+1)

- [ (f(xillt)-f(x))Dm(t)dt JI'(k+1)

.... 2j +,

= [ (f(x ill t) - f(x)) Dm(t) dt J~~~+l)

- [ (f(xilltill k1+1)-f(x))Dm(t)dt J~~~+l) 2

= [ (f(X ill t) - f(x ill till k1+1)) Dm(t) dt J~~~+l) 2

46 CHAPTER 2

But for each t E ~~~+1) the points x El1t and x EB t EB 1/2k+l both belong to the same interval of rank k. Consequently,

* ( 1) 1 p x EB t, x EB t EB 2k+l < 2k '

Therefore, we obtain from (2.3.12) that

(j) * 1 1 1* 1 1 IJ2 I::; w( k' J) IDm(t)1 dt = -we k' J) IDm(t)1 dt. 2 ~(2k+l) 2 2 ~(k)

J J

Summing this estimate over j, we have by (2.3.9) through (2.3.11) that

(2.3.13)

Since Lm ::; Ln + 1 by (1.4.11) and (1.4.13), we continue (2.3.13) to arrive at (2.3.8) .•

Theorem 2.3.3 implies the following result concerning uniform convergence of a given subsequence of partial sums of a Walsh- Fourier series:

2.3.4. Let {n;} == {2ki + mil, mj ::; 2ki, be an increasing sequence of natural numbers. If for some function 1 the condition

(2.3.14) .lim ~ (2~' ,1) Lni = 0 t--+oo I

holds, then the subsequence of partial sums {Sni(X,J)} converges to 1 unifonnly on [0,1).

PROOF. By (1.4.10) it is clear that Ln = fol IDn(t)1 dt ~ 1 for n > 1. Thus

(2.3.14) implies that ~(1/2ki, J) --+ 0 as i --+ 00. In particular,

Consequently, the proof is completed by an application of Theorem 2.3.3 .• Since estimate (2.2.3) holds for all natural numbers n, 2.3.4 contains the following

test for uniform convergence of Walsh-Fourier series which is an analogue of the Dini-Lipschitz test from the classical theory of Fourier series.

2.3.5. If the modulus of continuity of a function I satisnes

then the Walsh-Fourier series of I converges to I uniformly on [0,1).

PROOF. If 4 :s; 2k < n :s; 2k+l, then it follows from (2.2.3) that

Ln :s; log2 n :s; k + 1.

Hence the hypothesis of this theorem implies that (2.3.14) is satisfied by the sequence of natural numbers ni = i .•

For continuous functions, it is more natural to formulate Theorem 2.3.5 in terms of the usual modulus of continuity.

2.3.6. If the modulus of continuity of a function I satisfies

lim w (~'I) Inn = 0, n---+(X) n

then the Walsh-Fourier series of I converges to I uniformly on [0,1).

PROOF. This result follows directly from 2.3.5 if we notice that according to definition (2.3.3), the modulus of continuity is a monotone increasing function of 8 and satisfies w(28,f):S; 2w(8,f). Consequently,

and we finish the proof by appealing to (2.3.6) .• As a special case of Theorem 2.3.4 we see that if a subsequence of the Lebesgue

constants {Ln;} is bounded, then the corresponding subsequence of partial sums {Sn;(x, f)} converges to I uniformly if I is continuous on [0,1] or if I is uniformly p* -continuous on [0,1). (These results contain Theorem 2.3.1 as a special case.)

For the case when the subsequence {Ln;} is unbounded, it is not difficult to verify using the Banach-Steinhaus Theorem (see A5.3.3) that given any point x E [0,1) there is a function, continuous on [0,1], such that the corresponding subsequence of partial sums {Sni(X, f)} of its Walsh- Fourier series diverges at the given point x. We shall not give a more explicit proof of this fact, which is itself a useful exercise which involves application of the Banach-Steinhaus Theorem to the subsequence of functionals {Fi(f)} == {Sn;(x, I)}.

§2.4. Other tests for uniform convergence. We shall now establish a test for uniform convergence of 'Walsh- Fourier series

which is useful in the sense that from it one can deduce simpler and more readily applicable tests for uniform convergence.

48 CHAPTER 2

2.4.1. Let f be uniformly p*-continuous on [0,1) and consider the sum

(2.4.1)

If Tk(X) -+ ° uniformly on [0,1), as k -+ 00, then the Walsh-Fourier series of f converges to f uniformly on [0,1).

PROOF. To estimate the difference Sn(x, 1)- f(x), use identities (2.3.9), (2.3.11), (2.3.12) and inequality (2.3.10). For n = 2k + m, m ~ 2k we obtain

In this last, sum change variables in each integral from t to UJJ2j /2k+1. If we denote by Dm,j the constant value which the kernel D",(t) assumes on the int.erval 6jk), then we obtain

(2.4.2) ISn(.T, I) - f(x)1 ~ ~ (;k ' f)

2·-1 f ( 2j 2j +1) + I ~ Dm,j J,~~.+I) f(x fIJi ffi 2k+1 ) - f(x ffi t ffi 2k+1 ) Dm(t) dt I .

Observe that t 2: j2- k for t E 6;k). Consequently we have by (1.4.16) that

(2.4.3) j = 1,2, ... ,2k -1.

Moreover, Dm,o = m (see 1.4.3). Hence we can estimate the first term of the sum in

(2.4.2) by the quantity m2-k~(1/2k, I), and the remaining terms by using (2.4.3). We obtain

(2.4.4 ) ISn(x,1) - f(x)1 ~ 2~ (21k ,f)

Of course we recognize the sum inside this last integral as Tdx ill t) (see (2.4.1)). Consequently, for any 6 > ° we can choose k sufficiently large so that ITk (x ill t) 1 < 6

for all x and t. Hence the second part of (2.4.4) does not exceed the quantity

2k61D.~k+1)1 = 6/2. Since the first part of (2.4.4) can be made as small as one wishes by using uniform p*-continuity and the modulus (2.3.5), we have completed the proof of this theorem. •

This theorem gives another proof of the Dini-Lipschitz test formulated in 2.3.5 (or in 2.3.6). Indeed,

holds for all natural numbers k. In particular, the hypotheses of Theorem 2.3.5 imply those of Theorem 2.4.l.

Another corollary of Theorem 2.4.1 is the following result \vhich is an analogue of the Jordan test for trigonometric series.

2.4.2. If f is uniformly p* -continuous on [0,1) and is of bounded variation on [0,1) then its Walsh- Fourier series converges to f uniformly on [0,1).

PROOF. Recall that a function f defined on some interval (a, b) is said to be of bounded variation if the sums 2.:;=1 If(bj ) - f( aj)1 are bounded by some ahsolute constant for any collection {(a j, bj )} ;=1 of non- overlapping interyals lying in (a, b). The supremum of such sums taken over all such collections of non-overlapping intervals lying in ( a, b) is ca.lIed the variation of f on (a, b) and will be denoted by V;[JJ.

It is not difficult to verify that as j ranges over the integers 0,1, ... , 2k - 1, the points x ill (2j /2k+l) and x ill ((2j + 1)/2k+1) belong to different intervals of rank 1.: and thus generate non-overlapping intervals in [0,1). Consequently,

(2.4.5) 2~1 I ( 2j) ( 2j + 1) I /1 [ ~ f x ill 2k+1 - f x ill 2k+l :s; ~ 0 fl J=m

for all integers 0 :s; m < 2k.

Choose a non-decreasing sequence of natural numbers {md sHch that mk < 2k-l for each k and mk -7 00 as k -7 00 but w(1/2k,f)lnmk -70 as k -7 00. This is always possible if we choose {mk} tending to 00 sufficiently slowly, namely, if we set mk = eo«w(2- k ,f)-1).

Divide the sum Tk(X) (see (2.4.1)) into two pieces and apply (2.4.5). V-le obtain

50 CHAPTER 2

the estimate

* (1 ) mk 1 1 1 :::; W ?k' f L -;- + -+ 1 Vo (f]

~ j=l J mk

By the choice of the sequence mk it is now apparent that T" (x) tends to zero uniformly as k ---+ 00. It remains to apply 2.4.1 .•

We notice by (2.3.6) that Theorems 2.4.1 and 2.4.2 still hold if p*-continuity is replaced by continuity ( in the usual sense) on the interval [0,1]. Since every monotone bounded function is of bounded variation, we notice also that Theorem 2.4.2 is valid for such functions.

§2.5. The localization principle. Tests for convergence of a Walsh-Fourier series at a point.

A generalization of definition (2.3.5) is the concept of the integral mori1J,l'ILs of continuity, namely, the expression

(2.5.1 ) ~(1)(8,f) = sup t Jf(x EB h) - f(x)J dx. h<6 io

Such a modulus of continuity satisfies two basic properties.

2.5.1. If </> is continuous on [0,1] and w(8,f) is its usual modulus of continuity, then

PROOF. It is enough to notice that p*( x EB h, x) :::; 8 for h :::; 8 and thus from (2.3.5) and (2.3.6) we obtain

J</>(x EB h) - </>(x)1 :::; ;:;(8,</»:::; w(8,</».

The fact that for each fixed h the sum x EB h may not be defined for count ably many values of x (see §1.2) does not play any part in the integration .•

2.5.2. If f is Lebesgue integrable on [0,1) then

lim ~(J)( 8, f) = 0. 6 ...... 0

PROOF. By a property of the Lebesgue integral (see A5.2.1), for any c > 0 we can choose a function 4>, continuous on [0,1]' such that

11 If(x) - 4>(x)1 dx < c.

Since the integral is translation invariant under ffi, we obtain

11 If(x ffi h) - f(x)1 dx :::; 11 If(x ffi h) - 4>(x ffi h)1 dx + 11 If(x) - 4>(x)1 dx

+ 11 14>(x ffi h) - 4>(x)1 dx

:::; 2c + 11 14>(x ffi h) - 4>(x)1 dx.

Thus by 2.5.1 we see that

Since 4> is continuous, the expression w( 0,4» can be made as small as one wishes for sufficiently small O. This proves 2.5.2 .•

We shall now establish that the behavior on some interval of the partial sums Sn(x) of a Fourier series of an integrable function is totally determined, to within a quantity which tends to zero as n -t 00, by the values of that function on that interval.

2.5.3. If f is integrable on [0,1) and s is a natural number then

1 1 1 1 * (1) (1 ) Sn(x,f)- f(xffit)Dn(t)dt :::; --(-) w 2k ,f, L>.~,) 21t1os I

where k is determined by n = 2k + m, 1 :::; m :::; 2k.

PROOF. Use the integral representation of S71(x,f) (see (2.1.10)) to write

Thus it suffices to show that

(2.5.2) 11 1 1 *(1)(1 ) f(x ffi t)Dn(t) dt :::; --( -) w 2k ' f , [O,1)\L>.b') 21t1os I

n 2:: 2'.

52 CHAPTER 2

Combine formula (1.4.11) for the Dirichlet kernel with (1.4.13) which shows that

D2k(t) = a for t E [0,1) \ 6.~8) and k ~ s. Thus verify that Dn(t) = rk(t)Dm(t) for

t E [0,1) \ 6.~8). Notice by (1.4.14) that IDm(t)1 ~ 28- 1 for the same t and

2'-1

~~8) = U ~jk). j=O

Repeating the steps which lead to (2.3.12), we obtain

2k_1

I f f(x ffi t)Dn(t) dt I ~I L f f(x ffi t)Dn(t) elt I J[O,I)\A~') j=2' JAY)

2k -I ~ "28 - 1 f If(xffit)- f(xffitffi k1+1)ldt ~ JA(k+l) . 2

J=2' 2j

~ 28 - 1 fl If(x ffi t) - f(x ffi t ffi ')k~1 )1 elt Jo ~

= 2 8 - 1 11 If(t) - f(t ffi 2k~1 )1 elt

< 28 - 1c:,(1) (~ f) - 2k '

__ 1_c:,(1) (~ ) - 21~~8)1 2k ' f .

This verifies inequality (2.5.2) and completes the proof of this theorem. I An application of 2.5.3 gives two results which make up the localization principle

for Walsh-Fourier series.

2.5.4. If a function f, integrable on [0,1), vanishes identically on some dYAdic inter

val ~j8) then the Walsh-Fourier series of this function converges to zero unifol1lJly

on ~j8). Moreover,

(2.5.3) 1 *(1)(1 ) ISn(x,1)1 ~ 216.;8)l w 2k ,f, x E ~ (s) n ~ 28 ,

J '

where n = 2k + m, 1 ~ m ~ 2k.

PROOF. It is clear that x ffi t E ~;8) when x E ~;S) and t E 6.~8). Consequently, inequality (2.5.3) follows immediately from Theorem 2.5.3 and the hypotheses of this theorem. Moreover, uniform convergence to zero of the snms SI1(x, 1) follows directly from the property 2.5.2 of the integral modulus of continuity. I

2.5.5. If two integrable functions I and 9 coincide on some dyadic interval!::;. then their Walsh-Fourier series are uniformly equiconvergent on !::;., i.e., tIle difference of these series converges uniformly to zero on the interval !::;..

PROOF. It is enough to apply Theorem 2.5.4 to the difference I - 9 and recall from 2.1.1 that the Fourier series of the difference I - 9 is the difference of the Fourier series of the functions I and g .•

In particular, we see that if two integrable functions coincide on some open interval containing a point x, then the Walsh-Fourier series of these functions are equiconvergent at the point x, Le., at the point x either both these series converge or both these series diverge.

It is easy to see that the localization principle allows us to obtain local versions of the tests for uniform convergence which were proved in §2.3 and §2.4. The following result is a local version of Theorem 2.4.2.

2.5.6. If I is integrable on [0,1), uniformly p*-continuous on some dyadic illtCl"val !::;., and of bounded variation on !::;., then its Walsh-Fourier series converges to I uniformly on !::;..

PROOF. Clearly, the function

hex) = { ~(x) for x E !::;.,

for x E [0, 1) \ !::;.

is uniformly p*-continuous and of bounded variation on [0,1). Consequently, by Theorem 2.4.2 the Fourier series of h converges to h uniformly on [0,1). In particular, this series converges to I uniformly on !::;.. It remains to apply Theorem 2.5.5 to the functions I and h .•

We come now to a condition sufficient for convergence of a Walsh- Fourier series at a point which is an analogue of the Dini test for Fourier series.

2.5.7. The Walsh-Fourier series of an integrable function I converges at some point x to a, value c if the function (f(u) - c)j(u - x) is Lebesgue integra,bie near x, i.e., if

(2.5.4) l X +D I/(u) - cl d I I ,U < 00

x-Ii U - x

for some b > 0

PROOF. Since the indefinite Lebesgue integral is absolutely continuous, given c: > 0 we can choose s sufficiently large so that x E !::;.} s) C (x - b, x + b) and

(2.5.5) 1 I/( u) - cl d c: tt < -

6.(.) lu-xl 2· J

54 CHAPTER 2

If x E ~Js), t E ~~s), and x EEl t is defined then x EEl t E ~ jS) (see 1.2.1). Thus by the same kind of proof we used to establish translation invariance of the integral (see (2.1.2)), we can prove that

(2.5.6) f If(XEElt)-cl d f If(tt)-c l d JA~') I(x EEl t) - xl t = JA;') Itt _ xl tt.

But by (1.2.20), I(x EEl t) - xl :5 p*(x EEl t,x) = t. Thus it follows from (2.5.6) and (2.5.5) that

(2.5.7) f If(x EEl t) - cl dt < f If(x EEl t) - cl dt < ~. JA (·) t - JA (,) I(x EEl t) - xl 2

o 0

In particular, it follows from the estimate (1.4.16) for the Dirichlet kemel that

(2.5.8) f If(x EEl t) _ cIIDn(t)1 dt:5 f If(x EEl t) - cl dt < ~ JA~') JA~') t

for any n.

Apply Theorem 2.5.3 to the function f(x) - c and the interval ~~s). Fix sand choose ko 2: s so large that

_l_W(l) (~ r) < ~ 21~~s) I 2k >' - 2

for all k 2: ko. Thus by (2.5.8) and the estimate from Theorem 2.5.3 we have

e e ISn(x, f) - cl = ISn(x,j - c)1 < 2 + 2 = e,

This completes the proof of 2.5.7. I In particular, we see that the Walsh-Fourier series of a function f converges to

f(x) at a point x if the inequality

(2.5.9) If(u) - f(x)1 :5 cltt - xl'"

is satisfied for some 0' > 0 and fj > 0 and all u such that Itt - xl :5 fj. Condition (2.5.9) is obviously satisfied at each point where the function f is differentiable. Thus we have proved the following result:

2.5.8. If f is integrable on [0,1) then the Walsh-Fourier series of f converges to f( x) at every point x where f has a finite derivative.

In §2.3 we remarked that the subsequence of partial sums of order 2k playa fundamental role in questions about uniform convergence. Criteria for convergence of this subsequence at a given point are also of considerable interest and distinguished by unusual simplicity.


2.5.9. Let {.6.~k)} == {[Ok, ,Bk)}r;o be the sequence of dyadic intervals which sa.tisfy

x E .6.~k) for all k ~ 0 and let f be integrable on [0,1) with indefinite integraJ

F(x) = l x f(t)dt.

The subsequence of partia.l sums {S2k (x, J)} of the WaJsh- Fourier series of f at a point x converges to a number a if and only if

(2.5.10) lim F(,Bk) - F(Ok) = a. k->oo ,Bk - Ok

PROOF. For the proof it is enough to notice by (2.1.11) that

The limit in (2.5.10) is called the derivative with respect to binary nets {Nd (see §1.1), or the {Nd-derivative, and will be denoted by DV.,fdF(x). Hence

(2.5.11)

The usual concept of differentiation and differentiation with respect to binary nets are related in the following way:

2.5.10. IfF is differentiable at a point x with F'(x) = f(x) then F has a derivative with respect to binary nets and D{N'dF(x) = f(x).

PROOF. Since the derivative F'(X) = f(x) exists, we know that

and

as k -t 00. Subtracting the second expression from the first we obtain

This completes the proof of the theorem .• Combining 2.5.9 with 2.5.10, we are lead to the following test:

56 CHAPTER 2

2.5.11. The subsequence of partial sums {S2k (X, f)} of the Walsh-Fourier series of an integrable function f with indefinite integral F converges at each point where F is differentiable, in which case

In view of the theorem about almost everywhere differentiability of the indefinite Lebesgue integral (see A4.4.5), namely F'(x) = f(x) almost everywhere, the following result is a corollary of the preceding test.

2.5.12. If f is integrable on [0, 1) then the su bsequence of partial sums {S2 k (x, f) } of its lYalsll- Fourier series converges to f almost everywhere 011 [0,1).

§2.6 The Walsh system as a complete, closed system. Beginning with this section, we will systemat.ically use the following standard

notation for classes of integrable functions, each of which is a normedlinear space (see A5.1 and A5.2).

For any p 2 1 denote by LP = LP [0, 1) the set of measurable functions f for which the quantity

(2.6.1) ( I ) lip

IIfllp = llf(xW dx

is finite. This quantity satisfies all the properties of a norm if in LP we do not distinguish between functions which coincide almost everywhere (more details about this can be found in A5.2).

For the space Ll we shall frequently use the simpler notation L or L[O, 1). The space offunctions continuous on [0,1] will be denoted by C[O, 1] and its norm

by

(2.6.2) Ilflle = sup If(x)l· o~x9

We shall also use the symbols C[O, 1), respectively Cp.[O, 1), to denote the space of functions uniformly continuous, respectively uniformly p*-continuous (see §2.3), on [0,1). These spaces use the same norm (2.6.2).

By 1.1.5 the Walsh system is orthonormal on [0,1). As is the case with any orthonormal system, it is very important to verify that a function is uniquely determined by its Walsh-Fourier coefficients. Since changing a function on a set of measure zero does not affect the value of the integral which defines its Fourier coefficients, we can only speak of uniqueness with precision up to values on a set of measure zero. In the case of continuous functions this restriction is not necessary. We shall see that the theorem of uniqueness does hold for the Walsh system, namely, we shall establish the following result:

2.6.1. 1) If f E L[O,l) and fii) = ° for i = 0,1,2, ... , then f(x) ° almost everywhere;

2) if f E Cpo [0, 1) and !<i) = ° for i = 0,1,2, ... , then f(x) = ° for x E [0,1).

The theorem of uniqueness for an orthononnal system is connected with the completeness of that system. Recall that an orthonormal system of functions is sajd to be complete in the space Cpo [0, 1) ( or in LP[O, 1)), if there does not exist a non-zero function ( that is a function different from zero on some set of positive measure) which is orthogonal to each function in that system.

It is clear that the property of completeness is simply a reformulation of the theorem of uniqueness and thus Theorem 2.6.1 is equivalent to the following result:

2.6.2. The Walsh system is complete in the spaces Ll [0, 1) and Cpo [0,1).

We shall prove 2.6.1 and thus 2.6.2. PROOF. Since

11 f(t)w;(t) dt = 0,

it is evident that

i=0,1,2, ... ,

11 f(t)Tn(t) dt = ° for any Walsh polynomial Tn(t). Notice by (1.2.15) that for each fixed x the translated Dirichlet kernel Dn(x Ell t) is a Walsh polynomial. Consequently, it follows from the previous identity that

k = 0,1,2, ....

But the left side of this expression is a partial sum of the Fourier series of the function f at the point x ( see (2.1.10)). Hence

k = 0,1,2, ... ,

i.e.,

for every x E [0,1). But by Theorem 2.5.12

almost everywhere so we have f(x) = ° almost everywhere on [0,1). In the case when f is p*-continuous, it follows that f(x) = ° everywhere on [0,1) .•

It is clear that 2.6.1 1) holds if we substitute any space of integrable functions for the space L[O, 1), in particular, it holds for the spaces LP[O, 1), p ~ 1. Thus,

58 CHAPTER 2

2.6.3. The Walsh system is complete in the spaces LP[O, 1), P 2: 1.

In particular, the Walsh system is complete in the space L2[0, 1). For this space, a given system is complete if and only if it is closed (see A5.4.1). Never the less, it is easy to prove directly that the Walsh system is closed in L2[0, 1). Indeed, we will use Theorem 2.3.1 to prove that the Walsh system is closed in every LP, p 2: l.

Recall that a system of functions is said to be closed in the space LP if given I E LP and c > 0 there is a polynomial T( x) in this system which satisfies

(2.6.3) 11/(x) - T(x)lIp < c.

It is well known (see A5.2.1) that the collection of continuous functions is dense in the space LP[O, 1) for each p 2: 1. But by Theorem 2.3.1 any continuous function 1> can be approximated uniformly, hence in the LP [0, 1) norm, as closely as one wishes by the partial sums 52. (x, 1», i.e., by a polynomial in the Walsh system. Thus given I E LP[0,1) and c > 0 choose first a continuous function 1> such that II! -1>llp < c/2, and then a Walsh polynomial T such that 111> - Tllp < c/2. By the triangle inequality, it follows that (2.6.3) holds. In particular, we have proved the following result:

2.6.4. The Walsh system is closed in each space UfO, 1) p 2: 1 and in qo, 1).

Using other terminology, this result can be stated as follows.

2.6.4'. The set of Walsh polynomials is dense in each space LP[O, 1) P 2: 1 and ill qO,1).

Since the Walsh system is closed in the space UfO, 1), the Parseval identity holds (see A5.4.2).

2.6.5. If IE UfO, 1) then

f 11(iW = 11 I/(tW dt. ;=0 0

PROOF. For any polynomial Tn = E7=00'itl'i(t) and any function IE U[O,1) we have

(2.6.4) III - T.lll ~ l' (/(t) - t,a,w'(t)), dt

= 11 If(tW dt - 2 ~ O'J(i) + ~ IO'il 2

= 11 I/(tW dt - ~ 11(i)12 + ~(1ci) - 0';)2.

In particular, if O:i = !(i), i.e., the polynomial Tn is taken to be the partial sum Sn(J) then

(2.6.5) II! - Sn(J)II~ = 11 1!(tW dt - t li(iW· o i=O

Therefore, L:~=o li(i)12 < fol 1!(t)12 dt and it follows that the series L:~o li(i)1 2

converges and satisfies

(2.6.6) f li(iW S; 11 1!(tW dt. i=O 0

On the other hand, let c: > O. Since the Walsh system is closed we can choose a Walsh polynomial T such that II! - TII2 < -..ft. Therefore, by (2.6.4) we have

11 1!(t)12 dt - ~ l!(i)12 < c:

for any natural number n. We conclude by (2.6.6) that

11 1!(tW dt < c: + ~ l!(iW S; c: + ~ li(iW S; c: + 11 1!(tW dt.

Since c: was arbitrary, this completes the proof of the theorem .• We notice that Parseval's identity and (2.6.5) imply that the Walsh-Fourier series

of any function! E L2 [0,1) converges to it in the L2 [0,1) norm. Of course the Walsh system is not special in this regard; according to the Riesz-Fischer Theorem (see A5.4.2), this holds for any complete orthonormal system.

We close this section with some properties of the system {Wi2m(t)}~0 which will be used in Chapter 9. By (1.1.9), Wi2m(t) = wi(2mt). Thus the system {Wi2m(t)}~0

can be viewed on each interval fJ.jm) as a contraction of the Walsh system in the variable t by a factor of 2m • It is easy to verify that this contracted Walsh system inherits the properties of completeness and closure from the original system. Relationship (1.1.10) shows that this system is orthogonal on fJ.;m) , and can be

normalized by multiplying each function by 2m /2. Thus the following result is true:

2.6.6. On eadl interval fJ.;m) of rank m, the system {2m/2Wi2m (t)}~o is a complete

orthonormal system in the space LP(fJ.jm»).

§2.7. Estimates of Walsh-Fourier coefficients. Absolute convergence of Walsh-Fourier series ..

We shall obtain simple estimates of how rapidly Walsh-Fourier coefficients decay which will, in particular, show that Walsh- Fourier coefficients always tend to zero. These estimates are stated in terms of the modulii of continuity (2.3.3) and (2.3.5), and the integral modulus of continuity (2.5.1).

60 CHAPTER 2

2.7.1. Let 2k :::; 11 < 2k+1. Then the Walsh- Fourier coefficients of any f E L[O, 1) satisfies the inequality

(2.7.1)

PROOF. From the definition of the Walsh functions and from identity (1.1.5) it is evident that w n (2- k- l ) = -1 for 2k :::; 11 < 2k+l. Applying 2.1.2 for a = 2- k - 1

we obtain the following formula for the Walsh-Fourier coefficicnts of thc function f( t EB 2- k - 1 ):

Consequently,

(2.7.2) 21(11) = 11 (J(t) - f(t EB T k - I ») wn(t) dt,

and it follows that

l1(n)1 :::; ~ 11 If(t) - f(t EB Tk-I)I dt :::; ~~(I)(Tk, j). •

In view of 2.5.1, the following result for continuous functions is a corollary of 2.7.1:

2.7.2. TllC Walsh-Fourier coefficients of a function f E C[O, 1) satisfy

~ 1 1 If(11)I:::; '2w(2 k ,j),

and those of a function f E Cpo [0, 1) satisfy

~ 1 * 1 If(n)l:::; '2w(2 k ,j),

for 2k :::; n < 2k+1.

These estimates are in some sense exact (see Efimov [2]), and RubinsteIn [2], [3)). Notice also that Theorem 2.7.2 gives only an upper estimate for how rapidly the Walsh- Fourier coefficients of a function from C[O, 1) and Cpo [0,1) decay. One can ask what is the exact rate of decay for functions in these two classes. It turns out (see Bockarev [2J, [ID that if f E qO,l) and 1(11) = O(dn ) for some dn ! a with L:::I dn < 00, then f(.1:) is identically constant. On the other hand, given

L:~=l dn = 00, there exists a non- constant function fo E C[O, 1) such that 10(11) =

O( dn ). For non-constant functions from C p' [0, 1), the coefficients 1(11) can decay as slowly as one wishes.

Appealing to 2.5.2, another corollary of 2.7.1 is that Walsh-Fourier coefficients converge to zero:

2.1.3. If IE L[O,l) then lim fen) = 0. n_=

For various classes of continuous functions one can obtain estimates sharper than Theorem 2.1.2. Since any function from the Lipschitz class of order a satisfies w( 15, f) ~ C 15<>, it is clear that the Walsh-Fourier coefficients of such functions must satisfy

n -+ 00.

From the result of Bockarev cited above, it follows that this estimate holds for I E C[O, 1) and a ~ 1. For I E Cpo [0,1), a can take on any positive value.

Similar estimates can be obtained for classes of integrable functions, for example, for the integral Lipschitz classes defined with the integral modulii of continuity in place of the continuous one. We shall not give a more detailed statement of these results.

From inequality (2.7.1), one can also obtain estimates for Fourier coefficients of functions of bounded variation. We shall now prove the following result.

2.1.4. If I is a function of bounded variation on [0,1) with total variation VOl [fJ then

*(11 (1) 1I[ w 2k,1 ~2kVo/J.

PROOF. If h < 2-k and t E D.Jk1 then t(f)h E D.Jk1. Consequently, for such hand t we have

I/(t (f) h) - l(t)1 ~ sup I(x) - inf I(x). E~ (k) xE~ (k)

x j J

Moreover, it is clear by the definition of bounded variation (see §2.4) that

It follows, therefore, that

~(11 (2\,J) = sup {I II (t(f)h)-/(t)ldt h<I/2k Jo

2k_I

= sup L ( I/(t (f) h) - J(t)1 elt h<I/2k j=O J~Jk) 2k_l

~ L iD.jk11 j=O

( SUP f(x) - inf J(X») E~ (k) xE~ (k)

x j J

1 1 ~ 2k Vo [fJ. I

62 CHAPTER 2

2.7.5. If f is a function of bounded variation on [0,1) then

n = 1,2, ....

PROOF. Combining estimates 2.7.1 and 2.7.4 we see that

~ 1 1 If(n)1 :5 2k+l Vo [fl·

Since n < 2k+1, the proof of this result is complete. • We now consider absolute convergence of Walsh series. Suppose that a 'Valsh

series ~:o ajwj(x) converges absolutely at some point Xo. Since lajwi(xo)1 = lad for i = 0,1, ... it follows that ~:o aj is absolutely convergent and thus the Walsh series itself converges uniformly and absolutely everywhere on [0,1). Consequently, in contrast to the trigonometric case it does not make sense to study absolute convergence of Walsh series on proper subsets of [0,1). Moreover, when we speak of an absolutely convergent Walsh series it is understood that the series of its coefficients ~:o aj is absolutely convergent.

We shall identify several conditions on a function f sufficient t.o conclude that its Walsh-Fourier series is absolutely convergent, i.e.,

00

(2.7.3) L 11(n)1 < 00.

n=O

In order to state these conditions or prove the corresponding results, it is necessary to introduce a companion to the integral modulus of continuity defined in (2.5.1), namely, the L2_ modultL.!J of continuity of a function f which is defined by

(2.7.4) ~(2)(b,f) = sup IIf(x ffi h) - f(x)1I2. h<6

Analogous to 2.5.1, it is easy to see that

(2.7.5)

Moreover, it is also clear by the Cauchy-Schwarz inequality (see A5.2.2) that

(2.7.6)

The following gives a sufficient condition for absolute convergence.

2.7.6. If J E L2[0, 1) and

00

(2.7.7) L2k/ 2':'(2)(1/2k,J) < 00,

k=O

then (2.7.3) is satisfied.

PROOF. Let 2k :::; n < 2k+ 1 • Observe by (2.7.2) that the n-th Walsh-Fourier coefficient of the function J( t) - J( t E8 (1/2k+l» is precisely 2fc n). Hence by Parseval's identity 2.6.5, applied to the function J(t) - J(t E8 (1/2 k +1 », we obtain

Thus (2.7.4) implies

2 k +'_1

(2.7.8) 4 L IfcnW:::; 1':'(2)(l/2k,JW· n=2'

Use the Cauchy-Schwarz inequality (see A5.2.2) and then (2.7.8). We arrive at the estimate

7;: li(n)I'; t~'li(n)I') 'I' C~' 1') 'I'

:::; 2k/2- 1':'(2)( ;k' J).

Summing this estimate over k we obtain

flfcn)l:::; ~ f2k/2':'(2)(21k,J). n=1 k=O

Since the series on the right is finite by hypothesis, the proof of (2.7.3) is complete. I

2.7.7. If J is a function which satisfies

00

(2.7.9) L 2k/2':'(l/2k, J) < 00,

k=O

then (2.7.3) is satisfied.

PROOF. By (2.7.5) and (2.7.9), the hypotheses of Theorem 2.7.6 are satisfied. Consequently, the Walsh-Fourier series if f converges absolutely. I

64 CHAPTER 2

For continuous functions and the usual modulus of continuity, the condition analogous to (2.7.9) is written in the form

(2.7.10) ~ w(l/n,f) ~ n l / 2 < 00. n==1

2.1.8. lia function J E C(O, 1) satisfies (2.7.10) then its Walsh-Fourier coefficients satisfy (2.7.3).

PROOF. Since the terms of the series (2.7.10) are monotone decreasing, it IS

evident that

~ w(1/n, f) = ~ 2~1 w(l/n, f) ~ k-I w(1/2k, f) ~ nl/2 ~ ~ n l / 2 ~ ~2 2"/2' n==1 k==1 n==2 k - 1 k==1

Consequently, hypothesis (2.7.10) implies that the series L::'I 2k/2w(1/2k, f) converges. In particular, it follows from (2.3.6) that (2.7.9) holds, and thus 2.1.1 applies .•

Theorems 2.1.6 through 2.1.8 allow us to identify conditions on the rate of decay of the modulii of continuity, as a -+ 0, which are sufficient to conclude that a given function has an absolutely convergent Walsh-Fourier series. For example, the following result is true:

2.1.9. The lValsh-Fourier series of a function J converges absolutely if for a > 1/2 a.ny one of the following conditions is satisfied:

~(2)(a,f) = OW'), ~(a,J) = OW'), or w(a,f) = OW'),

as a -+ O.

PROOF. It is enough to not.ice that each of these condi tions forces convergence of the corresponding series in Theorems 2.1.6 through 2.1.8 .•

The following result is anot,her consequence of Theorem 2.1.6.

2.1.10. If J is of bounded variation on [0,1) and satisfies the condition

(2.7.11) f J~(1/2k,f) < 00,

k==O

then the Walsh-Fourier series of J converges absolutely.

PROOF. Clearly,

(1 1 IJ(t ffi h) _ J(t)12 dt) 1/2

< (sup IJ(t ffi h) - J(t)1 fl IJ(t ffi h) - J(t)1 dt)1/2 tE[O,l) Jo

Specializing to the case h < 1/2k we obtain

(2.7.12) ~(2) (~ f) < 2k ' -

* (1 ) * (1) (1 ) w 2k ' f w 2k ,f .

But 2.7.4 implies

Substituting this into (2.7.12) we obtain

~(2) (21k ,f) ~ ;k~ elk ,f) vi[J].

Consequently,

Since by (2.7.11) the series on the right side of this last inequality converges, it follows that the hypotheses of Theorem 2.7.6 hold. In particular, the function f satisfies (2.7.3) .•

2.7.11. Iff is a function of bounded variation whose modulus of continuity satisfies

C:;(o,j) = 0 (In ~)-o)

for some (\' > 2 then its Walsh-Fourier series converges absolutely.

PROOF. The hypotheses imply that

Thus we can use Theorem 2.7.10 .• We notice that the hypothesis of this theorem is satisfied when the function .f is

of bounded variation and belongs to the Lipschitz class of order (\' > O. Here is a criterion for absolute convergence of a Walsh-Fourier series which uses

the concept of convolution, which was introduced in §2.1.

66 CHAPTER 2

2.7.12. Let f E L[O, 1). Then the Walsh- Fourier series of f converges absolutely if and only if f can be written as the convolution of two functions 9 and h whidl belong to the space L2 [0,1).

PROOF. Suppose first that f = 9 * h for some g, h E L2 [0,1). By Parseval's identity

<Xl <Xl

L Ig(nW = Ilgll~ < 00, L Ih(nW = Ilhll~ < 00.

n=O n=O

Thus by (2.1.9) and the Cauchy-Schwarz inequality we have

<Xl <Xl

L l[(n)1 = L Ig(n)h(n)1 ::; IIgl1211 h ll 2 < 00.

n=O n=O

Conversely, suppose that the Walsh-Fourier series of f converges absolutely, i.e., I:~=o l[(n)1 < 00. Choose by the Riesz-Fischcr Theorem (see A5.4.3) functions 9 E L2[0, 1) and h E L2[0, 1) whose Walsh-Fourif'r coefficients satisfy

(2.7.13) g(n) = /1[(n)l, hen) = sgn[(n)/I[(n)l,

and notice that <Xl <Xl <Xl

n=O n=O n=O

Consider the convolution 9 * h. By (2.1.9) and (2.7.13) we have

(2.7.14) (g * h)(n) = [(n), n = 0,1,2, ....

Since by Theorem 2.6.1 functions in L[O, 1) are uniquely determined by their WalshFourier coefficients, it follows from (2.7.14) that the functions f and g*h are identical in the space L[O, 1) .•

§2.8. Fourier series in multiplicative systems. In §1.5 we defined a class of multiplicative systems which contained the Walsh

system as a special case. All the results about 'Walsh-Fourier series which appear in the previous sections of this chapter have analogues for Fourier series with respect to these multiplicative systems. We shall explicitly mention here only the simplest of these results. Other results along these lines can be obtained as corollaries of the theorems in Chapter 6 about multiplicative transformations.

A feature which distinguishes these general multiplicative systems from the Walsh system, which is itself a special case, is that in general they are made up of complex valued functions.


The Fourier series of a function f E L[O, 1) with respect to the system {Xn(x) }~=o defined by (1.5.10) (or, more briefly, the {Xn(x)}-Fourier series of f) is the series

00

(2.8.1) L l(n)Xn(X), n=O

where

(2.8.2)

According to the second identity of (1.5.7), the partial sums of the series (2.8.1) can be represented by using the Dirichlet kernels of this system (1.5.19) in the following way:

n-l 1 1

Sn(X) = L 1 f(t)Xn(t)Xn(X) dt = 1 f(t)Dn(x e t) dt. (2.8.3) ;=0 0 0

Analogous to formula (2.1.11), it is easy to see by (1.5.21) that the partial sums of order mk (see (1.5.4» satisfy the identity

(2.8.4) Sm.(x,f)= (lk) [ f(t)dt, 10j I J6~·)

x E O(k) J '

where the intervals o?) are defined by (1.5.13). As in the Walsh case, this identity can be used to establish the following analogue of Theorem 2.1.3:

2.8.1. A series in the system {Xn(x)} is the Fourier series of some f E L[O,I) if and only if the partial sums of this series of order mk satisfy identity (2.8.4)'

Identity (2.8.4) implies

(2.8.5) Sm.(x,f)-f(x)= (lk) [(f(t)-f(x»dt, XEOJ~k). 10j I J6~·)

Consequently, we are lead to the following result:

2.8.2. The sequence of partial sums {Sm.(x, f)} of the {Xn(x)}-Fourier series of a function f continuous on [0, 1] converges to f uniformly on [0, 1) and satisfies

(2.8.6) ISmk(x,f)-f(x)lS;w(~k,f), xE[O,I).

As we did in the Walsh case in §2.3, this result can be transferred to the class of discontinuous functions which are continuous with respect to a metric on [0,1) corresponding to the operation e, analogous to the metric (1.2.19). We leave it to the reader to develop for the system {Xn(x)} the corresponding concepts of the metric pp, uniform pp-continuity, the generalized modulus of continuity :'p( 0, f), analogous to identity (2.3.5), and to formulate in these terms the generalized version of Theorem 2.8.2.

We shall prove analogues for the system {Xn(x)} of Theorems 2.5.9 and 2.5.12.

68 CHAPTER 2

2.8.3. Let {b~k)} == {[ok,.Ih)}k::1 be the sequence of half open intervals of tile

form (1.5.13) which satisfy x E b~k) for all k ~ 0. If F(x) = J; J(t) dt for some J E L[O, 1), then the subsequence of partial sums {Smk(X, f)} of the {Xn(x) }-Fourier series of J converges at a point x to a number a if and only if

(2.8.7) lim F«(Jk) - F(Ok) = a. k ..... oo (Jk - Ok

PROOF. By formula (2.8.4)

Sm.(x,f) = _1_ f J(t)dt= F«(Jk)-F(ad , Ib~k)1 J6~·) (Jk - Ok

and the theorem follows immediately. • The limit on the left side of identity (2.8.7) can be viewed as a derivative with

respect to a net formed by the intervals b~k). In contrast to the nets {Nk} introduced in § 1.1, these nets are not binary.

Analogous to Theorem 2.5.10 it is easy to show that at each point x where the derivative F' (x) exists, the limit (2.8.7) exists and equals F' (x). Thus since F'(x) = J(x) the following result is a corollary of Theorem 2.8.3.

2.8.4. If J E L[0,1) then the subsequence of pCl.rtial sums {Sm.(X, f)} of the {Xn(x)}- Fourier series of J converges to J(x) almost everywllCre on [0,1).

This result shows us that the multiplicative systems are complete.

2.8.5. The system {Xn(x)} is complete in the spClces V[O, 1) for every p ~ 1 Clnd in qo, 1).

PROOF. The proof proceeds along the same lines as that of Theorem 2.6.2. Namely, the condition

11 J(t}xn(t) dt = 0, n = 0,1,2, ... ,

implies

11 J(t)Dmk(xet)dt=O

for x E [0, 1) and k = 0,1, ... , because for each fixed x the function Dmk (x e t) can be considered as a polynomial in the system {Xn(X)}. By (2.8.3) this last identity means that

Smk (x) = ° k = 0, 1, 2, ....

Hence we conclude by Theorem 2.8.4 that

J(x) = lim Smk(x) = ° k ..... oo

for almost every x E [0,1). In the case that J is continuous, we also have that J( x) = ° everywhere .•

Analogous to Theorem 2.6.4 we can prove the following result.


2.8.6. The system {Xn(x)} is closed in each of the spaces LP[a, 1), p ~ 1, and in C[a, 1).

In particular, Parseval's identity holds for multiplicative systems.

Chapter 3

GENERAL WALSH SERIES AND FOURIER-STIELTJES SERIES. QUESTIONS ON UNIQUENESS OF

REPRESENTATIONS OF FUNCTIONS BY WALSH SERIES

In this chapter we shall consider general Walsh series, i.e., series whose coefficients are not necessarily Walsh-Fourier coefficients of some function. For the study of these series, the function which is the sum of series obtained by term by term integration of the given series plays an important role. This function allows us to formulate necessary and sufficient conditions for a given series to be a Walsh-Fourier series or a Walsh-Fourier-Stieltjes, i.e., a Walsh series with coefficients of the form

aj = 11 Wj(x)dtP(x).

(This integral is understood as a Stieltjes integral (see A4.3).) In the process we shall show that any Walsh series can be interpreted as a Walsh-Fourier-Stieltjes series if we generalize the Stieltjes integral suitably. This generalization of the Stieltjes integral will be based on binary nets which were introduced in §1.1.

In §2.5 (see 2.5.9) we already noticed the connection between convergence of a. Walsh series and questions of differentiability of functions with respect to binary nets. In this chapter this connection will be used frequently. Because of this, we shall mention below several results from the theory of differentiation with respect to binary nets. Our study of uniqueness of representation of functions by Walsh series will essentially be based on these results.

In §2.6 we established Theorem 2.6.1 which showed that a function is uniquely determined by its Walsh-Fourier coefficients. Here we examine uniqueness theorems of a different type. Namely, we shall be interested in questions concerning whether the coefficients of a given Walsh series which converges in some sense are uniquely determined by the sum of this series, and also in questions about reconstruction of the coefficients from this sum if such uniqueness holds. In particular, when the sum of this series is integrable we shall consider the question of whether these coefficients coincide with the Fourier coefficients of this sum.

Notice first of all that if the partial sums Sn(x) of the series (1.4.1) converge in LP[O,I) norm for some p ~ 1, or in C[O, 1) norm, and S(x) represents the limit of these partial sums, then (1.4.1) must be the Walsh-Fourier series of the function S( x). Indeed, in this case for every i ~ 0 we can view aj as a Fourier coefficient of

70

GENERAL WALSH SERIES AND FOURIER-STIEL TJES SERIES

each partial sum Sn(x) for n ~ i. Thus

lai -11 S(t)Wi(t) dtl = 111 (Sn(t) - Set)) Wi(t) dtl

::;11 ISn(t) - S(t)1 dt

= IISn - Sill -+ 0 as n -+ 00 .

71

Consequently, ai = SCi). By Holder's inequality (see A5.2.2), these steps are still valid when the LP norm replaces the L1 norm. The case e[a, 1) is even simpler.

Therefor, uniqueness holds when the Walsh series converges in the norm of the spaces mentioned above. Moreover, this illustrates the leading role that WalshFourier series play among the entire class of Walsh series.

The situation is more complicated when the Walsh series only converges pointwise or is summable in some sense. In §3.4 we shall show that convergence almost everywhere is not sufficient for a Walsh series to be uniquely determined by its sum. To show this we shall construct an example of a Walsh series which converges to zero almost everywhere whose coefficients are not identically zero, and consequently, cannot be the Fourier series of its sum.

If we consider only the Walsh series which converge everywhere, or everywhere except possibly on a countable set of points, then uniqueness holds. In fact, we shall show in §3.2 that if such a series converges to an integrable function then its coefficients can be reconstructed from its sum by means of the Fourier formula.

Similar uniqueness theorems can be established for certain methods of summability. We shall not discuss such questions here, except the case when the some subsequence of partial sums of the series converges, which in itself represents a kind of summability method.

§3.1 General Walsh series as a general StieItjes series. It is convenient to have a way to characterize a given Walsh series by means of a

sequence piecewise constant functions.

3.1.1. Let {k j} be an increasing sequence of natural numbers and let {rP j} be a

sequence of functions which are constant on each interval L).~j) of rank k j. Suppose further that

(3.1.1)

for each interval L).~j). Then there exists one and only one Walsh series S whose partial sums SZ"j (x) coincide with rPj( x)l.

1 The sequence {.pj} is a martingale (see for example [20]).

72 CHAPTER 3

PROOF. By 1.3.2 each of the functions ¢j can be written uniquely as a polynomial of the form

2kj -1

(3.1.2) ¢j(X) = L aiWi(X). i=O

It remains to see that for each it > j the coefficients of order i ::; 2kj - 1 of the polynomial representation of ¢it coincide with ai given by (3.1.2), i.e., that the identity

2kjl -1

¢it(X) = ¢j(X) + L aiWi(X) i=2 k j

holds. For this we need only notice that the coefficients of these polynomials are in fact

Walsh-Fourier coefficients. Thus we need to verify that ¢jl (i) = ¢j(i) for i < 2kj.

But for each i < 2kj the function Wi(t) is constant on the interval ~~j) (see 1.1.3) with constant value Wi,m' Hence it follows from (3.1.1) that

it> j.

In particular,

¢it(i) = 11 ¢it(t)wi(t)dt

2kj -1

= L Wi,m f . ¢it(t)dt m=O la<":'l)

2kj -1

= L Wi m f ¢j(t) dt m=O 'la<":'j)

Given a Walsh series (1.4.1), we shall denote the indefinite integral of its partial sums S2k(X) by

(3.1.3)

GENERAL WALSH SERIES AND FOURIER-STIEL TJES SERIES

It is clear by (1.4.7) that for points of the form j /2k we have

j/2 k

'l/;kH(j/2k) = 1 S2k+1 (t)dt

i-I = 2: ( S2 k +1 (t) dt

m=oltl.!::) i-I

= 2: ( S2k(t)dt = 'l/;k(j/2k). m=oltl.!::)

73

Hence beginning with the index i = k, the sequence {'I/;i(X)}~o determined by (3.1.3) is constant at each dyadic rational point of the form j /2k, and thus converges at such points. Therefore, the limit 'I/;(x) = limk_oo'l/;k(x) exists for each dyadic rational point x. Moreover, 'I/;(x) = 'l/;k(X) for x = j/2k, j = 0,1, ... ,2\ and consequently if x E [m/2\ (m + 1)/2k) then

(3.1.4) S2k(X) = ~(k) ( S2k(t) dt = 'I/;«m + 1)/2:~)- 'I/;(m/2k). Illm Iltl.!::) Illm I

Notice that the function '1/;( x) just defined can be viewed as the term by term integral of the Walsh series (1.4.1). Moreover, '1/;(0) = o.

Consider now the reverse problem. Namely, let I/J( x) be a function defined on the set of dyadic rational points. Thus for each interval ll~) = ll~~l) U ll~~~~ we have

(3.1.5) I/J«m + 1)/2k) -1/J(m/2k) I/J«m + 1)/2k) -1/J«2m + 1)/2k+1)

2Ill~~1)1 Ill~)1

If we set

+ .:...;.I/J(~( 2_m_+~1 )c.!..../2--;k7-:+ 1-:-,) :---,-I/J~( m---,/,--2-.!..k) 21ll (k+I) I

2m+1

I/J ( ) = I/J«m + 1)/2k) -1/J(m/2k) k x Ill~)1 '

for x E ll~), 0 S m S 2k - 1, k = 0,1,2, ... , then we obtain

{ I/Jk(x)dx= { I/Jk+1(X)dx+ { 1/Jk+I(x)dx= ( 1/Jk+I(x)dx. ltl.!::) ltl.~':..+l) ltl.~':..+;~ ltl.!::)

Hence by 3.1.1 the sequence {l/Jk(X)} uniquely determines a series whose partial sums satisfy S2k(X) = I/Jk(X), i.e., whose partial sums satisfy the identity

S ( ) _ I/J«m + 1)/2k) -1/J(m/2k) 2k x - (k)

Illm I

74 CHAPTER 3

for x E ~~). At the same time these partial sums must satisfy (3.1.4), where 1f;(x) represents the term by term integral of the series S. Consequently,

Applying this equation to the dyadic interval ~~k) = [0,1/2k) and using the identity 1f;(0) = 0, we see that <p(1/2k) - <p(0) = 1f;(1/2k). Continued applications of this equation to the intervals ~~k), ~~k), ••• , ~~~l eventuates in

<p(m/2k) - <p(0) = 1f;(m/2k).

Since k is arbitrary, we conclude that <p(x) - <p(0) = 1f;(x) for all dyadic rational points x.

Thus we have proved the following result:

3.1.2. To each Walsh series (1.4.1) there corresponds a function 1f;(x), defined at each dyadic rational x E [0,1] (1f;(0) = 0) which represents the formal term by term integral ofthe series (1.4.1) and satisfies (3.1.4). Conversely, to each function 1f;(x) defined at all dyadic rationals x E [0,1] there corresponds a unique Walsh series whose partial sums are connected with the given function 1f;(x) by means of(3.1.4) and whose term by term integral is a series which coincides with 1f;(x) -1f;(0).

Thus there is a 1-1 correspondence between Walsh series and functions 1f;(x) defined on the dyadic rationals. We shall call the function 1f;(x) which satisfies (3.1.4) the function associated with the Walsh series (1.4.1).

The limit, as k ---+ 00, of the right side of (3.1.4) is precisely the definition of the derivative D{JI!d1f;(x) of the function 1f;(x) with respect to the binary net {Nt}. Thus equation (3.1.4) leads directly to a generalization of Theorem 2.5.9.

3.1.3. Suppose that a function 1f;(x) and a Walsh series (1.4.1) are related to one another in the sense of 3.1.2. Then the subsequence {S2.} of partial sums of this Walsh series converges at a point x if and only if the function 1f;( x) is differentiable with respect to the binruy net {Nk } in which case

(3.1.6)

Notice that in order for the derivative (2.5.11) to exist at a point x it is only necessary for the differentiable function to be defined at the endpoints of the intervals ~~k) which contain x. Thus for the situation described in 3.1.3 it is sufficient that the function 1f;(x) be defined at all dyadic rational points in the interval [0,1].

We generalize the concept of differentiation with respect to binary nets by introducing the upper and lower derivative with respect to binary nets. These are defined by

(3.1. 7)

GENERAL WALSH SERIES AND FOURIER-STIELTJES SERIES 75

and

D ol,() 1· . f .,p(P!) - .,p( a~) -{Nk}'f' x = lmm (k)'

k--+oo 16% 1 where {6~k) == [a~,p!)} is the sequence of all dyadic intervals which contain x. Recall that the limit supremum and limit infimum of a sequence of real numbers {ad is defined by

lim sup ak = lim (sup ai ) , k--+oo k--+oo i~k

lim inf ak = lim (inf a i ) . k--+oo k--+oo i~k

By using (3.1.4), we can strengthen (3.1.6) in the following way:

(3.1.8) limsupS2k(x) = D{Nk}.,p(X), liminf S2h(X) = D{Nk}.,p(X), k--+oo k--+oo

where the function .,p( x) and the Walsh series S are related to each other in the sense of 3.1.2.

We shall also need the concept of {Nd-continuity of a function, i.e., continuity with respect to binary nets. We shall say that a given function .,p( x) defined on the set of dyadic rational points is {Nk }-continuOU8 at a point x E [0,1] if the endpoints of a sequence {~~k) = [a~, p!)} of dyadic intervals whose closures contain the point x satisfy

(3.1.9) k -t 00.

Notice that if x is itself a dyadic rational then there are two sequences of dyadic intervals whose closures contain x, a left one and a right one, and that condition (3.1.9) must be fulfilled by both these sequences in the case of {Nd-continuity at this point.

3.1.4. Let .,p(x) be a function wbich is defined at all dyadic rational points in tbe interval [0,1]. Tben .,p(x) is {Nd-continuous at a point x E [0,1] if and only if tbe corresponding Walsb series determined by (3.1.4) satisfies

(3.1.10) S2k(X ± 0) = o(2k), k -t 00

at the point x.

PROOF. If x E 6~k) = [a~,p!) then it is clear by (3.1.4) that

(3.1.11) S2k(X)Tk = .,p(P;) - .,p(a~).

On the other hand, if x belongs only to the closure of ~~k) but not to the dyadic interval itself, then x must be the right endpoint of ~~k). Since S2k (x) is constant on this interval, it follows from (3.1.11) that

S2k(X - O)Tk = .,p(P;) - .,p(a~).

Therefore, this identity and (3.1.11) hold for all k ~ 0 and the proof of the theorem is complete. •

76 CHAPTER 3

3.1.5. If the coefficients ai of a Walsh series L aiwi( x) converge to zero as i ---7 00

then (3.1.10) holds uniformly for x E [0,1].

PROOF. It is obvious that

2k -1

IS 2 k (X)12- k :::; 2-k L lail· i=O

But the right side of this inequality is an arithmetic mean of the coefficients ai. Since ai ---7 a as i ---7 00 it follows that this arithmetic mean must also converge to zero as k ---7 00 (see Theorem 4.1.3 from the next chapter) .•

The following is a corollary of 3.1.4 and 3.1.5.

3.1.6. If the coefficients of a Walsh series converge to zero then the associated function 1jJ(x) is {Nd-continuous at each point in the interval [0,1].

Theorem 2.1.3 and formula (3.1.4) imply the following result:

3.1.7. A Walsh series (1.4.1) is the Walsh-Fourier series of some function f integrable on [0, 1) if and only if the function 1jJ( x), defined at dyadic rational poin ts as the sum of tile term by term integral of this series, coincides at these points with the indefini te in tegral F( x) = J; f (t) dt.

In particular, it is clear that given any Walsh-Fourier series, the function 1jJ associated with it is everywhere {Nd- continuous on [0,1].

Theorem 3.1.7 can be extended to Fourier-Stieltjes series in the following way:

3.1.8. A Walsh series (1.4.1) is the Fourier- Stie1tjes series of some function ef, continuous from the left and of bounded variation on the interval [0,1), if and only if the function 1jJ( x), defined at dyadic rational points as the sum of the term by term integral of this series, satisfies

(3.1.12) 1jJ(x) = ef(x) - ef(O)

for all dyadic rational x E [0, 1]. In the case when ef( x) is continuous from the right, condition (3.1.12) is replaced by

(3.1.13) 1jJ(x) = ef(x - 0) - ef(O)

PROOF. We shall suppose that ef is continuous from the left. The case when ef is continuous from the right is handled similarly.

Write the function ef as the difference of two non-decreasing functions, say ef = efl -ef2, where each efi is also continuous from the left (see A4.3.4). Let mes1>! and mes1>2 be the corresponding Lebesgue-Stieltjes measures and recall (see A3.3) that

(k) _ (m+1) (m) mes1>! (~m ) - ef1 2k - ef1 2k '

GENERAL WALSH SERIES AND FOURIER-STIEL TJES SERIES 77

and

mes~2(~}:)) = ¢;2 (m2t 1) - ¢;2 (;)

for any dyadic interval ~}:) = [m/2\(m + 1)/2k ). Since ¢; = ¢;1 - ¢;z, it follows that

(3.1.14)

Suppose that the function ¢; satisfies (3.1.12). Then we can write (3.1.14) in the form

(3.1.15)

As we have already noticed, the coefficients ai of a Walsh series can be viewed as the Fourier coefficients of its partial sums Sn(x) when i < n. Thus

Since the function Wi( x) has a constant value Wi,m on each interval ~~), it follows from (3.1.4) and (3.1.15) that

Zk_ 1 Zk-1

= L wi,mmes~l(~}:)) - L wi,mmes~2(~}:))' m=l m=l

But the expression on the right is precisely the Lebesgue-Stieltjes integral of the step function Wi(X) with respect to the function ¢; (see A4.3.8). Consequently,

(3.1.16) ai = (LS) 11 Wi(X) d¢;(x).

Conversely, suppose that the given Walsh series is the Fourier- Stieltjes series of the function ¢;, i.e., that its coefficients are determined by formula (3.1.16).

78 CHAPTER 3

Analogous to the derivation of formula (2.1.11), we can use (3.1.14) and A4.3.8 to see that for any x E ~~),

S2'(X, d</l) = 11 D2.(t ffi x) d<fi(t)

= l~t)1 i~) d¢>(t)

= -k- (mes.p, (~~») - mes.p2(~~»)) I~m I

= l~t)1 (</I(m2~1) -</I(~)). In particular, we conclude by Theorem 3.1.2 that 1jJ(x) = ¢>(x) - </1(0), i.e., (3.1.12) holds as promised. •

By applying 3.1.8, we can establish the following theorem:

3.1.9. Suppose that the Walsh series (1.4.1) is the Fourier-Stieltjes series of some function </I. Then

(3.1.17) lim S2.(x,d</l) = </I'(x) k-=

for almost every x in [0,1) and in fact at each point x where the derivative of </I exists.

PROOF. Since </I is a function of bounded variation it is differentiable almost everywhere (see A4.3.5). But at each point x where </I is differentiable, the derivative D{Nd</l(x) with respect to binary nets also exists (see Theorem 2.5.10). On the other hand, 3.1.8 implies that at each dyadic rational point the function </I

differs from 1jJ, the term by term integral of the given series, by at most a constant. Consequently, identity (3.1.17) follows directly from (3.1.6) .•

The function 1jJ we considered above, which is the term by term integral of a Walsh series, allows one to view any Walsh series as a generalized Fourier-Stieltjes senes.

We shall now consider the generalized Riemann-Stieltjes integral. Let 1jJ(x) be any function defined on the dyadic rationals in the interval [0,1)

and let f(x) be a function defined everywhere on [0,1). Consider the sequence of partitions of the interval [0,1) generated by the nets {Nd, namely, the k-th partition of [0,1) will be the partition consisting of the nodes of the net {Nd, i.e., consisting of the dyadic intervals ~~) for m = 0,1, ... ,2k - 1. Corresponding to the k-th partition we shall define the Riemann-Stieltjes sums

(3.1.18)

where ~m is any point belonging to the interval D.~). If for some constant I, Ik ~ I as k ~ 00 in the sense that given e > 0 there

is a natural number ko such that for all k > ko the sums h, independent of the choice of ~m E D.~), satisfy the inequality Ilk - II < e, then we shall say that f(x) is integrable on [0,1) with respect to the function ¢>( x) through the sequence of nets {Nd, or more briefly, {Nk}-integrable with respect to lj;(x). We shall denote the corresponding integral by

The class of functions f integrable in this sense is obviously a vector space. We shall not require any deep properties of this integral I since it is only used to define Fourier-Stieltjes coefficients. For this it is sufficient to notice that any function which is constant on all intervals of the form D.~) for some natural number k is integrable (in the sense above) with respect to any function lj;( x). In particular, the integral

(3.1.19)

is defined for all i ~ O. The Walsh series whose coefficients have the form (3.1.19) will be called the

generalized Fourier-Stieltjes series of lj;(x). We are now prepared to formulate the interesting result that any Walsh series

can be viewed as a generalized Fourier-Stieltjes series.

3.1.10. Suppose a Walsh series (1.4.1) and a function lj;(x) are related to one another in the sense of 3.1.2. Then

ai = {Nd 11 wi(x)dlj;, i ~ o.

PROOF. By the proof of Theorem 3.1.8 we have

for any i < 2k. But the expression on the right is a Riemann- Stieltjes sum of the type (3.1.18) for the function Wi (x). Moreover, in this case the value h of this sum does not depend on the choice of the points ~m' Since h equals ai for all 2k > i, it follows that ai is the limit of these sums and thus coincides with the integral (3.1.19) .•

80 CHAPTER 3

The proof of this theorem shows that the partial sums Sn (x) of any Walsh series (1.4.1) can be written in an integral form analogous to the equation (2.1.10) which was valid for Walsh- Fourier series, namely,

§3.2. Uniqueness theorems for representation of functions by pointwise convergent Walsh series.

We shall establish several properties about functions which depend on the behavior of their {Nk}-derivative and then use these properties to prove a uniqueness theorem about Walsh series which converge everywhere except perhaps on some countable subset of the interval [0,1).

We shall begin by proving the following auxiliary result related to the behavior of the partial sums of order 2k of a Walsh series.

3.2.1. Suppose that the partial sums of a Walsh series satisfies (3.1.10) for all points x in the closure of some dyadic interval ~~) (at the endpoints one needs only assume the condition from within). Then for some p > k there are two non-overlapping

intervals 6.~! and 6.~! of rank p which are subsets of ~~) such that p

S2;(X) ~ S2k(X)

for all x E ~ (p) U ~ (p) and all i satisfying k < i ~ p. mp m~

PROOF. Denote the constant value which the function S2i(X) assumes on the

interval ~~ by s~2. By Theorem 1.4.2,

(HI) (i+l) s(i) = S2m + S2m+l

m 2 (3.2.1)

Consequently, S2i+1 (x) ~ S2i (x) surely holds on at least one of the intervals ~~;;,1 i+l or ~2m+l.

Let ~~;1) denote one of the two intervals of rank k + 1 whose union is ~~) on

which S~;I) ~ s~) (if this inequality is satisfied on both halves of ~~) then choose

the left-most one). Similarly, choose ~~;2) from the two intervals whose union is

~~;1) on which the corresponding inequality is satisfied for the partial sums of order 2k+2. Continuing this argument we construct a sequence of nested intervals { A (k+j)}oo J: h· h

Urnj j=1 lor w IC

(3.2.2) j = 1,2, ... , mo == m.

Each interval ~}:+ j) has a neighbor ~0+ j) of the same rank which satisfies J ffij

(k+ ") However, the intervals {.0._ J} are not nested.

We shall show that ther';\s a number jo ;::: 1 such that

(3.2.3)

Suppose to the contrary that

(3.2.4 ) s~+j) > s~, mj

j = 1,2, ... ,

and thus in particular that

(3.2.5) (j == s~+l) - s~ > o. mt

Then by (3.2.1) we have

s(k+j) + s~+j) = 2S(k+j-l) mj mj mj_l' j = 1,2, ... ,m + 0 == m.

Successively applying this identity, (3.2.4) and (3.2.5) we obtain

It follows that s~+j2-(k+j) < s~2-(k+j)_{j2-(k+l) and s~+j2-(k+j) does not tend to J J

zero as j --* 00. Let x be the point to which the intervals .0.~t j) shrink. (This point either belongs to the intersection of these intervals or is eventually an endpoint of these intervals from some point on.) Then contrary to hypothesis, condition (3.1.10) fails at the point x. Hence (3.2.4) cannot hold for all j ;::: 1 and we have proved that there is a jo 2: 1 for which the inequality (3.2.3) holds.

Let p = k + J. and let .0. (p) and.0. (p) represent the intervals .0.( k+ jo) and .0. ~+ jo) . o ffip m' m Jo . p ffiJO

Using (3.2.2) it is easy to see that these intervals satisfy all the necessary proper-ties. I

This result allows us to establish the following theorem:

82 CHAPTER 3

3.2.2. Suppose the partial sums of a Walsh series satisfy condition (3.1.10) at all points in the interval [0,1]' and satisfy

(3.2.6) limsupS2k(X) ;::: 0 k-+oo

for all points x E [0,1), except perhaps for points in some countable set E. Then

(3.2.7) x E [0,1), k;::: O.

PROOF. Evidently, it suffices to show that for any dyadic interval .0.;:) the following inequality holds:

(3.2.8)

Suppose to the contrary that there exists an interval .0.;:) which satisfies

(3.2.9)

Apply 3.2.1 to this interval. Thus choose two intervals, which we shall denote by .0.(0) and .0.{!» on which S2i(X) :::; S2k(X) for certain indices i. Apply 3.2.1 to each of the intervals .0.(0) and .0.{!), choosing intervals .0.(00), .0.(01), .0.(10).0.(11) such that .0.(00) U .0.(01) C .0.(0) and .0.(10) U .0.(11) C .0.(1). Continuing this process for s steps we obtain 2' non-overlapping intervals .0.(~1~2"'~')' where en = 0 or 1, such that if P. is the rank of these intervals then S2i(X) :::; S2k(X) for x E .0.(e1~2""') and for all indices i which satisfy k < i :::; P.. There is one of these intervals .0.(~1'2""') which corresponds to each choice of a sequence {en}~=1 of zeroes and ones. Moreover, any such sequence can be interpreted as the dyadic expansion of some number a with 0 :::; a :::; 1. Thus the set of such sequences, hence the collection of intervals {.0.(.,.2 ...•• ) }~1 has the power of the continuum. But each sequence {.0.("'2 ... ") }~1 is nested and shrinks to some point in the interval [0,1] which belongs to the closures of all these intervals. Moreover, this point cannot be related like this to more than two such sequences of nested dyadic intervals. Consequently, among the uncountable collection of such nested sequences there is a sequence which shrinks to a dyadic irrational point Xo which does not belong to the countable set E. In particular, condition (3.2.6) holds for x = Xo. On the other hand, by construction

i > k,

because the point Xo is a dyadic irrational and therefore must lie in the interior of each interval from a nested sequence of the form {.0.(e1e2 ... e.)}~1 which shrinks to Xo. Consequently, it follows from (3.2.9) that limsuPi-+oo S2i(XO) :::; S2k(XO) < 0, contradicting inequality (3.2.6) which, as we have already observed, holds for the point Xo. Consequently, (3.2.8) holds for any dyadic interval .0.;:) and the proof of this theorem is complete. I

Theorem 3.2.2 is equivalent to the following:

GENERAL WALSH SERIES AND FOURIER-5TIEL TJES SERIES 83

3.2.3. Let 1j;(x) be a function which is defined at all dyadic rational points in the interval [0,1]. Suppose further that 1j; is {Nk}-continuous everywhere on [0,1] and satisfies the inequality

(3.2.10)

for all points x E [0,1), except perhaps for points in some countable set E. Then 1j;(x) is non-decreasing on the set of dyadic rational points in [0,1].

PROOF. By 3.1.2 construct a Walsh series corresponding to 1j;(x) whose partial sums satisfy (3.1.4). This Walsh series evidently satisfies (3.1.10) by Theorem 3.1.4 and the fact that 1j;(x) is {Nk}-continuous everywhere on the interval [0,1]. This Walsh series satisfies (3.2.6) by the first identity in (3.1.8) and hypothesis (3.2.10). Hence we can apply Theorem 3.2.2 to this Walsh series verifying that its partial sums satisfy (3.2.7). It follows, therefore, from (3.1.4) that 1j;«m+ 1)/2k) ~ 1j;(m/2k) for any m satisfying 0:::; m :::; 2k -1 for some k ~ O. Since each pair a, b of dyadic rationals, the interval [a, b) can be written as a finite union of non-overlapping, contiguous dyadic intervals, it follows that 1j;(b) ~ 1j;(a) .•

This test for monotonicity of the function 1j;( x) is used in the proof of the following theorem which is the main result of this section.

3.2.4. Let 1j;(x) be a function which is defined at all dyadic rational points in the interval [0,1). Suppose further that 1j; is {Nk}-continuous everywhere on [0,1) and satisfies the inequality

(3.2.11)

for all points x E [0,1), except perhaps for points in some countable set E, where J(x) is everywhere finite-valued and Lebesgue integrable on [0,1). Then

(3.2.12) 1j;(x) = 1j;(0) + 1x

J(t)dt

for all dyadic rationals x.

PROOF. Let e > O. Since J E L[O, 1) we can choose (see A4.4.6) a lower semicontinuous function u(x) which satisfies the following properties:

a) u(x) > -00 everywhere on [0,1); b) u(x) ~ J(x) everywhere on [0,1); c) u(x) E L[O, 1) and fa1 u(t) dt < e + J01 J(t) dt. Set

U(x) = 1xu(t)dt, F(x) = 1 xJ(t)dt.

By property c) we have

(3.2.13) U(l) - F(l) < e.

84 CHAPTER 3

We shall show that since u( x) is lower semicontinuous everywhere on the interval [0,1) we also have

(3.2.14) x E [0,1).

Fix a point x. By property a) we know that u(x) > -00. For each A < u(x) choose by the definition oflower semi continuity (see A4.1) 8 > a such that u(t) > A for all t E [0,1) which satisfy It - xl < 8. Thus for all dyadic intervals D..~k) which satisfy

x E D..~k) C (x - 8, x + 8) we have

Setting D..~k) == [Q'~, 13:), it follows that

This means that the lower {N"d-derivative of U(x) at the point x is no less than A, and consequently, (3.2.14) holds. In particular, we see by properties a) and b) that

(3.2.15) xE[O,l),

and

(3.2.16) xE[O,l)

Recall for any sequence of real numbers {ak} and {bk} that

limsup(ak - bk ) ~ liminf ak -liminf bk . k-+oo k-+oo k-oo

Since the function I is finite-valued, it follows from inequalities (3.2.11) and (3.2.15) that

for every x E [0,1) \ E. Since U(x) - 1jJ(x) is obviously {Nd-continuous, it follows from Theorem 3.2.3 that the difference U(x) - 1jJ(x) is non-decreasing on the set of dyadic rationals in [0,1).

Similarly, we can find a lower semi continuous function -v( x) corresponding to the function - I( x) whose indefinite integral - V( x) satisfies the inequalities

(3.2.17) -V(l) + F(l) < c,

GENERAL WALSH SERIES AND FOURIER-STIELTJES SERIES

and

I.e.,

D{Jvd(-V(x» ~ -f(x),

D{Nd Vex) ~ f(x),

x E [0,1),

x E [0,1).

As we did above for the difference U(x) -1jJ(x), we can show that

D{Nk} (1jJ(x) - Vex)) ~ D{Nk}1jJ(X) - D{Nd Vex) ~ 0

85

for all x E [0,1) \ E. In particular, it follows from Theorem 3.2.3 that the difference 1jJ(x) - Vex) is also non-decreasing on the dyadic rationals in the interval [0,1).

Since both the functions U(x) -1jJ(x) and 1jJ(x) - V(x) are non-decreasing on the set of dyadic rationals it is easy to see that

(3.2.18) Vex) ~ 1jJ(x) -1jJ(0) ~ U(x)

for all dyadic rational points x. Moreover, the choice of the functions U(x) and Vex) directly imply

Vex) ~ F(x) ~ U(x), x E [0,1],

and (3.2.13) and (3.2.17) clearly imply

o ~ U(x) - Vex) ~ 2c.

Consequently, it follows from inequality (3.2.18) that

IF(x) - (1jJ(x) -1jJ(0))1 ~ 2c

for all dyadic rationals x. Since c > 0 was arbitrary, we conclude that (3.2.12) holds for all dyadic rational points x .•

We are now prepared to formulate a very general uniqueness theorem which contains as corollaries several useful results.

3.2.5. Suppose the partial sums of a Walsh series satisfy condition (3.1.10) at every point in the interval [0,1) and satisfy

(3.2.19) liminf S2k(X) ~ f(x) ~ limsupS2k(X) k-OC) k-oo

for all points x E [0,1), except perhaps for points in some countable set E, where f( x) is everywhere finite-valued and Lebesgue integrable on [0,1). Then this Walsh series is the Walsh-Fourier series of the function f.

PROOF. This theorem is essentially a translation of Theorem 3.2.4 into the language of series. Indeed, by 3.1.2, 3.1.4, and equations (3.1.8), the function 1jJ(x) associated with the given Walsh series is {Nd-continuous everywhere on [0,1) and satisfies inequality (3.2.11). Thus Theorem 3.2.4 applies to 1jJ(x) and it follows that 1jJ(x) satisfies (3.2.12) for each dyadic rational point x. Recall that the function associated with a Walsh series satisfies the condition 1jJ(0) = O. Therefore, we conclude by Theorem 3.1.7 that the given Walsh series is the Walsh-Fourier series of the function f .•

The following theorem is a corollary of Theorem 3.2.5.

86 CHAP1ER3

3.2.6. Suppose that a Walsh series (1.4.1) converges at all but countably many points in [0,1) to a finite-valued function f E L[O, 1). Then this series is the WalshFourier series of the function f.

PROOF. Since !aiwi(x)! = !ad for i = 0,1,2, ... , it is clear that if the Walsh series converges at even one point then a; -+ ° as i -+ 00. Thus by Theorem 3.1.5 the partial sums of this Walsh series satisfy condition (3.1.10). Moreover, at any point where this series converges, condition (3.2.19) is satisfied. Therefore, the proof of this result is completed by an application of Theorem 3.2.5. I

Theorem 3.2.6 contains the following result which is a uniqueness theorem in the fundamental sense of the word.

3.2.7. If two Walsh series

00 00

Laiwi(X), Lbiw;(x) i=O ;=0

converge everywhere, except perhaps on some countable subset of [0,1), to a finitevalued function then these series are identical, i.e., ai = bi for all i = 0,1, ....

PROOF. The difference of these two series is a Walsh series with coefficients ai - bi

which converges (except perhaps on some countable set) to the zero. Applying Theorem 3.2.6 to the function f(x) = 0 we see that the coefficients aj - bi must be the Walsh-Fourier coefficients of the zero function, i.e., ai - bi = 0 for i = 0,1, .... I

§3.3. A localization theorem for general Walsh series. In this section we shall use a technique called the formal product of a Walsh series

(3.3.1 )

with a Walsh polynomial

p

(3.3.2) P(x) = L bnwn(x) n=O

which is defined to be the series

(3.3.3)

where each coefficient Ce is the sum of the coefficients of the products wm(x)wn(x) which equal wee x) that result from taking the formal product of (3.3.1) with (3.3.2). Recall from (1.2.17) that wm( x )wn( x) = WmEBn( x). Since n EEl (e EEl n) = e (see §1.2),

it follows that the coefficients Cl of the formal product (3.3.3) are determined by the equations

(3.3.4) P

Cl = L bnalE!)n. n=O

3.3.1. Suppose the coefficients of (3.3.1) satisfy the condition

(3.3.5) lim am = 0 m-+oo

and let (3.3.3) be the series obtained from a polynomial (3.3.2) by formula (3.3.4). Then (3.3.3) and the series

00 00

(3.3.6) L(amWm(X)P(X)) = P(x) L amwm(x) m=O m=O

are uniformly equiconvergent, i.e., their difference converges unifonnly to zero.

PROOF. Fix k so large that p < 2k. Then the indices n from the sums (3.3.2) and (3.3.4) also satisfy the inequality n < 2k. Notice for each fixed n < 2k that the transformation f -+ f ~ n is a permutation of the collection of integers {f: 0 :5 f :5 s2k -I} for any fixed natural number s, i.e., the given transformation is a 1-1 map from this collection onto itself. This last remark follows easily from the definition of the operation ~ (§1.2). Consequently,

82k-1 82k-1

(3.3.7) L atE!)nWlE!)n(X) = L alWt(x). 1=0

It is also easy to verify that

(3.3.8)

Therefore, we obtain from (3.3.7) and (3.3.2) that

s2k-1 82k-1 P

(3.3.9) L CtWt(x) = L L bnalE!)nWn(X)WtE!)n(x) 1=0 1=0 n=O

82k-1

= P(x) L alWt(x). 1=0

88 CHAPTER 3

Let N be any natural munber and choose s such that

(3.3.10)

Then (3.3.9) implies

N N

(3.3.11) 1 2::>t Wl(X) -P(x) L:atWl(X) 1

(=0 (=0

N N

=1 L: ClWt(X) - P(x) L: atwt(x) 1

N N

:s L: let! - IIPlic L: latl·

Let c; > O. If N and s2 k (see (3.3.10)) are sufficiently large then lall < c; for all e ~ s2k. Thus by (3.3.4) and (3.3.8) the last line of display (3.3.11) does not exceed the value c; (2k 2:~=0 Ibnl + 2kIlPllc), i.e., the difference in the first line of display (3.3.11) converges uniformly to zero as N --+ 00 .•

An application of this theorem gives the following localization theorem for Walsh senes.

3.3.2. If (3.3.1) is a Walsh series whose coefficients satisfy condition (3.3.5) and if [a, (3) is a half open interval with dyadic ra.tional endpoints then there exists a Walsh series 2::::'=0 a;,. Wm (x) whose coeHicients satisfy a;" --+ a as m --+ 00 which is uniformly equiconvergent with the series (3.3.1) on [a, (3) and which converges uniformly to zero outside [a, (3).

PROOF. The interval [a, (3) can be written as a finite union of dyadic intervals

D.; k) for some k. Thus by 1.3.2 the characteristic function Xro-,,8) (x) of this interval can be represented by a Walsh polynomial P( x). Apply Theorem 3.3.1 to this series and this polynomial. Thus choose a series 2::::'=0 a;" Wm (x) which is uniformly equiconvergent with the series 2::::'=0 am w m (x)xro-,,8)(x). Evidently this last series is identically equal to the series (3.3.1) for x E [a, (3) and converges to zero outside [a, (3).

Notice that the differences am - a;" are the coefficients of a series which converges on [a, (3). Consequently, limm --+ oo( am - a;") = O. Thus it follows from (3.3.5) that a;" --+ a as m --+ 00. Therefore, the series 2::::'=0 a;" Wm (x) satisfies the required conditions .•

3.3.3. Let (3.3.1) be a Walsh series whose partial sums satisfy condition (3.3.5) and suppose that a subsequence {52';} of partial sums converges to zero on some interval (a, b) C [0,1). Then the full sequence of partial sums of the series (3.3.1)


converges to zero on (a, b). In fact, this series converges to zero uniformly on each subinterval [a,.8) of ( a, b) with dyadic rational endpoints.

PROOF. We need only verify the last part about uniform convergence on an interval [a,.8) c (a, b) with dyadic rational endpoints. Apply Theorem 3.3.2 to the series (3.3.1) and the interval [a,.8). Thus the partial sums {S;k;(X)} of the series L::=o a;;' wm(x) converges to zero on [a,.8) since they are uniformly equiconvergent with the series (3.3.1). But outside [a,.8) this sa.me subsequence of partial sums also converges to zero since by Theorem 3.3.2, the series L::=oa;;'wm(x) converges to zero for x f/. [a,.8). Therefore, the partial sums of the series 2::=0 a;;'wm(x) satisfy the relationships

liminf S;k(X) :s; lim S;dx) = O:S; limsupS;k(x) k--+oo 1--+00 k--+oo

for all x E [0,1). Applying Theorem 3.2.5 to the function I( x) = 0 we see that a;;' = 0 for m = 0,1, .... In particular, the series L::=o a;;'wm(x) is identically zero, whence it converges uniformly to zero. Since this series is uniformly equiconvergent with the series (3.3.1), we conclude that (3.3.1) converges uniformly to zero on [a, .8) .•

§3.4. Examples of null series in the Walsh system. The concept of U-sets and M-sets.

We shall construct a Walsh series which converges to zero almost everywhere on [0,1) but whose coefficients are not identically zero. In the theory of orthogonal series, such series are called null series.

We begin with the following lemma:

3.4.1. Let E be a union of intervals llJn) of rank n ~ O. Then the function

(3.4.1) { 0 forxE[O,I)\E,

Pn(x; E) = (n) W22n+j2n(X) forxEllj CE

is a Walsh polynomial of the form

22n+1 _!

(3.4.2) Pn(x; E) = L b~n)Wi(X), i=22n

where

(3.4.3)

Moreover, the set EI == {x : P n (x; E) = + I} is a union of intervals of rank 2n + 1 and

(3.4.4 ) , 1

mesE = "2 mesE.

90 CHAPTER 3

PROOF. The second half of this lemma concerning the set E' and especially (3.4.4) follows directly from the definition (3.4.1).

Since j :5 2n - 1, the index of a Walsh function which appears on the right side of (3.4.1) cannot exceed 22n + (2n - 1)2n < 22n+l. Thus the function Pn(x; E) is constant on each interval A(2n+1) of rank 2n + 1 (see 1.1.3). Therefore, Theorem 1.3.2 implies that Pn(x; E) is a Walsh polynomial of order no greater than 22n+l_1. In particular, the upper limit of the sum in (3.4.2) is correct.

On the other hand, recall from 1.1.4 that ft.(2n) wi(x)dx = 0 on each interval A(2n) of rank 2n for i ~ 22n. Consequently, we have by (3.4.1) that

(3.4.5) [ Pn(x; E)dx = O. It.(2n)

Let i < 22n and denote the constant value Wi(X) assumes on each interval D.~2n) by wi,i' Viewing the coefficients of the polynomial Pn(x; E) as its Fourier coefficients, it follows from (3.4.5) that

1 22n_l

r Pn(x;E)w;(x)dx = L w;,i r Pn(x;E) dx = 0, 100 . 1" ~2n) 1=0 u J

This verifies that the lower limit of the sum in (3.4.2) is correct.

It remains to verify estimate (3.4.3). We have

(3.4.6) b~n) = Pn(i) = 11 Pn(x;E)w;(x)dx

2n_l

= L W;,i r Pn(x;E)w;(x)dx. It.(n)

i=O j

It is clear by (3.4.1) that

(3.4.7) D.~n) C [0,1) \E, i = 1,2, ....

If A ~n) C E and 22n < i < 22n+1 then write i in the form i = 22n +f2n + k, for some 1 -o :5 f :5 2n - 1, 0 :5 k :5 2n - 1. Use relationship (1.2.17), equation (1.1.10) for

the system {Wi2n(X)}~0' and the fact that for k < 2n each Walsh function Wk(X)

is constant on the interval 6.~n). We obtain

for any 22n :::; i < 22n+l. Substituting (3.4.7) and (3.4.8) into (3.4.6) we see that

Ib(n)1 =1 Pn(X; E)w;(x) dx 1= ' l , 1 { 2-n if 6. (n) C E

• ~ln) 0, if 6.~n) C [0,1) \ E

for i = 22n + i2n + k, which establishes (3.4.3) .• We are now prepared to prove the fundamental result of this section.

3.4.2. There exists a Walsh series whose coefficients are not all zero which converges to zero everywhere on [0, 1) except on some closed set of Lebesgue measure zero.

PROOF. We shall use Lemma 3.4.1 to construct a sequence of sets {Fk}k=O and corresponding polynomials {Pnk(x, Fk)}k=O in the following way.

Set Fo = [0,1) and no = 1. Choose by Lemma 3.4.1 a polynomial

23 -1

Pno == Pl(x; [0,1)) = L b~l)w;(x) ;=22

and set Tl(x) == 1 + Pno(x; [0,1)).

By 3.4.1 it is clear that Pno(x; [0,1)) = 1 on some set which is a union of intervals of rank nl = 3 = 2no + 1. Denote this set by Fl. Then we have by construction that Tl(X) = 2 for x E FI, mesFl = 1/2, mesFo = 1, Fl C Fo and Tl(X) = 0 on the set [0,1) \ Fl'

We shall now show how to execute the inductive step from k to k + 1. Suppose we have already chosen a sequence of sets

(3.4.9) {Fd~o, H c Fk-l C ... C Fo, mesF; = 2- i

92 CHAPTER 3

(where each Fi is a union of intervals of rank ni = 2ni-l + 1) and a sequence of polynomials {Pn.(x; Fi)}f,:-J of the form (3.4.2) such that if

k-l

Tk(X) == 1 + L2 j p n;(x;Fi) j=O

then

(3.4.10)

Apply Lemma 3.4.1 to a function Pnk(x; Fk) of type (3.4.1) (i.e., with E = Fd to verify that Pn.(x; Fk) can be written in the form (3.4.2) and moreover, that the polynomials Tk(x) and Pnk(x; Fd have no common terms with the same index. In fact, if we set

(3.4.11)

then it is clear that the polynomial Tk(X) is a partial sum ofthe polynomial Tk+l (x). From the construction of the polynomial Pnk (x; Fk) it is not difficult to verify that the set Fk+l = {x : Tk+l(X) = 2k+l} satisfies Fk+l C Fk, Tk+l(X) = 0 for all x E [0,1) \ Fk+l, that mesFk+l = 1/2mesFk = 2-(k+ 1) and Fk+l is a union of intervals of rank nk+l = 2nk + 1.

This completes the inductive step, and it follows that we can construct an infinite sequence of sets Fk and polynomials Tk(X) which satisfy properties (3.4.9) and (3.4.10).

It is evident by (3.4.11) that the Tk+l (x)'s can be viewed as a partial sums of some Walsh series, and that these partial sums have the form

2 2n .+ 1 _l

S2 2n k+ 1 (X) = L aiwi(x) = Tk+l(X), i=O

By (3.4.10) we also have that

(3.4.12) x E [0,1) \ Fk+l'

Set F = n~o J\ where Fk represents the closure of Fk • Notice that in this case, mesFk =mesFk, so we have by (3.4.9) that mesF = O. Moreover, by (3.4.12) we have

(3.4.13) x E [0,1) \ F.

Notice that nk > 2nk_l > 2k and ai = 2kb~n.). Consequently we have by (3.4.3)

that lail = 12kb)nk)1 :::; 2k2-nk :::; 2k2-2k for 22nk :::; i < 22n.+1 = 2nHl and ai = 0

GENERAL WALSH SERIES AND FOURIER-STIELTJES SERIES 93

for 2nk ::; i < 22nk • In particular, the coefficients ai of the constructed Walsh series converge to zero as i -t 00. Since this series also satisfies (3.4.13), we can apply Theorem 3.3.3 on each subinterval [a,,B) of the open set (0,1) \ F, provided the endpoints a,,B are dyadic rationals. It follows that the constructed Walsh series converges everywhere outside F to zero, i.e., it is the Walsh series we search for .•

In the theory of orthogonal series, the following concept is defined: A set E is called a set of uniqueness or U-set for some system {<Pn} if the only

series in this system which converges to zero outside E is identically zero, i.e., its coefficients are all equal to zero.

If there is a series in the system {<Pn} whose coefficients are not all zero which converges to zero outside some set E, then the set E is called an M-set for the system {<Pn}.

In this terminology Theorem 3.4.2 shows us that there is a closed non-empty M -set for the Walsh system which is of Lebesgue measure zero.

It is not difficult to verify that any subset E of [0,1) of positive Lebesgue measure is an M-set for the Walsh system. For this it is enough to look at the WalshFourier series of the characteristic function of a closed set F of positive measure which satisfies FeE. We leave it to the reader to verify the details.

On the other hand, Theorem 3.2.5 applied in the special case when f(x) = ° shows us that any countable set is aU-set for the Walsh system.

Among the subsets of [0,1) which contain uncountably many points there are both M-sets (which we verified above) and U-sets. The first uncountable U-set for the Walsh system was constructed by A. A. Sneider (see the commentary for this chapter). We shall not give an account of his construction here. We only notice that for trigonometric series, there is a subtle theory of sets of uniqueness which contains, in particular, a profound characterization of certain perfect sets into the classes of M- sets and U-sets (see [2], Chapter 14). For the Walsh case, this theory is not yet sufficiently developed, although it is already clear that the solution to this problem will require just as complicated and delicate methods as the trigonometric case does (see the commentary).

Chapter 4

SUMMABILITY OF WALSH SERIES BY THE METHOD OF ARITHMETIC MEANS

As we saw in §2.3, there exist continuous functions whose Walsh- Fourier series diverge at a given point. We shall show in Chapter 9 that there exist integrable functions whose Walsh-Fourier series diverge everywhere on [0,1). Thus we see the necessity of identifying various methods of summability which allow a function to be recaptured from its Fourier series. In Chapter 2 we considered one such method of summability, convergence of Walsh- Fourier series through the sequence of partial sums of order 2n. Here we shall examine another widely used method of summability which is called the method of arithmetic means, the first order method of Cesaro, or more briefly, the (C, 1) method.

We shall show that the Walsh-Fourier series of every continuous functions is uniformly (C, 1) summable and that any Fourier- Stieltjes series, in particular, any Walsh-Fourier series of an integrable function, is (C,l) summable almost everywhere.

§4.1. Linear methods of summability. Regularity of the arithmetic means.

This section is of an introductory nature. Recall that by a method of summability we mean a method which in general defines a sum of a divergent series.

The method of arithmetic means, or the (C, 1) method is the following one. Let {Sn} represent the partial sums of some series of numbers. Instead of insisting that the sequence Sn converges, we only look for convergence of the arithmetic means, or (C, 1) means, of these sums. These means are defined by

n

If limn ..... oo Un exists and equals U then we say that the sequence {Sn}, or the series itself, is (C, 1) summable (or summable by the method of arithmetic means) to the number u.

The (C, 1) method is a special case of a class of methods called linear. These methods are defined in the following way.

Let

A = (~~~"':""'" ~~~ ... ::: ) anI ... ank ...

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

94

SUMMABILITY OF WALSH SERIES 95

be an infinite matrix whose rows, anI, a n 2, ... , ank, ... , are fixed sequences of real numbers for each n = 1,2, .... ( We shall denote both this matrix and the method of summation it defines by A.)

Given a sequence {Sn}~=1 we define its n-th order means by the method A (or A means ) to be the sum

00

O"~A) == L>nkSk, k=1

when this sum converges. (In the event that this sum does not converge for some n, we say that the A means are not defined.)

If the A means O"~A) of some sequence {Sn}~=1 are defined for n = 1,2, ... , and

iflimn ..... ooO"~A) = 0" then we shall say that the sequence {Sn} (or the series whose partial sums are given by Sn) is summable by the method A to the number 0".

Such methods are called linear because, as can easily be seen, if two sequences {S~} and {S~} are summable by the method A to 0"' and 0"", respectively, then any linear combination {aS~ + jJ S~} of these sequences is also summable by the method A and is summable to aO"~ + jJO"~.

The (C, 1) method is a linear method of summation. Indeed, its means can be defined by using the matrix

(

1 0 0 ............... )

:~;::;::<::><o::: . In each row of this matrix there are only finitely many non-zero entries. Thus the (C, 1) method belongs to the class of finite linear methods of summation.

Another method of summation used in previous chapters is the method obtained by taking limits of the subsequence of partial sums {S2n}. This also is a finite linear method of summation. Indeed, the matrix which determines this method of summation is the one whose n-th row consists of a 1 at the 2n-th place and zeroes elsewhere.

Of greatest practical interest among the linear methods of summation are the so-called regular methods of summation, i.e., those for which if the sequence {Sn} converges to a number S in the conventional sense then the sequence is summable to the same number S.

We shall prove the following theorem:

4.1.1. A linear metbod of summation generated by a matrix A is regular if tbe following conditions are satisfied:

1) For eacb k = 1, 2, ... , lim ank = 0;

n ..... oo

96 CHAPTER 4

2) the series 2:~1 ank is absolutely convergent for each n = 1,2, .... In fact, if

00

En == Llankl, k=1

then En ::; C for n = 1,2, ... , where C is an absolute constant which does not depend on ni

3) if An == 2:~1 ank then An ~ 1 as n ~ 00. Conditions 1) through 3) were first identified by Toeplitz, who proved that these

conditions are not only sufficient but also necessary for the method A to be regular. We prove here only that the Toeplitz conditions are sufficient.

PROOF. Suppose that a sequence {Sn} converges to a number S. Then Sn can be written in the form Sn = S + en, where en ~ 0 as n ~ 00. Thus

00 00 00 eX)

a~A) = L ankSk = L ankS + L a .. /.:ek = SAn + L ankek· k=1 k=1 k=1 k=1

By condition 3), SAn converges to S. Hence it remains to prove that

00

lim '""' ankek = o. n~oo~ k=1

Let 10 > O. Choose N so large that lekl < e/(2C) for k > N, where C is the constant given by condition 2). Then

00 N 00

1 L ankek 1::;1 L ankek 1 + 2~ L lankl· k=1 k=1 k=N+l

(4.1.1)

Since N is fixed, we see by property 1) that the first sum on the right side of (4.1.1) converges to zero as n ~ 00. Consequently, this sum is less than e/2 for n sufficiently large. By condition 2), the second sum is no greater than e/2. Therefore, 1 2:~1 ankek 1 < 10 for n sufficiently large. We conclude that 2:~1 ankek ~ 0 as n ~ 00 .• 4.1.2. Suppose A is a finite method of summation whose matrix contains only nonnegative entries. Suppose further that A satisfies condition 1) in Theorem 4.1.1 and An == 2:~1 ank = 2:~!1 ank = 1. Then the method A is regular.

PROOF. This result is an immediate corollary of Theorem 4.1.1. Indeed, since the entries A are non-negative, condition 2) follows from condition 3) .•

4.1.3. The (C, 1) method is regular.

PROOF. Above, we identified the matrix which corresponds to this method of summation. It is clearfrom its form that condition 1) of 4.1.1 is satisfied. Moreover, An = 2:~=1 l/n = 1. Thus 4.1.3 follows from 4.1.2 .•


§4.2. The kernel for the method of arithmetic means for Walsh-Fourier series.

We shall denote the arithmetic means (or (C, 1) means) of the partial sums of the Walsh-Fourier series of some integrable function f by

(4.2.1)

Substituting the integral representation (2.1.10) of Si(X, f) into (4.2.1), we obtain

(4.2.2) tIn fl

I7n(X, f) = 10 f(x EB t)- L Di(t) dt = 10 f(x EB t)Kn(t) dt, o n ;=1 0

where Kn(t) == .!. t Di(t) is the kernel of the (C, 1) method. n i=1

We shall study the kernel Kn(t). By (1.4.11) it is clear that

2· 2·+m (4.2.3) nKn(t) = L Di(t) + L Di(t)

m

= 2k K 2.(t) + L(D2.(t) + wZ.(t)Di(t)) ;=1

for n = 2k + m, 1 ~ m ~ 2k. Specializing to the case m = 2k we find that

( 4.2.4)

or

(4.2.5)

From this formula we obtain a closed form for the kernel KZk(t) and prove that it is non-negative.

4.2.1. For each k ~ 0 the identity

( 4.2.6)

for t E .6. (k). o ,

fortE.6.~~), r=0,1, ... ,k-1j

otherwise

98 CHAPTER 4

holds. In particular, the kernel [{2' (t) is non-negative for all t E [0,1), all k ~ 0, and satisfies

( 4.2.7)

PROOF. Identity (4.2.6) is verified by induction on k. It is obvious when k = 0, since in this case

Suppose (4.2.6) holds for some k ~ O. We shall show it holds for k + 1 in place of k by using formula (4.2.5).

. h (k) (HI) U (HI) d (k) (HI) U (HI) b' . Notice t at.6.o =.6.0 .6.1 an .6.2" = .6.2"+1 .6.2"+1+1' By com mmg (4.2.5) and (4.2.6), we obtain

for any t E .6.~k). To evaluate [{2>+I(t) for t E .6.~~) recall that D2.(t) = 0 for

such t's. Thus on each interval .6.~~), for r = 0,1, ... , k - 1, the value of [{2'+~ (t) is completely determined by the first term on the right side of identity (4.2.5). Consequently,

It remains to observe by (4.2.5) that [{2.+1 (t) takes on the value 0 everywhere that [{2.(t) does. This completes the inductive step from k to k + 1 and verifies that (4.2.6) holds for all k. In particular, the kernel is non-negative and (4.2.7) holds as promised .•

Successive applications of (4.2.4) lead to the identity

k-I k-I

( 4.2.8) 2k [{2.(t) = 1 + L 2iD2i(t) + L2iw2i(t)[{2i(t). ;=0 i=O

We establish one more formula for the kernel [{net) which generalizes (4.2.3). Let

k k-i

( 4.2.9) "'''' 2i '" 1 n(i) - "''''"2i n = L.....-"j '''k =, - L.....-"J , i = 0,1, ... , k. j=O j=O

SUMMABILITY OF WALSH SERIES

Using this notation, (4.2.3) can be written in the following way:

nKn(t) = 2k K2k(t) + n(t) D2k (t) + w2k(t)n(t) Kn(1)(t).

99

Applying this same formula to n(1)Kn (1) (t), and continuing in this manner, we see by the choice of the coefficients e i that

(4.2.10) nKn(t) = 2k K2k(t) + n(l) D2k(t) + W2k(t)(2k-lek_lK2k-l(t)

+ n(2)ek_lD2k-1 (t) + (W2k-1 (t))~k-l n(2) Kn(2)(t))

k k k k

= 'L>i2j II (W2i(t))eiK2i(t) + Lejn(k-i+I) II (W2i(t))«iD2i(t) j=O i=i+l i=O i=i+l

k k

= L ej2iWn_n (j) (t)K2i (t) + L ejn(k-H l)wn_n(n(t)D2i (t). j=O j=O

This formula allows us to establish the following property for the kernel Kn(t):

4.2.2. For all n ~ 1,

This property of the kernel Kn(t) is called quasi- positivity. PROOF. Combining (4.2.10) and (4.2.7), and bearing in mind that n(i) :::; 2k- i+1 ,

we obtain

§4.3. Uniform (e,1) summability of Walsh-Fourier series of continuous functions.

In parallel with our study of uniform convergence in §2.3, we shall look not only at functions continuous in the usual sense, but also p* -continuous functions for which the modulus of continuity is defined by identity (2.3.5). We shall obtain uniform (e,1) summability of Walsh-Fourier series of p*-continuous functions as a special case of a result which estimates the rate of approximation to a function by the (e, 1) means of its Walsh- Fourier series in terms of its modulus of continuity.

4.3.1. Let (;;( S, f) be the modulus of continuity of I defined by identity (2.3.5). Then the (e, 1) means O'n(x, f) of the Walsh-Fourier series of the function I satisfies

k

(4.3.1) 100n(x,f) - l(x)1 :::; eL 2i- k(;;(2- i,f) i=O

100 CHAPTER 4

for all2k < n ~ 2k+l, where C is an absolute constant.

PROOF. Using (4.2.2) and the fact that Jol Kn(t) dt = 1, we obtain the following integral expression for the difference we must estimate:

O'n(X, I) - f(x) = 11 (I(x EEl t) - f(x))Kn(t) dt.

Let n = 2k + m, 1 ~ m ~ 2k. Substitute the expression for Kn(t) from (4.2.3) and use (4.2.8) to obtain

(4.3.2) 111 O'n(x'l)-f(x) = - (I(xEElt)-f(x))dt n 0

k-l 1

+ .!. L 2; 1 (I(x EEl t) - f(x ))D2i(t) dt n ;=0 0

1 k-1 '11 + - L 2' (I(x EEl t) - f(x ))W2i (t)K2i (t) dt n ;=0 0

+ m f\f(xEElt)-f(x))D2k(t)dt n io + m f\f(x EEl t) - f(x ))W2. (t)Km(t) dt. n io

We estimate the last term first. Toward this, notice that for m ~ 2k the kernel Km(t) is a sum of Dirichlet kernels each of which is constant on the intervals tl.)k)

(see 1.4.3). Consequently, for each j = 0,1, ... , 2k- 1 , Km(t) is also constant on the interval tl.;k). We shall denote this constant value by K~,j.

Recall that W2k(t) = Tk(t) = 1 for t E tl.~~+1) and W2k(t) = -1 for t E tl.~~!~>' where tl.)k) = tl.~~+l) U tl.~~!~). Also observe for each t E tl.~~+1) that t EEl 2-(k+I)

belongs to tl.~~!!) but the points xEElt and X EEltEEl 2-k - 1 belong to the same interval of rank k. Hence

1 * 1 If(x EEl t) - f(x EEl t EEl 2k+1 )1 ~ w( 2k ' I).

Therefore, we can estimate the last term of (4.3.2) as follows:

III (I(xffit)-f(x»W2k(t)I<m(t)dt I 2k_l

=1 L I<~!j 1 (k) (I(x ffi t) - f(x »W2k (t) dt I j=O Aj

2k_l

=1 .?= K~~j l(~+l)(I(X ffi t) - f(x» - (I(x ffi t ffi 2k~1) - f(x» dt I }=O 2,

2k_l

=1 L K~~j l(k+l) (I(x ffi t) - f(x ffi t ffi 2;+1» dt I }=O 2,

2k_l

:::; L IK~~jl·I6.~~+l)I·~(Tk,J) j=O

Similarly, we can estimate the third term on the right side of (4.3.2) by applying the same reasoning to I<2;(t) instead of I<m(t). We obtain

( 4.3.4) i = 0, 1, ... ,k - 1.

The terms which have the kernels D2;(t) as integrands can be estimated by using the facts that each such kernel is supported on the interval6.~i) and that for t E 6.~i) the points x and x ffi t always belong to the same interval of rank i. Thus

( 4.3.5) i = 0,1, ... , k.

Finally, the integral in the first term on the right side of (4.3.2) can be estimated directly using the modulus of continuity ~(1, J). Substituting estimates (4.3.3)

102 CHAPTER 4

through (4.3.5) into (4.3.2) we conclude that

In particular, (4.3.1) holds with C = 3 .• An immediate corollary of this result is the following theorem:

4.3.2. If a function I is p* -continuous on [0,1) (in particular, if it is continuous in the classical sense on the interval [0,1]), then the Walsh-Fourier series of I is (C, 1) summable to I uniformly on [0,1).

PROOF. The hypotheses imply that (;;(1/2 i , f) -+ 0 as i -+ 00. Consider the method of summability which is determined by the matrix

1 0 0 0 ............... 1/2 1/2 0 0 ...............

A= 1/4 1/4 1/2 0 ...............

1/2 0

It is evident that this method satisfies the hypotheses of Theorem 4.1.2. Hence the summability method A is regular. Since the sequence {(;;(2-i,J)}~o converges to 0, it follows that

also converges to zero as k -+ 00. Since the sum on the right side of (4.3.1) is

precisely 20"kA ) - 2-k(;;(1, f), we conclude by Theorem 4.3.1 that O"n(x, f) converges to I uniformly on [0,1) as promised .•

4.3.3. Suppose 0 < 0: :::; 1. If (;;(2- i , f) = 0(2- i O') as i -+ 00 (in particular, if I belongs to the Lipschitz class on [0, 1) of order 0:) then

{ O(1/nO')

//O"n(x, f) - I(x)// = O«lnn)/n) for 0 < 0: < 1,

for 0: = 1.

SUMMABILITY OF WALSH SERIES

PROOF. Substitute the estimate

~(~ f) < c l 2.. 2;' - 2;<>

into (4.3.1) to see that the difference I1l7n (x,f) - f(x)1I is dominated by

and

1 k . 1 1 2k L 2' . 2;<> :::; C2 2(k+1)<>

i=O

1 k ; 1 k+l k -"'2 ·_=-<C3 -2k L.J 2; 2k - 2k+l ;=0

for 0 < Q < 1,

for Q = 1

103

where 2k < n :::; 2k+1. Since such integers n, k satisfy k :::; C4 ln n, these inequalities establish the promised estimates. •

§4.4. (C, 1) summability of Fourier-StieItjes series. Ir. this section we again use the technique of differentiation through binary nets,

this time to establish that every Fourier- Stieltjes series in the Walsh system (in particular, every Walsh- Fourier series) is (C, 1) summable almost everywhere on [0,1).

We begin with two lemmas connected with the dyadic metric p*(x, t) defined in (1.2.19).

4.4.1. Let I = (a, b) C [0,1) be an interval and set d(x) = min{p*(x, a), p*(x, b)}. Then d(x) :::; 2111 for all x E I.

PROOF. Let 6. (k) be the dyadic interval of minimal rank which is contained in the interval I. Thus for any x E I, the interval 6.~k-l) which contains x also contains either a or b. Hence min{p*(x, a), p*(x, b)} :::; 1/2k - 1 . But III;::: 1/2k since 6.(k) is a subset of I. Consequently, d(x) :::; 1/2k - 1 :::; 2111 .• 4.4.2. Let E be a closed subset of [0, 1) and define a function p by

Then the series

p(x) == p*(x, E) == inf p*(x, y). yEE

00 1 '" 2j p( x EEl --:- ) L.J 21 j=l

converges for almost every x in E.

PROOF. Let {In} be the collection of intervals contiguous to E, i.e.,

00

U In = (0,1) \ E. n=l

104 CHAPTER 4

It is clear that 2:::"=1 IInl ~ 1. Fix 0 < h < 1 and consider the function p(x EB h). If we denote the set {x = y EB h : y E E} by E EB h then we have by translation invariance of Lebesgue integration that

(4.4.1) [ p( x EB h) dx = [ p( x ) dx JE JEtBh

00

= L [ p(x)dx n=l J(EtBh)nln

= L + L == Sl(h) + S2(h). n:IInl<h n:llnl2:h

For each interval In define the function dn (x) as in 4.4.1 and notice that p( x) :<::::

dn(x) for all x E In. Hence by lemma 4.4.1 we have p(x) :<:::: 211,,1 for every x E In. Moreover, mes((E EB h) U In) :<:::: IInl. Consequently,

[ p(x) dx :<:::: 21In12. J(EtBh)nln

Therefore,

Sl(h):<:::: L 21 In1 2. n:llnl<h

Applying this estimate to h = 2- j for j = 1,2, ... results in

( 4.4.2) t. 2' S, (2-') '" t. 2' CI~'-; 2II. I')

~ t,2II.1' C"~.I-' 2') 00 00

< 2 L IInI 2IInl- 1 = 2 L IInl :<:::: 2. n=l n=l

To estimate the remaining sum S2(h) in (4.4.1), fix IInl ~ h and observe by (1.2.20) that

lyEB h - yl:<:::: p*(YEBh,y) = h.

Thus it is clear that if y E E and x = y EB h lies in In then x is no further from the endpoints of the interval In than h, i.e., mes((E EB h) n In) ~ 21hl. Moreover, it is obvious that p(x) :<:::: p*(y EB h, y) = h for x E (E EB h) n In, where x = y EB hand y E E. Consequently,


Using this estimate for h = 2-i and summing over j we are lead to the following inequality:

00

::; L 41I nl ::; 4. n=1

Therefore, it follows from (4.4.1) through (4.4.3) and a theorem of B. Levy (see A4.4.2) that

00 00

j=1 j=1

Thus the function 2:;1 2ip(x EB -fr) is integrable on E, in particular, almost everywhere finite on E .•

The next result is the main technical lemma of this section. We shall use it to prove that each Fourier-Stieltjes series in the vValsh system is almost everywhere (C, 1) summable.

4.4.3. Let 1/J(x) be a function of bounded variation on [0,1) with 1/J'(x) = f(x) almost everywhere. For each fixed x, let vat (1/J(u) - f(x)u) represent the variation of the function 1/J(u) - f(x)u as u ranges over the interval [0, t). Then

( 4.4.4)

for almost everywhere x in [0,1), where Kn(t) represents the kernel with respect to the Walsh system of the (C, 1) method of summation.

PROOF. Let 'I] > ° and choose by Lusin's Theorem (see A4.1.2) a closed set El C [0,1) such that mesE1 > 1-'1]/2, and such that the function f( x) is continuous on the set E 1 . We may also choose the set El so that it contains no points from the set P U{U;1 PEB2- j }, where P is some countable set which contains all the points

106 CHAPTER 4

of discontinuity of the function V~(1,b(u)). We may also suppose that El contains no dyadic rational points.

By Theorem A4.3.7 we know that

lim -hI V;+h (1,b(u) - f(x)u) = 0 h ..... O

for almost every x E [0,1). Consequently, it follows from 2.5.10 that

(4.4.5) 1 It~

lim -(k) V : (1,b(u) - f(x)u) = 0 k ..... O lb.., I a",

for almost every x E [0,1), where as usual {b.~k) = [a~,,8:)}bO represents the sequence of dyadic intervals which contain x. Apply Egoroff's Theorem (see A4.1.1) to this limit of functions. Thus choose a closed set E2 such that mesE2 > 1 - TJ/2 and

( 4.4.6) lim ~(k) V It: (1,b(u) - f(x)u) = 0 uniformly for x E E2 . k ..... oo lb.., I 0",

Let E = El n E2. Then E is closed, mesE > 1 - 7], and E contains no dyadic rational points and no points from the set PU{U~l P EB 2-i } mentioned above. The function f(x) is continuous on E so there is a constant AI) such that

( 4.4.7) If(x)1 5 AI), x E E.

Moreover, if we set

( 4.4.8) 1 It' Mk == sup sup -(-) Va'''' (1,b(u) - f(x)u)

.~k .,EE lb..," I '"

then we have by (4.4.6) that

( 4.4.9) lim Mk = o. k ..... oo

Since TJ was arbitrary, it suffices to prove (4.4.4) for almost every point x from the set E.

Use (4.2.6) to write

( 4.4.10)

K2k (x EB t)dVot (1,b(u) - f(x)u) = 2k - 1 1 dVot (1,b(u) - f(x)u) 2 "ffi~~k)

k-l

+ 2:=2k - r - 2 1 dV~(1,b(u)-f(x)u). r=O "ffi~;!)

Fix x E E and let x E /j.~k) = [Q~, 11:). Then with the exception of a countable set of points, x EB /j.~k) = /j.~k) (see 1.2.1). By (4.4.8), the first term on the right side of (4.4.10) is dominated by

hence must converge to zero by (4.4.6). Change the index in the sum on the right side of (4.4.10), setting k - r = j. Since with the exception of a countable set of

points, 2-j EB /j.ak ) = /j.~!~i' for j = 1,2, ... k, we see that this sum can be written in the form

~ t2i 1 dV~ (,p(u) - f(x)u). 4 j=l x$2-i$.o.~·)

In particular, it remains to show that the sum

(4.4.11)

converges to zero almost everywhere on E. Fix c > 0 and choose by (4.4.9) and the continuity of f on E an index q = q(c)

so large that

(4.4.12)

and

( 4.4.13) If(y) - f(x)1 < c, x, y E E, Ix - yl ~ Tq.

Divide the sum (4.4.11) into two pieces:

q k

(4.4.14) S = L + L == SI + S2. i=1 j=q+l

It is easy to see that V~ (,p(u) - f(x)u) ~ vot (,p(u)) + If(x)lt. Therefore, we have by (4.4.7) that

SI ~ t2j ( r . dV~ (,p(u)) + 1/j.(k)llf(X)I) j=l JX$2-1$.o.~.)

::; (m~ 1 dVot (,p(u)) + A1J· 2- k ) t2 j . 1:::;,:::;q x$2-i$.o.~k) i=1

108 CHAPTER 4

By the choice of E, the function vat ('!/J( u)) is continuous at each point of the form t = x EB 2- j , for x E E. Hence for each fixed q, the right part of this last inequality ( and thus S1 itself) converges to zero as k -+ 00. Consequently,

(4.4.15)

for k sufficiently large. We now estimate the sum S2. For q < j $ k and x E E let Yj = Yj(x) represent

a point from E for which the minimum minYEE p*(x EB 2- j , y) is attained. Thus Yj can be thought of as a point in E nearest to x EB 2- j as measured by the metric p*, i.e., p*(x EB 2-j ,E) = p*(x EB 2- j ,Yj). Notice that this minimum is attained since E is closed, contains no dyadic rational points, and thus by 1.2.5 the function p*(x EB 2-j , y) is continuous in Y on E. But x E E. This means that

( 4.4.16) *( - j ) <' *( rr j) 1 p x EB 2 ,Yj ..:::. P x EB ~ ,x = 2j·

Hence it follows from (1.2.20), (1.2.19), and the fact) > q that

(4.4.17) Ix - Yjl $ p*(x, Yj) = A(g(X) EB g(Yj) EB g(Tj) EB (Tj))

$p*(xEBTj,Yj)+Tj $2·Tj $Tq.

The integral in each term of the sum S2 can be estimated by observing that the variation of a sum of functions does not exceed the sum of the variations of these functions:

1 dVot ('!/J(u) - f(x)u) $ 1 dVot ('!/J(u) - f(Yj)u) xEB2-iEBL!.~k) xEB2-i EBL!.~k)

+ 1 If(Yj) - f(x)1 dt xEB2- j EBA~k)

== II + h·

By (4.4.13) and (4.4.17), 12 $ E:2- k. To estimate II, let )1 $ k be the maximal rank of a dyadic interval which contains both points x EB 2- j and Yj. By (4.4.16) and 1.2.4 it is clear that

( 4.4.18)

Since the rank jI is maximal, the points x EB2- j and Yj belong to different intervals of rank jI + 1. Consequently, we have by 1.2.4 in the notation of 4.4.2 that

( 4.4.19)

Since j ~ k, the interval A~1d which contains Yj also contains the interval

Since Yj E E, we obtain from (4.4.8), (4.4.12), (4.4.18), and (4.4.19) that

where A~{tl = [a~~, ,8~~) denotes the interval which contains Yj. Combine these estimates of II and 12 to verify the inequality

Applying these inequalities for all j which satisfy q < j ~ k, we obtain

k

52 = '" 2j ·1 dVot (1fJ(u) - f(x)u) ~ _' A(k) j=q+l x$2 J $""'0

k

~ L 2j e(2- k + 2p(x E9 2-j » j=q+l

00

~ 2e + 2e L 2ip(x E9 2- j ).

i=l

In view of (4.4.15), we at last can estimate the sum (4.4.14):

00

5 = 51 + 52 ~ 3e + 2e L2ip(x E9 2-i ). j=1

Since e was arbitrary, it follows that the sum 5 converges to zero at every point x from the set E for which the series on the right side of this last inequality converges. By 4.4.2 this happens for almost every x in E. We conclude that (4.4.4) holds for almost every x. •

110 CHAPTER 4

4.4.4. Let 'ljJ(x) be a function of bounded variation on [0,1) with 'ljJ'(x) = f(x) almost everywhere. Then the Fourier-Stieltjes series of the function 'ljJ( x) in the Walsh system is (C, 1) summable to f( x) almost everywhere on [0,1).

PROOF. Analogous to formula (4.2.2) it is easy to see that the (C, 1) means of the given Fourier-Stieltjes series can be written in the form

We need to show

lim (11 Kn(XEBt)d'ljJ(t)-f(x») = lim 11 Kn(xEBt)d('ljJ(t)-f(x)t)=O n--+oo 0 n--+oo 0

almost everywhere on [0,1). We shall show that this expression holds at every point which satisfies (4.4.4) and (4.4.5).

Fix such a point x E [0,1). Use identity (A4.3.6) for Stieltjes integrals and apply formula (4.2.10) to the kernel I\n(t), where n is written in the form (4.2.9). Since n(i) ::; 2k - i+I, we obtain the following estimate:

Set

and

III I<n(xEBt)d('ljJ(t) -f(x)t I::; l1IKn(xEBt)ldvot('ljJ(U)-f(X)U)

::;.!.. tei2i 11 K2j(XEBt)dVot ('ljJ(tt)-f(x)u) n i=O 0

+.!.. tei2i 11 D2j(xEBt)dVot ('ljJ(u)-f(x)u) n i=O 0

= E(n) + E(n) - I 2·

11 k

9i(X) = 0 D2j (x EB t) dVot ('ljJ(u) - f(x)u) = 2i V!; ('ljJ(u) - f(x)u).

Notice as j - 00, that hAx) - 0 by (4.4.4), and that 9i(x) - 0 by (4.4.5). Moreover, the sums E~n) and E~n) can be viewed as n-th means of the sequences {hi(x)} and {9i(x)} for the finite linear method of summation induced by the matrix whose n-th row is given by ajn = ei2i In for j ::; k and 0 for j > k. Since

SUMMABILITY OF WALSH SERIES III

An = Ej=o ajn = 1 for all n, it follows from 4.1.2 that the sums ~~n) and ~~n) also converge to zero. This completes the proof of 4.4.4 .•

It is clear that each Walsh-Fourier series of a function f E L[O, 1) can be viewed as a Fourier-Stieltjes series (in the Walsh system) of the absolutely continuous function t/J(x) = fo" f(t) dt. Hence Theorem 4.4.4 contains the following result as a special case:

4.4.5. The Walsh-Fourier series of a function f E L[O, 1) is (C,l) summable to f(x) almost everywhere on [0,1).

Chapter 5

OPERATORS IN THE THEORY OF WALSH-FOURIER SERIES

§5.1. Some information from the theory of operators on spaces of measurable functions.

In this chapter, and the next, we shall obtain several results about Walsh-Fourier series by using properties of operators which take one space of measurable functions to another. We begin with definitions and some simple properties of the class of operators we wish to use.

We shall define the distrib1Ltion function 1 )"f(Y) of a function f, measurable on [0,1), by

(5.1.1) )"f(Y) == mes{x : If(x)1 > V}·

We shall establish a formula which allows us to express the LP[O,l) norm of f in terms of the function).. f(Y):

(5.1.2)

First, write the right side of (5.1.2) in the following form:

P 100 yP- 1 )..f(y)dy=p 100

yP-l (11 X{x:If(Xll>y}(x,Y)dX) dy,

where, as usual, XE represents the characteristic function of the set E. Apply Fubini's Theorem (see A4.4.4) to this last integral and rewrite it using the fact that for a fixed x we have

X{x:lf(xll>Y}(X, y) = { ~ We obtain

for Y < If(x)l, for y ?:: If(x)l.

p 100 yp - 1 (11

X{x:lf(xll>Y}(X' y) dX) dy = 11 (p If(Xll yP-l dY) dx

= 11 If(x)IP dx.

1 More precisely, this is the distribution function of 11(x)l.

112

OPERATORS IN THE THEORY OF WALSH-FOURIER SERIES 113

This completes the proof of (5.1.2). Besides linear operators in the following, we shall use the pointwise upper bound

of a sequence of linear operators. Such an operator T turns out to be sublinear (i.e., convex downward). This means that if f and 9 belong to some linear space of functions which make up the domain of T then in the space of measurable functions which make up the range of T we have

(5.1.3) IT(f + g)1 ::; ITfl + ITgl, IT(c!)1 = kllTfl almost everywhere.

It is easy to see that if T is a sublinear operator and f, 9 are functions in the domain of T then

{x: ITfl::; y} nix : ITgl ::; y} c {x: IT(f + g)1 ::; 2y}.

By taking complements, we see that

{x: IT(f + g)1 > 2y} C {x: ITfl > y} U{x: ITgl > v}·

In particular, using the notation of (5.1.1) we arrive at

(5.1.4)

If an operator T takes the space LP [0,1) into itself for some 1 ::; p ::; CXJ and satisfies the inequality

(5.1.5) liT flip::; Gil flip (i.e., in the case that T is a linear operator it is a bounded operator in the usual sense), then we shall say that T is of strong type (p,p).

If

(5.1.6)

for y > 0 and f E LP [0,1), then we say that T is of weak type (p, p), and we shall refer to (5.1.6) as the inequality of weak type (p,p) for the operator T.

5.1.1. If a sublinear operator T is of strong type (p,p) with constant G, then it is of weak type (p, p) with the same constant.

PROOF. We need only observe that

yP ATf(Y) = [ yP dx::; t ITf(x)IP dx. • J{x:ITfl>y} Jo

Notice that this last inequality contains the famous Chebyshev inequality:

(5.1. 7) mes{x: Ifl > y} ::; y-P 11 If(x)IP dx.

We shall need the following special case of the Marcinkiewicz Interpolation Theorem:

114 CHAPTER 5

5.1.2. Let 1 :::; PI < P2 < 00 be two numbers and suppose a sublinear operator T satisfies

for y > 0, f E LPj [0, 1), and j = 1,2, where the Cj's are absolute COl]stants independent of y and f. In the case when P2 = 00, suppose that T is of strong type, namely, IITfiloo :::; C2 11flloo. Then

for all f E UfO, 1), PI 0, set

fY(x) == { f(x), 0,

and fy(x) == f(x) - fY(x).

if P2 < 00,

if P2 = 00.

if If(x)1 :::; Ay,

if If(x)1 > Ay,

Since T is a sublinear operator and f(x) = fy(x) + fY(x), we have by (5.1.4) that

Thus in the case when P2 < 00 we have by hypotheses that

ATf(2y) :::; Ci1y-Pl 11 Ify(x)IPl dx + q2 y - P2 11 IfY(x)iP2 elx

= Ci'y-P' 11 X{x:lf(x)I>Ay}(X,y)lf(x)iPl dx

+ q2 y- P211 X{x:lf(x)I~Ay}(X,y)lf(x)iP2 dx.

Multiply this inequality by p2PyP-l and integrate the resulting product with respect to y from ° to 00. Apply Fubini's Theorem in the same way we did to prove (5.1.2).

Thus verify that

Substituting the value for A chosen at the beginning of this proof establishes the promised inequality.

In the case P2 = 00 we have IIf'lIoo ::; Ay = Ci1y. Hence liT Plloo ::; C2 11fY il00 ::; y and we have by definition (5.1.1) that ATf'(Y) = 0. Consequently, in the string of inequalities above we can simply omit the terms involving P2 when P2 = 00 .•

These observations lead to a useful result about sequences of operators.

5.1.3. Let {Tn }::'=1 be a. sequence of linear operators such that the sublineal' oper-ator

f(x) --+ Tf(x) = supITnf(x)1 n2:1

is of weak type (p,p) for some 1 ::; p < 00. If limn __ oo Tn</J(x) = </J(x) almost everywhere on [0,1) for all functions </J in some dense subset of LP[O, 1), then

(5.1.8) lim Tnf(x) = f(x) n--oo

almost everywhere on [0,1)

for all f E LP[O, 1).

PROOF. Each point x for which (5.1.8) fails to hold satisfies the inequality

limsupITnf(x) - f(x)1 > 0. n--oo

116 CHAPTER 5

Hence each such point belongs to the set

. 1 Pn == {x : hmsup ITd(x) - f(x)1 > -}

k ..... oo n

for some natural number n. If P represents the set of aU points x where (5.1.8) fails then P = U::"=l Pn . Since Pn +1 :J Pn implies mesP = limn ..... oomesPn, we will conclude that mesP = 0 if we show mesPn :5 lin for n = 1,2, .... In particular, it suffices to show that

(5.1.9) mes{x: limsupITnf(x) - f(x)1 > c} < c. n ..... oo

Toward this, notice first that

lim sup ITnf(x) - f(x)1 :5 lim sup ITnf(x) - Tn.p(x)1 n-+oo n-+oo

+ limsup IT".p(x) - .p(x)1 + If(x) - .p(x)l·

Repeating the argument which lead to (5.1.4), it is not difficult to see that

(5.1.10) mes{x : lim sup ITnf(x) - f(x)1 > c}

n ..... oo :5 mes{x : lim sup ITnf(x) - Tn.p(x)1 > c/3}

n ..... oo + mes{x : lim sup ITn¢(x) - .p(x)1 > c/3}

n ..... oo + mes{x : If(x) - .p(x)1 > c/3}.

Choose the function .p from the dense set which is mentioned in the hypotheses of this theorem. Thus the second term on the right side of (5.1.10) reduces to zero. Applying inequality (5.1.7) to the function .p(x) - f(x), we can estimate the third term by

(5.1.11) 3P

mes{x: If(x) - .p(x)1 > c/3} :5 cP Ilf - ¢II~·

Finally, the set in the first term is evidently a subset of

{x: sup ITn(f(x) - .p(:r»1 > c/3}. n~l

Since the operator T is of weak type (p,p) (see (5.1.6», it follows that the first term on the right side of (5.1.10) can be estimated by

(5.1.12) mes{x : lim sup ITn(f(x) - ¢(x»1 > c/3}

3P :5 mes{x : IT(f(x) - .p(x))1 > c/3} :5 CP cP IIf - .pll~·

Substituting (5.1.11) and (5.1.12) into (5.1.10), we obtain

3P mes{x: limsupITnf(x) - f(x)1 > e:} ~ (1 + CP)-lIf - <1>11:.

n-+oo e:P

Thus complete the proof by choosing the function so near f, in the norm of LP[O, 1), that

£H1/p

Ilf - <l>lIp < 3(1 + CP)1/p· •

§5.2. The Hardy-Littlewood maximal operator corresponding to sequences of dyadic nets.

In the theory of trigonometric series the Hardy-Littlewood operator plays a major role. This operator assigns to each integrable function f a "maximal function" defined by

1 l x+h Mf(x) = sup 2h If(t)1 dt.

h>O x-h

In the theory of Walsh series a similar role is played by an operator based on the dyadic maximal f1tnction

(5.2.1) Af(x) = sup ~(k) ( If(t)1 dt, k~O lD.x I Jt:.~k)

where {D.~k)}f::o is the sequence of dyadic intervals which contain x, i.e., the sequence of nodes from the binary nets Nk (see §1.1) which shrink to x.

We shall establish a number of properties for the operator A.

5.2.1. The operator A is of weak type (1, 1), namely, for all y > 0 and f E L[O, 1) the inequality

(5.2.2) 1 11 mes{x: Af(x) > y} ~ - If(t)1 dt. y 0

is satisfied.

PROOF. We introduce the notation

(5.2.3) Ey = {x : Af(x) > y}.

Suppose first that 0 < y ~ fo1If(t)1 dt. Since by (5.2.1) it is clear that for all x

Af(x) ~ 11 If(t)1 dt,

118 CHAPTERS

our assumption implies that Ey = [0,1) and thus (5.2.2) is obviously satisfied. Suppose now that

(5.2.4) y> 11 IJ(t)1 elt.

In this case we shall show that Ey can be written as a finite or count ably infinite union of dyadic intervals tl j , perhaps of different ranks, such that the following conditions are satisfied:

{

1)tljntlj=0forj=/:-i,

2) Ey = U~l tlj, (5.2.5)

3) y < I~jllj IJ(t)1 elt ::; 2y for all tl j •

For each x E E y, consider the sequence of intervals {tl~k)} which contain x, and choose from among them the interval of smallest rank which satisfies

~(k) ( IJ(t)1 elt > y. Itlx I JA~k)

Denote this interval by tlx. Notice by (5.2.4) that tlx =/:- tl~O) = [0,1) and by minimality that we can find a dyadic interval tl; from the sequence {tl~k)} of rank one less than that of tlx such that tlx C tl;, Itl;1 = 2ltl xl, and

1~;ll~ IJ(t)1 elt ::; y.

Hence

I~xllx IJ(t)1 dt = 1~;llx IJ(t)1 elt::; 1~;ll; IJ(t)1 dt::; 2y.

Moreover, by construction we see that tlx is a subset of E y. Hence Ey = UXEEv tl x. In particular, it remains to choose from the family

( 5.2.6)

a sequence of non-overlapping intervals whose union is E y •

Toward this, recall that a pair of dyadic intervals are either non- overlapping or one is a subset of the other. Choose from (5.2.6) all intervals of rank 1, if they exist, then all intervals of rank 2, e.t.c., with the stipulation that at the n-th stage we choose only those intervals from (5.2.6) of rank n which are not contained in the intervals of lower rank already chosen. In this way we obtain a sequence {tl j} of non-overlapping intervals which satisfy all the conditions in (5.2.5). Using these conditions, we verify the inequality

00 00 1 1 1 1 1 11 mesEy = L Itljl ::; L - IJ(t)1 dt = - IJ(t)1 dt ::; - lI(t)1 dt. j=l j=l Y Aj Y E. Y 0

Thus (5.2.2) is proved .•

5.2.2. The operator A is of type (00,00), namely,

IIAfiloo ~ IlJiloo. PROOF. This result follows directly from definition (5.2.1) and the fact that

_1_ ( _1_ (k) _ 1~(k)1 J~(k) lJ(t)1 dt ~ 1~(k)1 IIfllool~ 1- IIflloo

for every ~(k) .• Combining 5.2.1 and 5.2.2 with the interpolation theorem 5.1.2 for PI = 1 and

P2 = 00 leads to a proof of the following theorem:

5.2.3. The operator A is of type (p,p) for all 1 < p ~ 00, namely,

Along with the operator A we shall also consider the operator H, which is defined for each function f E L(Q, 1) by

(5.2.7)

where S2k (x, f) represents the partial sum of the Walsh- Fourier series of f of order 2k.

By (2.1.11), we have

Hf(x)=sUP~(k) If f(t)dtl· k~O I~x 1 J~~k)

Thus it is clear that Q ~ H f(x) ~ Af(x) for all x E [0,1). In particular, the following result is a corollary of 5.2.3.

5.2.4. The operator H is of type (p,p) for alII < p ~ 00, namely,

§5.3. Partial sums of Walsh-Fourier series as operators. Below we shall use the so-called "modified" Dirichlet kernel, which is defined by

the identity

(5.3.1)

where Dn(t) is the usual Dirichlet kernel in the Walsh system which we defined in (1.4.8). The kernel D~(t) can be used to generate the expression

(5.3.2) S~(x, f) = t f(t)D~(x EEl t) dt, Jo

120 CHAPTERS

which is called the modified partial sum. Using (5.3.1), (1.2.15), and (2.1.10), we see that

(5.3.3)

If the dyadic expansion of an integer n has the form n = 2:7=0 c j 2 j , where Ck = 1,

and we denote the truncated expansion by n j = 2:~=k- j cj2 j, then the definition of the Walsh functions (see § 1.1) can be used to write the Dirichlet kernel in the following way:

n-I

Dn(t) = L Wi(t) i=O

2k -I L Wj(t) + ck-I

i=O

nj -1 n-I

+ Ck-j L w;(t) + ... + co L w;(t) i=nk_l

2.- 1 _1

= D2k(t) +Ck-I L W2 k +i(t)+ ... ;=0

2 k - j -I

+ck-j L Wnj_l+j(t)+···+cOWnk_l(t) i=O

2k - I _1

= D2k(t)+ck-IW2k(t) L Wi(t)+ ... i=O

k 2k - j _1 k

+ Ck-j II (W2m(t)tm L Wi(t) + ... + co II (W2 m(t))ym wo (t) m=k-j+1 i=O m=l

k k

= L Ck-j II (W2 m (t))em D2 k-j (t) j=O m=k-j+l

k k

= LCj II (W2m(t))e m D2j(t). j=O m=j+l

Multiply this expression by wn(t) = rr:=0(W2m(t))em and recall that w?(t) = 1 for all i. We obtain the following formula for the kernel (5.3.1):

k j

D~(t) = LCj II (W2m(t))e m D2j(t). j=O m=O

But each function D 2; (t) is different from zero only on the interval .6.~j), and on this interval W2m(t) = 1 for m < j. Consequently, the modified Dirichlet kernel can be written in the form

k k

(5.3.4) D~(t) = LCjW2;(t)D2;(t) = LcjD;;(t). j=O j=O

The following important property shows why the modified kernel D~(t) is more convenient to work with than the usual kernel Dn(t).

5.3.1. Ifx tf. .6.(0) for some dyadic interval .6.(0), s ~ 0, tben tbe kernel D~(x EEl t) takes on a constant value for a11 t E .6.(0).

PROOF. By (5.3.4), it is enough to prove the theorem for the functions

for all j. The function D;;(t), and thus the function D;;(x EEl t), is constant on each interval .6.U+l) of rank j + 1. Hence the theorem is obvious if the rank of the interval .6.(0) satisfies s ~ j + l.

If s < j + 1, i.e., s ::; j, then the conditions t E .6.(0) and x tf. .6.(8) imply that

x EEl t ~ .6.60). Hence by (1.4.13) we have D 2; (x EEl t) = O. In particular, the kernel D;;(x EElt) is again constant with respect to t on .6.(0) .•

vVe shall now consider the maps which take an integrable function J to the partial sums Sn(x, I) or to the modified partial sums S~(x, I). These maps are evidently linear operators, defined on the space of integrahle functions, and we shall denote them by Sn and S~. These operators enjoy the following properties:

5.3.2. Tbe operators Sn and S~ are of weak type (1,1) and of strong type (p, p) for alII < p < 00 witb constants wbich do not depend on n.

PROOF. Notice first of all that if one of the operators Sn and S~ are of weak or strong type (p, p) then the other one is also of the same type. Indeed, if Sn is of strong type (p,p) with constant Cp then since Iwn(t)1 = 1 it is obvious that

Hence S~ is also of strong type (p,p) with the same constant Cpo On the other hand, if S~ is of strong type (p,p) then by (5.3.3), with the function J in place of wnJ, we have

A similar argument shows that Sn is of weak type (p, p) if and only if the operator S~ is also. Consequently, it suffices to prove the theorem for one of the operators Sn or S~. For the most part, it is more convenient to work the operator S~.

122 CHAPTER 5

We begin by noting that Parseval's formula (see 2.6.5) for the Walsh system implies IISn(J)1I2 ~ IIJII2, i.e., Sn is of strong type (2,2) with constant 1. In view of what we have already proved, we see that S~ is also of type (2,2) with the same constant, i.e.,

(5.3.5) IIS~JII2 ~ IIJ112.

We shall now prove that S~ is of weak type (1,1), i.e.,

(5.3.6) ell mes{x: IS~(x, 1)1> y} ~ - IJ(t)1 dt, Y 0

where C does not depend on n. Fix J E L[O, 1). Since for 0 < Y ~ Jo1 IJ( t)1 elt the inequality (5.3.6) is evidently satisfied, with constant C = 1, we may suppose that y > J: IJ(t)1 elt. Define the set Ey by (5.2.3) and decompose it into a union of dyadic intervals D.j which sa.tisfy conditions (5.2.5). Using these intervals, define two auxiliary functions:

(5.3.7) {_I f J(t)dt

g(x) = lD.jl J~j J(x)

for xED. j C Ey,

for x E [0, 1) \ Ey

and hex) = J(x) - g(x). If x E Ey = U~1 D.j, then it follows from (5.2.5) that

(5.3.8) Ig(x)1 ~ I~il Lj

IJ(t)1 dt ~ 2y.

If x ft Ey then use the definitions of the set Ey and the maximal operator (5.2.1) to verify the inequality

(5.3.9)

for all dyadic intervals D.~k) containing x. But Theorem 2.5.10 implies

lim ~(k) f J(t)elt = J(x) k--->oo lD. x I J ~~k)

for almost every x E [0,1). Consequently, we ha.ve by (5.3.9) that IJ(x)1 ~ y holds almost everywhere on [0,1) \ E y . This inequality together wit.h (5.3.8) and (5.3.7) shows that

Ig(x)1 ~ 2y almost everywhere on [0,1).

OPERA TORS IN THE THEORY OF WALSH-FOURIER SERIES 123

Therefore,

(5.3.10)

Moreover, (5.3.7) implies that

L. Ig(t)1 dt =1 L. J(t) dt I:::; L. IJ(t)1 dt. I I I

In particular, we have

(5.3.11) 11 Ig(t)1 dt = L { Ig(t)1 dt + ( Ig(t)1 dt :::; 11 IJ(t)! dt. ° j 11:>.j AO,I)\E, °

Combine the Chebyshev inequality (5.1.7) for p = 2, inequality (5.3.5), written for the function g, and also inequalities (5.3.10) and (5.3.11). We obtain

mes{x: !S!(x,g)! ~~}:::; (~)211!S!(t,g)!2dt

:::; ~ 11 !g(tW dt Y 0

:::; ~ 11 !g(t)! dt :::; ~ 11 !J(t)1 dt. Y 0 Y 0

We shall now prove an analogous inequality for the function hex). Notice that hex) = 0 for x E [0,1) \ Ey and II:>.. h(t) dt = 0 for any tl. j C Ey. Fix x f/ Ey and notice by 5.3.1 that the kernel D:tx EB t) takes a constant value on tl.j which we shall denote by D!(x EEl tl.j). It follows that

S!(x, h) = 11 hCt)D:Cx EB t) dt

= ~ L. h(t)D:(x EB t) dt J I

= ~D:CxEBtl.j) L. h(t)dt=O. J I

Consequently, {x : S!(x, h) # O} C Ey. In view of the third condition of (5.2.5), it follows that

mes{x: S!(x, h) > y/2} :::; mes{x : S!(x, h) > O} :::; L !tl.j ! j

:::; L ~ ( !Jet)! dt :::; ~ t !JCt)! dt. i Y 1aj Y 10

124 CHAPTER 5

Use these inequalities together with (5.3.12) and (5.1.4) applied to the operator S~ and the functions 9 and h. Thus verify that

911 mes{x: IS~(x,J)1 > y}::; - lJ(t)ldt. Y 0

This proves that the operator S~ is of weak type (1,1) with constant 9. We have already noted (see (5.3.5)) that the operator S~ is of strong type (2,2), so by 5.1.1, it is also of weak type (2,2) with constant 1. Hence by the interpolation theorem 5.1.2, S~ is of strong type (p,p) for alII < p::; 2, i.e.,

Furthermore, we see by 5.1.1 that the constant Cp depends only on the constants 1 and 9 and thus does not depend on n.

It remains to extend this last inequality to the case 2 < P < 00. To accomplish this we pass to the dual space (see A5.2.2). Let f E UfO, 1) for 2 < p < 00 and <fJ E L[O, l)q, where l/p + l/q = 1. Thus 1 < q < 2 and by the inequality already proved, we have

( 5.3.13)

By Fubini's Theorem (see A4.4.4), Holder's inequality (see A5.2.2), and (5.3.13) we obtain

(5.3.14)

111 S~(x,J)<fJ(x)dx 1=111 (11 f(t)D~(x EBt)dt) <fJ(x)dx I

=111 f(t) (11 D~(x EB t)<fJ(x) dX) dt I

=111 f(t)S~(t,<fJ)dt I ::; IIfllpIIS~(<fJ)lIq ::; Cqllfllpll<fJllq'

Since F( <fJ) = Jo1 S~ (x, J)<fJ( x) dx is a functional on the space Lq [0, 1) with norm precisely IIS~(x, f)lIp (see A5.3.2), we see by (5.3.14) that this norm satisfies the inequality (see A5.3.1)

But this means that the operator S~ is of strong type (p, p) for 2 < p < 00. Thus the theorem is proved. I

§5.4. Convergence of Walsh-Fourier series in LP[O, 1). Theorem 5.3.2 contains the following important result:

5.4.1. If IE V[O, 1) for some 1 < p < 00 then

n -t 00,

i.e., the partial sums of the Walsh-Fourier series of I converge to I in the LP norm.

PROOF. We shall use the fact (see Theorem 2.6.4) that the Walsh polynomials are a dense subset of LP[O, 1).

Notice for any polynomial T and any integer n which exceeds the degree of this polynomial, that Sn(x, T) = T(x). For such an n we have by Theorem 5.3.2 that

Consequently,

Therefore, if we choose the polynomial T such that liT - Ilip < c:J(1 + Cp) then III - Sn(f)lIp < c: for any integer n which exceeds the degree of the polynomial T. Since c: was arbitrary, this proves the theorem. •

We shall now show that the preceding theorem fails to hold for p = 1. Namely, the following result is true:

5.4.2. There exists a Walsh-Fourier series which diverges in tile norm of the space L[O,I).

PROOF. We shall show that the function we search for can' be defined by the senes

(5.4.1)

First, the series (5.4.1) converges in the norm ofL[O, 1) and thus defines a function IE L[O, 1), since by (1.4.13)

11 IW2k3(x)D2k3(x)1 dx = 11 D2k3(X) dx = 1.

Next, each term in (5.4.1) is by (1.2.17) a Walsh polynomial

126 CHAPTER 5

and for different k's the polynomials Pk(X) do not contain any common terms. Hence it is not difficult to verify

Therefore,

~ 1 I(i) = k2 '

k 3 1 2 +mk-1

IS2k3 +mk (x,f) - S2 k3 (X,f)1 =1 k2 L w;(x) 1 ;=2 k3

1 mk-l 1 =1 k2 W2k 3 (X) L w;(x) 1= pIDmk(x)1

;=0

for any m" which satisfies 2,,3 + m" :::; 2k3+1 • Specializing this inequality to a number m" given by Theorem 2.2.2 which also satisfies 2k"-1 < m" < 2,,3, we sec that 11 1 log2 m" k3 - 1 k

o IS2k3+m.(X,f) - S2.3(X,f)ldx?: 1.2 4 > 4k2 > 8

for k ?: 2. In particular, the Walsh-Fourier series of the function I cannot converge in the L[O, 1) norm .•

Chapter 6


§6.1. Existence and properties of generalized multiplicative transforms. Let 1 ::; p < 00. A complex valued function f(x) is said to belong to LP(O,oo)

if Iooo If(x)IP dx < 00. The norm of f(x) in the space LP(O, 00) will be denoted by IIfllp and is defined by

IIfllp = (100 If(x)\P dX) lIP.

The limit of a function f(x, a) as a -4 00 in the LP(O, 00) norm will be denoted by lim pf(x, a), i.e., the equation

a-+oo

f(x) = lim pf(x,a) a-+oo

means that IIf(x) - f(x,a)lIp -4 0, a -4 00.

Let f(x) E Ll(O,oo). For the collection 'P = ( ... ,P-j, ... ,P-l,Pl, ... ,Pj, ... ) given by (1.5.22), and the corresponding function x(x, y) given by (1.5.33), define integral transforms of f(x) by

(6.1.1) Flf](y) == j(y) = 100 f(x)x(x,y)dx

and

(6.1.2) F·IJ](x) == rex) = 100 f(t)x(x, t) dt.

We shall call these direct multiplicative transforms. Notice first of all that if the collection (1.5.22) is symmetric, i.e., P_j = Pj for all

j = 1,2, ... , then

(6.1.3) FIJ](y) = F*IJ](y)· The transforms (6.1.1) and (6.1.2) are analogues of the classical Fourier transform.

For a function f E LP(O, 00) with 1 < P ::; 2, the direct multiplicative LPtransforms will be defined by

(6.1.4) FIJ](y) == i(y) = lim p' r f(x)x(x, y) dx Q-foOO Jo

and

(6.1.5) F*lf](x) == rex) = lim pI r f(t)x(x, t) dt a'--'oo io

where IIp + IIp' = 1. In view of the similarity of definitions (6.1.1) and (6.1.2), we need only look a.t

the properties of the transform (6.1.1).

127

128 CHAPTER 6

6.1.1. The transform F(fJ is additive and homogeneous, i.e., iff = I:J=l ajiJ and

jj represents F[Jil for j = 1,2, ... , k, then

k k

= LajF[Jil = Lajjj. j=l j=l

(See 2.1.1.) PROOF. This property follows immediately from the fact that the integral is both

additive and homogeneous .•

6.1.2. Translation of the a.rgument of f(x) by the quantity ffih for h > 0 corresponds to multiplication of the transform F(fl by X(h, y), i.e.,

100 f(x Ell h)x(x,y)dx = x(h,y)F[fl·

Similarly, translation byeh corresponds to multiplication by X(h, y).

(See 2.1.2.) PROOF. Make the change of variables1 v = x ffi h and use (1.5.34) together with

the fact that (x ffi h) e h = x = ve h. We obtain

100 f(xffih)x(x,y)dx= 100

f(v)x(v8h,y)dv

= 100 f(v)x(v,y)x(h,y)dv = x(h,y).F[fl·

Similarly, since (x 8 h) ffi h = x = v ffi h we have

100 f(xeh)x(x,y)dx = 100

f(v)x(vffih,y)dv

= 100 f(v)x(v, y)x(h, y) dv = x(h, y)F[JJ. I

We shall say that a function f(x) is the 1'-adic non- symmetric convolution of a

function It (x) wi th another function f2 (x) and will denote it by f( x) = (11 ; h)(.1:) if

f(x) = 100 h(vffix)f2(V)dv.

1 As we have seen, the transformation z - z EI1 h is defined 011 [0,00) only up to a countable set but is measure preserving and 1-1 off this countable set.

GENERALIZED MULTIPLICATIVE TRANSFORMS 129

The non-symmetric convolution of the function h(x) with the function h (x) will be denoted by g(x):

6.1.3. If Ji(x) E L1(0, 00) for j = 1,2 then the convolutions I(x) = (h -; h)(x) and g( x) = (12 -; Id(x) exist, belong to Ll (0,00) and satisfy the inequalities

IIllh ~ IlhlhllhllI and IIglh ~ IIhlhllhlh·

PROOF. The proof is similar to that given in Chapter 2 for (2.1.7). Namely, notice that the functions WI (x, v) = h (v EB x )h( v) and W2(X, v) = h( v EB x )h (v) are measurable. Moreover, at each point where h(v) (respectively, h(v» is finite, i.e., almost everywhere, the identities

respecti vely,

hold. Consequently,

Thus we conclude by Fubini's Theorem that

100 1100 h(v EBx)h(v)dv I dx ~ 100 100

Ih(v EBx)h(v)ldvdx

= 100 100 Ih(v EBx)h(v)ldxdv

= IIhlhllhlh·

Similarly,

100 1100 h(vEBx)h(v)dv I dx ~ Ilhlhllhlh- •

6.1.4. If Ji(x) E Ll(O, 00) for j = 1,2 then the convolutions defined above satisfy

F[J) = F[h)F[h) and F[g) = F[JdF[h)·

130

In particular, F(J] = F[g ].

PROOF. By Fubini's Theorem and 6.1.2, we have

Similarly,

F[f] = f(y) = lco (100 fI(Vffi:r)h(V)dV) x(x,y)dx

= 100 h(v)dv 100 It(vffix)x(x,y)dx

= 100 h(v)x(v,y)dvF[It] = F(Jt]F[h].

F[g] = g(y) = 100 (100 h(v ffi X)fI(V)dV) X(x, y)dx

= 100 It (V) dv 100 h (v EfJ x) x( x, y) dx

= 100 It(v)x(v,y)dvF[h] = F(Jt]F[h]·

Consequently, F[f] = F[g] .•

CHAPTER 6

In this chapter it is convenient to modify somewhat the concepts of continuity and the moduli of continuity which were set forth in the first two chapters. We shall call a function f( x) 'P- continuous at a point x E [0,00) if

(6.1.6) sup If(x ffi h) - f(x)l-t 0 as r -t 00. O-:;'h<I/mr

It is not difficult to show that continuity at a point in the sense of (6.1.6) is equiva.lent (for the case mr = 2r) to continuity with respect to the metric p* which was defined in (1.2.19).

The sequence defined by the relationship

(6.1.7) Wr(f, ~) = wr(f) = sup ( max If( x ffi h) - f( x )1) mr zE[O,co) O-:;'h<I/m r

will be called the 'P-modulus of continuity of the f~mction f( x).

6.1.5. If f(x) E LI(O, 00) then the transform

F(J](y) = f(y)

is pI -continuous on [0,00), where pI = ( ... ,Pj, ... ,PI, P-I, ... ,P_ j, ... ).

PROOF. By property (1.5.34) applied to the function X(x,y) we have

icy (J) h) - i(y) = 100 f(x) (x(x, y (J) h) - x(x, v») dx

= 100 f(x)x(x,y) (x(x,y) -1) dx.

131

Choose an integer No so large that for a given c > 0 and all h > 0 the following inequality holds:

I [00 f(x)x(x,y) (x(x,y) -1) dx I~ 2 [00 If(x)ldx < C. iNa iNo

Let r > 0 satisfy m-r+l > No. Then for all h such that ° ~ h < l/mr and all o ~ x ~ No, we have by (1.5.31) that X-k = 0 for k ~ r, h'-k = 0 for k = 1,2, ... , and hk = ° for k ~ r - 1. Consequently,

( h) (2 . ~ x-khk + Xkh'-k) (2 . ~ X_khk) X x, = exp 1l'l ~ = exp 1l'Z ~ -- = 1. k=l Pk k=l Pk

This means that

l Na f(x)x(x,y) (x(x,y) -1) dx = 0,

i.e., Ii(y (J) h) - i(y)1 < c for 0 ~ h < l/mr . Thus the transform F(jj is plcontinuous on [0,00) .•

Similarly, one can prove that the transform F*[fl is P-continuous on [0,00).

6.1.6. H a function f(x) E LI(O,OO) is P-continuous at the point x = ° and tile transform :r[f] = i(y) is real and non-negative, i.e., J(y) ~ ° for y E [0,00), then J(y) E LI(O, 00) and

100 J(y) dy = f(O).

PROOF. Since i(y) ~ 0, we can write

To prove that the limit of this integral exists as A ----+ 00 it is enough to show it converges through some subsequence of values An for example, for Ar = m r .

132

By (1.5.38) and (1.5.21), we have

lAr 1cy)dy = lmr 100

f(x)x(x,y)dxdy

= 100 f(x)dx lmr

x(x,y)dy

= 100 f(x)D(x,mr)dx

= 11 f(x)D(x, m r) dx

tlmr (l/mr = mr Jo f(x)dx = mr Jo f(OEBx)dx.

Consequently,

CHAPTER 6

100 Imr I 11mr 1cy)dy=lim j(y)dy=limmr f(OEBl~)dx=f(O). I

o r-oo 0 r-oo 0

We shall now prove that the LP-transform defined by (6.1.4) exists.

6.1.7. Let f E LP for some 1 0, r be a natural number, n = [bm r ], ov(r) be the half open interval [v/m r , (v + l)/m r ), and o~(r) = [v/m_ n (v + l)/m_r). Define numbers

and a function

av = ( f(x) dx, J6• (r)

v = 0,1, ... ,

n-1 ( ) q,n(x) = L avX x, _v_ . v=o m-r

In 1.5.6 we proved that the system

{<Pv,r(x)} = { r.!-X (x,~)}OO ym-r m-r v=O

is uniformly bounded and orthonormal on the interval [0, m_ r ) with

l<PvAx)l:::; l/Jm_r = M.

GENERALIZED MULTIPLICATIVE lRANSFORMS 133

Set n-1

iP(X) = L a/lI/J/lAx) = MiPn(x) /1=0

and apply a theorem of lliesz ([17], p. 237). We obtain

where

and

Thus iPn(x) can be estimated by

Hence it follows that

But Holder's inequality implies

la/lIV:S f If(x)IP dx ( f dx)V-1 = ;-1 f If(x)IP dx. 16v (r) 16v (r) m_r 16v (r)

Consequently,

(6.1.9)

for all A :S m-r .

134 CHAPTER 6

Now, we shall show that

(6.1.10) lim q}n(x) = 16 f(y)x(y,x)dy = !(x,b). r-+oo 0

Indeed, by 1.5.4 we have x(y,x) = x(x,lI/m_ r ) for all y E c5~(r) and x < m_ r •

Thus

I q}n(X) -16 f(y)x(y,x)dy I

=1 I: ( f(y) (x(x, _11_) - x(x,y)) dy -16 f(y)x(y,x)dy I 11=0 16~(r) m-r n/m_r

:5 ( If(y)1 dy -+ 0, r -+ 00. J6'n(r)

Therefore, if we let r -+ 00 while A remains fixed, we see by (6.1.9) and (6.1.10) that

A 6 ( 6 ) l/(p-l) 1 11 f(y)x(x,y)dy ( dx:5 l lf(x)IPdx

In particular, if we let A -+ 00, we obtain

(6.1.11) 00 6 ( 6 ) l/(p-l) 1 11 f(y)x(x, y) dy Ipl dx:5 llf(x)IP dx

Apply (6.1.11) to the function

fo(x) = { ~(x) We obtain

for 0 :5 x :5 a,

for a < x < 00.

(00 16 I 10 I 0 fo(y)x(x,y)dy IP dx

(00 16 r I = 10 I 0 f(y)x(x,y)dy- 10 f(y)x(a:,y)dyll' dx

= 100 I!(x, b) -!(x, a)IP' dx

( b) l/(p-l)

:5 l lfo (x)IP dx

( b) l/(p-l)

= l lf(x)I P dx ,

GENERAUZED MULTIPLICATIVE TRANSFORMS 135

(6.1.12) 00 ( b ) l/(p-l)

ll1cx,b)-1cx,a)IP'dx~ llf(x)IPdx

Since f(x) E LP(O,oo), the right side of (6.1.12) converges to zero as a,b -+ 00. Since LP' (0,00) is complete, it follows that l( x, a) converges in the LP' (0,00) norm to some function l(x) E LP'(O,oo). This means that the LP-transform of any f(x) E LP(O,oo) exists for 1 < p ~ 2. Moreover, if we let b -+ 00 in (6.1.11), we obtain

(100 Il(x)IP' dX) IIp' ~ (100 If(x)IP dX) IIp

This verifies inequality (6.1.8) .•

§6.2. Representation offunctions in LI(O, (0) by their multiplicative transforms.

As we saw in §6.1, the direct multiplicative transform

(6.2.1)

exists for every function f( x) E LI (0,00). However, as in the case of the exponential Fourier transform, inversion

(6.2.2) f(x) = 100 l(y)x(x,y)dy

does not hold for every function I ( x) ELI (0, 00 ). Thus it is interesting to ascertain what kinds of conditions, on the functions f(x) and fey), imply the inversion formula (6.2.2).

Before we begin identifying such conditions, we shall establish a result about a method of summability for the integral (6.2.2) which plays the same role that the methods of Fejer and Abel- Poisson do for the proof of inversion of the exponential Fourier transform.

6.2.1. Let I(x) E L1(0, (0) and l(y) be its multiplicative transform (6.2.1). Then

f(x) = lim fm r fey)x(x,y)dy, r ........ oo 10

at each point x ofP-continuity of the function f(x).

PROOF. Since Iooo If(x)1 dx < 00, we can apply Fubini's Theorem to the integral

136 CHAPTER 6

to change the order of integration, i.e.,

Imr(J,x) = 100 f(t)dtlmr x(t8x,y)dy= 100

f(t)D(t8 x ,m r )dt.

Recall from (1.5.38) and (1.5.21) that

if a :S t8x < l/m r ,

if l/m r :S t 8 x < 00.

Thus

Imr(J,x)=m r ( [ f(t)dt) =mr tim. f(xffiu)du. Jo~5.tex<l/m. Jo

Hence, it follows that if x is a point of P-continuity for f(x) then

111m. lim Im.(J,x) = lim mr f(x ffi u)du = f(.7:) .•

r-<XI r-CX) 0

6.2.2. Let f(x) E Ll(O,OO) and suppose that its multiplicathre transform fey) belongs to Ll (0,00). TlIen equation (6.2.2) holds at each point x of P-continllity of the function f( x).

PROOF. Since fey) E Ll(O,oo), the integral in (6.2.2) converges absolutely and thus

100 1m. fey)x(x,y)dy = lim f(y)x(x,y)dy.

o r->oo 0

Applying 6.2.1 thus verifies 6.2.2 .• Theorems 6.1.6 and 6.2.2 contain the following corollary.

6.2.3. If x = a is a point of P-continuity of a function f(x) E Ll (0,00) and if the transform fey) is real and non-negative, then equation (6.2.2) holds at each point of P-continuity of the function f( x).

The Plancherel identity holds for multiplicative transforms:

6.2.4. Suppose that f( x) E Ll (0,00) is bounded and P-continllous, and fey) is its multiplicative transform. Then

PROOF. Consider the P-adic non-symmetric convolution of the function f(x) with itself, i.e.,

We know by 6.1.3 that 1i4>/h ~ IiJlli and by 6.1.4 that

F[~](y) = F[j]F[j](y) = IF[j](y)1 2 2: 0, Y E [0, (0).

Moreover, since J( x) E Ll (0,00) is bounded, the value

is finite, and since J( x) is P-continuous, the function ~(x) is P-continuous at the point x = 0. Hence it follows from 6.1.6 that F[ ~](y) E Ll (0,00). Applying 6.2.3, we obtain

~(x) = 100 J(v EEl x)J(v)dv

= 100 F[~](y)x(x, y) dy

= 100 IF[j](yWx(x, y) dy.

Substituting x = ° into this equation, we see that 6.2.4 holds .• We shall presently establish (6.2.2) in the case when the P- modulus of continuity

(see (6.1.7» of the function f(x) E Ll(O, (0) satisfies a particular growth condition. Before we do this we make some preliminary estimates.

6.2.5. Let n be a natural number, ° < a < 1, and J(x) E Ll (0,00). Then

I n +o< lim icy)x(x, y) dy = 0.

n-+oo n

PROOF. In the following, we shall drop the index P on Dn(x)p and Xn(X)P but retain the index pIon Xn(x)p •. By Fubini's Theorem,

r+ a

InU, a) = in j(y)x(x, y) dy

= I n+o< (100 f(t)X(t;Y)dt) x(x,y)dy

roo r+ a

= io J(t) in x(t8x,y)dydt

= 100 J(t) (D(t8 x,n+a)-D(t8 x,n») dt.

138

Apply (1.5.38). In the case 0 ~ t 8 x < 1 we obtain

(6.2.3)

CHAPTER 6

D(t 8 x, n + 0:) - D(t 8 x, Ii) = Dn(t 8 x) + O:Xn(t 8 x) - Dn(t 8 x)

= O:Xn(t 8 x)

and in the case 1 ~ k ~ t 8 x < k + 1 we obtain

(6.2.4)

Make the change of variables t = Z EB x and use (6.2.3) and (6.2.4) to see that

In(f,o:) = 100 J(zEBx) (D(z,n+o:)-D(z,Ii») dz

= {I J(ZEBx)O:Xn(z)dz + f (HI J(zEBX)xn({Z}) f'" Xk(v)p,dvdz Jo k=I Jk Jo

=0: t J(ZEBX)Xn(z)dz + f t J(k+zEElx)xn(Z) f'" Xk(v)p,dvdz Jo k=I Jo Jo

= 11 (O:J(ZEBX) + ~J(k+ZEBX) l cr Xk(V)p,dV) Xn(z)dz

= 11 F(f,o:,z)Xn(z)dz.

Notice that

lIIF(f,o:,z)ldZ~ 11 (O:IJ(Z EEl x)1 + ~IJ(k+ZEBX)IO:) dz

= 0: 100 IJ(z EEl x)1 dz < 00,

i.e., F(f,o:,z) E LI(D, 1). But by a theorem of Mercer (see [2], p. 66), the Fourier coefficients of an integrable function in any bounded orthonormal system {X n (z ) } converge to zero. Consequently,

lim In(f,o:) = lim t :F(f,o:,z)Xn(z)dz = o .• n-+oo n-+oo Jo

6.2.6. Let rand n be whole numbers with r 2: 1 and mr-I < n < m r . If <p(x) is constant on the intervals bv(r - 1) = [v/mr-I, (v + l)/m r -d and J(x) is Pcontinuous, then

l I / mk 1 I J(x)<p(x)Xn(x) dx I~ -wr-l(f)Ah o mk

for k = 0,1, ... ,1' -1, where Mk = maxO~X9/mk 1¢>(x)l.

PROOF. Write the integral we wish to estimate in the form

(6.2.5) 11/mk (mr-,/md-ll f(x)¢>(x}xn(x)dx = L f(X)¢>(X}xn(x)dx.

o ,,=0 6v (r-l)

Choose a point x" = v/mr-l + ° from each bAr -1) at which the function f(x,,) is defined and set M" = 1¢>(x,,)I. Since

r Xn(x)dx=O, J6 v(r-l)

v = 0,1, ...

(being a sum of Pr-th roots of unity) and since ¢>( x) is constant on b,,( r - 1) we have

I r f(x)¢>(X)Xn(X)dxl=1 r (f(x)-f(x,,))¢>(X)Xn(X)dxl J6 v (r-l) J6v (r-l)

~ r If(x)-f(xv)IMv dx J6v(r-l)

11 / mr - 1

= Mv 0 If(x" Ef) u) - f(x,,)1 du

Mv ~ --wr -l(f).

mr-l

Substituting this estimate into (6.2.5) finishes the proof of this theorem .•

6.2.7. Let l' and n be whole numbers with l' 2: 1, and mr-l < n < m r . Determine coefficients n_j by n = n-l + n-2ml + ... + n-rmr-l' If fELl (0, 00) then

where Wr-l(f) is the P-modulus of continuity of f(x).

PROOF. By Fubini's Theorem we can write

In,r(f,x) = i~-l (100 f(t)X(t.Y)dt) x(x,y)dy

= 100 J(t) i~_l x(t8 x ,y)dydt

= 100 f(t)(D(t 8 x, n) - D(t 8 x, l'hr -r)) dt.

140 CHAPTER 6

By (1.5.38), we have

In,r(f,X) = r f(t)(Dn(tex)-Dmr_,(teX») dt. }0'5,t8X<1

Moreover, by (1.5.20) we can represent the Dirichlet kernel in the following form:

0=1

Hence it follows from the equation

that

where

{ mv Dm.(z)= 0

for 0 ::; Z < l/mv,

for l/mv ::; Z < 1, 1/ = 1,2, ...

+ ~mk-1 (r f(t)(h(tex)xn_rn>r_l(teX)dt) , k=1 }098x<l/mk_l

________________ ~~n-k-l------~ (h(z) = Xn_r+lmr_d···+n_k_lmk(z) 2: X.krnk_l(Z),

'k=O

for k = 1,2, ... , r - 2. But for each 1::; k ::; r - 2 the function ¢lk(Z) is constant Oil

the intervals over - 1), 1/ = 0,1, ... , and satisfies l¢lk(Z)1 ::; n-k. Consequently, we can use 6.2.6 to write

Moreover,

I [ f(t)Xsmr_l (t e x) dt I~ -l-Wr_l (f). JO~tex<l/mr mr-l

Therefore,

1 r-2 1 IJn,r(f,x)1 ~ mr-l(n-r -l)--Wr-l(f) + Lmk-I--n-kWr-I(f)

mr-l k=l mk-l

= (n-l + n-2 + ... + n-r)Wr-I(f) .•

We shall now prove a theorem about inversion of the multiplicative transform of a function f(x) E LI(O,OO).

6.2.8. Let P = { ... ,P-j, ... ,p-I,PI, ... ,Pj, ... }. H f(x) E LI(O,OO) and its Pmodulus of continuity satisfies

(6.2.6)

then (6.2.2) holds at each point ofP-continuity of f(x).

PROOF. Let x be a point of P-continuity of f(x). By 6.2.1 it suffices to show

lim l a icy)x(x, y) dy = 0,

r-+oo fflr-l

a E [mr-l, mr)'

By 6.2.5 we may suppose that a is an integer, i.e., a = n E [mr-l,mr ). But 6.2.7 allows us to write

in icy)x(x,y)dy ~ (n-l + n-2 + ... + n-r)Wr-I(f) mr_l

~ (PI - 1 + ... + Pr - l)Wr-I(f)

= (t Pk - 1') Wr-l(f). k=l

Consequently, (6.2.6) implies

lim in icy)x(x,y)dy = O .• r-+oo m,._l

The following result is a corollary of 6.2.8.

142 CHAPTER 6

6.2.9. Let P = { ... ,P-j"",P-I,PI, ... ,Pj, ... } and suppose that L:~=IPk ~ cr for some c > O. If f(x) E LI(O, 00) satisfies

(6.2.7) Wr-I(f) = o(l/r),

then (6.2.2) holds at each point ofP-continuity of f(x).

It is useful to notice that if SUPnPn < 00 then hypothesis (6.2.7) is equivalent to the condition wr(f) = 0(1/ In mr)' This is clear from the relationship

r

lnmr = 'L)npk ~ rln(suPPk) = cIr, k=I k

and the fact that Pk ~ 2 implies

where 0 < C2 < CI'

We shall establish a connection between finit.f'ness of the transform j(y) and the behavior of the function f( x). For this we will use the following analogue of a theorem of Young (see [30], Vol. I, p. 160).

6.2.10. Let {wn(x)}~=o be a complete, ortlJOnormal system in V[a,b] and f(l:), g( x) be square integrable on [a, b]. If

then the series

converges uniformly on [a, b] and

for all x E [a, b].

PROOF. Let n

sn(t) = L CkWk(t). k=O

We obtain from the Cauchy-Schwarz inequality that

11X J(t)g(t)dt- ~Ck l x

g(t)wk(t)dt 1=llx g(t)(f(t)-sn(t))dt 1

( b )1/2( b )1/2 ::; 1lg(tWdt l IJ(t) - Sn(tW dt

By Parseval's identity, the right side of this inequality converges to zero as n -+ 00.

Thus the series

converges uniformly for x E [a, bJ to the function fax J(t)g(t) dt .• The following is an analogue of a theorem of Kotel'nikov [1 J about representation

of functions with finite Fourier spectrum by its values at equally spaced points.

6.2.11. If J(x) E Ll(O, (0) is P-continuous on the positive real axis [0,(0) and its multiplicative transform iCy) satisfies i(y) = ° for y > a and some a < 1nn then the function J(t) can be reconstructed from its values at the points tk = kl1nr for k = 0,1, ... by means of the formula

1 00 k k J(t) = - LJ(-)D(- 8t,m r ),

1n r k=O 1n r 1nr

where the kernel D(u,O is defined by (1.5.38).

PROOF. By 1.5.6 the system of functions V'v,r(Y) = II ym:;.x(I/I1nr, y) is orthonormal and bounded on the interval [0, 1n r ). Thus by a result of Vilenkin [2J the

system {1Pv,r(Y)}~o is complete in L2[0,1nrJ. Since iCy) = ° for y ?:: 1nr we can associate with iCy) a series in the system {1jJv,r(Y)}~o, i.e.,

00

icy) ,..., L bv1jJv,r(Y), ",=0

where

(6.2.8) 1mr ~ 1 1mr ~ l/ bv = J(y)1jJv,r(Y) dy = ~ J(y)X(-, y) dy o y1n r 0 mr

for l/ = 0,1, .... On the other hand, i(y) belongs to Ll (0,00) since it has compact support, and therefore by 6.2.1 we have

(6.2.9) f(x) = lim i(y)x(x, y) dy = icy)x(x, y) dy. 1m. 1mr k-OCi 0 0

144 CHAPTER 6

Now, (6.2.8) and (6.2.9) imply that

1 V bl/ = ~f(-),

ymr mr v = 0,1, ....

Consequently,

(6.2.10) ~ 00 ___ 00 1 v v fey) '" 2: bl/tPl/,r(Y) = 2: -f(-)x(-, v)·

1/=0 1/=0 mr mr mr

Since the system {tPl/,r(X)} is bounded and the function g(y) = X(x,y) belongs to L2(0, m r ), we have by 6.2.10 that the series for the product J(y)x(x, y) can be obtained by integration. Therefore, it follows from (6.2.9) that

1mr 00 1 f(x) = x(x, y) 2: -f( ~ )x(~, y) dy

o 1/=0 mr mr mr

1 00 v 1mr v =-2:f(-) X(-8x,y)dy, mr 1/=0 mr 0 mr

I.e., 1 00 v ----:-;v----

f(x) = - 2: f(-)D(- 8 x, rhr). • mr 1/=0 mr mr

A consequence of 6.2.11 is the following.

6.2.12. Let f(x) E LI(O,oo) be 'P-continuous on [0,00) with finite spectrum, i.e, its multiplicative transform satisfies J(y) = ° for y ~ m r . Then f(x) is a step function, constant on the intervals of rank r of the form 81/(r) = [v /m r , (v + 1 )/m r ),

v = 0,1, ....

PROOF. The proof is an easy consequence of 6.2.11 and the fact that

for ° :::; (v/m r ) 8 x < l/mn

for l/m r :::; (v/m r ) 8 x < 00 .•

6.2.13. If f(x) E LI(O,oo) is a step function, constant on the intervals of rank r of the form 8~(r) = [v/m_r,(v + l)/m_ r ), then it has finite spectrum, i.e., its multiplicative transform satisfies fey) = ° for y ~ m- r ·

PROOF. Since the step function f(x) is const.ant on the intervals 8~(r), we can write it in the form

00 1 v f(x) = 2:cl/-D(mr, - 8 x),

1/=0 m-r m-r


for some constants c". Thus

If we change variables by v = (v /m-r) e t, i.e., v /m_r = v ED t and t = (v /m_ r) e v we obtain

00 11/m-r v fey) = LC" X(-ev,y)dv

,,=0 0 m-r

00 v 11/m-r = LC"X(-,Y) X(v,y)dv.

,,=0 m-r 0

But rl/m-r t/m_r

Jo X(v,y)dv = Jo X[y)(v)dv = 0

for y ~ m_r . Thus fey) = 0 for y ~ m_r .•

Under the condition that the collection 'P is symmetric, i.e., 'P' = 'P, we shall establish theorems which are in a sense duals of 6.2.12 and 6.2.13.

6.2.14. Let f(x) E LI(O, 00) and suppose that f(x) = 0 for x ~ m r . Then the transform fey) is a step function constant on the intervals 6,,(r) of rank r.

PROOF. Since f(x) = 0 for x ~ m r , we can write the transform fey) in the form

where fk(X) = f(k + x) and C[y)(fk) = 0 for k ~ m r . Consequently,

mr-l

!(y) = L c[y)(ik)n({y}). k=O

But for k::; mr -1, the functions n({y}) are step functions and are constant on 6,,(r) .•

146 CHAP1ER6

6.2.15. Let I(x) E LI(O, 00) be 'P-continuous on [0,00). !fits transform ley) is a step function, constant on the intervals Du( r) of rank r, then I( x) = 0 for x 2 m r .

PROOF. Suppose to the contrary that there exists a point Xo 2 mr such that I(xo) =I- O. Since I(x) is 'P- continuous there is an interval [xo,a) C [f,f+ 1) for some natural number f 2 mr such that I( x) =I- 0 for all x E [xo, a).

Consider the function I[xo)(t) = I([xol + t) defined for t E [0,1). Choose a

natural number v such that cu(f[xo) '" O. For Y E [v, v + 1) the transform ley) can be written in the form

ley) = hey) + hey) + hey),

where

mp-l

= L Cu(fk )Xk(Y) k=O

is a step function of rank r,

mn-l

hey) = L Cu(!k)n(y), k=m r

and 00

hey) = L Cu(fdXk(Y), k=mn

for n chosen so that mr :::; m n-l :::; [xol < m n . Since the functions Xk(y) are linearly independent and cu(f[xo) '" 0, there exist neighboring intervals 81'(n) and 81'+1 (n), belonging to the same intervaI8p(r) C [f,f+ 1) ofrank r, on which hey) takes two different values. Since hey) is periodic of period limn, it follows that h(y)+ hey) is not constant on 8ji(r). This contradicts the hypothesis of this theorem .•

GENERALIZED MUL TIPLICA TIVE TRANSFORMS 147

§6.3. Representation of functions in V(O, 00), 1 < P ::; 2, by their multiplicative transforms.

We noticed in Theorem 6.1.7 that given any function f E V(O, 00), for 1 < P ::; 2, the multiplicative transform fey) exists and belongs to LP' (0,00). In the case of zero-dimensional, locally compact groups, inversion of the L2 -transform was proved in [1] (see p. 84). We shall prove that inversion holds for the LP- transform for any 1 < P ::; 2.

Throughout this section we shall suppose that the sequence P is symmetric, i.e., that P-k = Pk for k = 1,2, .... We first consider integrability of the kernel D( u, 0 defined in (1.5.37).

6.3.1. For all e ;::: 0, the kernel D( u, e) belongs to V(O, 00) for all 1 < P < 00.

PROOF. We have

The first term can be estimated by

Let £ ;::: 1. Use (1.5.38) to write

Consider the function

g(x) = { ~

We have

for 0::; x < {e}, for {O ::; x ::; 1.

(W t Jo Xt(x) dx = Jo g(x)xe(x) dx = ce(g),

148 CHAPTER 6

i.e., JoW Xl( x) dx is the €-th Fourier coefficient of the function g( x) with respect to the system {Xk (x)}. Consequently,

(6.3.1) 100

ID( u,~)IP dtt ::s: e + f: ICl(9)IP o 1=1

00 mr-l

= e + L L ICl(g)IP r=11=mr _1

00 Pr-1 (l_r+1)mr_l-l

= e + L L L ICt(g)IP. r=ll_ r=1 l=l_rmr_1

To estimate the coefficients Ct(g), let € E [L r m r-l, (Lr + l)mr-l -1], Lr -j. 0 and introduce the notation

Then

21ri€_r q=exp-

Pr

l w 11Or_1 j{Or j{O Ct(g) = Xt(x) dx = Xl(X) dx + Xt(x) dx + Xe(x) d:c.

o 0 {Or-I {Or

Since m r -l ::s: € < mr implies

i "/mr

Xt(x)dx=O, (II-l)/mr

v = 1,2, ... ,r - 1,

and the interval [0, {Or-d can be represented as aunion of intervals of the form

[(v - 1)/mr, v/mr]' we have JolOr-1 Xt(x) dx = O. It follows that

flOr flO Ct(g) = J( xc(x) dx + J( xe(x) dx

{Or-I {Or

er-l (j{Or_IHA+l)/mr ) jlO = L xc(x)dx + xe(x)dx

~=O {e}r-I +~/mr {Or

1 er-1 jW = - L q~ + Xt(x)dx

mr ~=O {Or

1 1 - qer j{O = ---+ Xt(x)dx.

mr 1 - q lOr

GENERALIZED MUL TIPLICA TIVE TRANSFORMS

Therefore,

1 ( )1 1 I sin«7rLrEr)IPr) I 1 c( 9 < - . +-.

- mr sm«7r£-r)IPr) mr

Substituting this estimate into (6.3.1) we obtain

= = tP ~ m r -1 P~ 1

:5 <, + 2 ~ mP pf. Pr ~ £p r=1 r-1 (=1

= 1 <tP+A~-. - <, ~ p-1

r=1 m r - 1

149

This last series converges since P > 1. We conclude that D( u, 0 E LP(O, 00) for any E ~ 0 .•

6.3.2. Let f E V(O, 00) for some 1 < P :5 2 and

icy) = lim p l l a f(x)X(x,y)dx,

a-+oo 0

i.e., icy) E V'(O,oo)

where lip + lip' = 1. Then

(6.3.2) d 1= l x fey) = dx 0 fey) 0 x(y,t)dtdy,

and

(6.3.3) d f= r f(x) = dx Jo icy) Jo x(y,t)dtdy

both hold for almost every x E [0,00).

PROOF. We introduce the notation

icy, a) = 1a f(x)x(x, y) dx.

150

For any ~ E [0,00) we have

By hypothesis it follows that

(6.3.4) lim lI[(y) -ley, a)llp' = o. a-+oo

Consequently, for all ~ E [0,00) we have

1(~ 1e~ f(y)dy = lim f(y,a)dy,

o a-+oo 0

i.e.,

f( fey)dy = lim fe r f(t)x(t,y)dtdy Jo a-+oo Jo Jo

= lim rf(t) f(x(t,y)dydt a-+oo Jo Jo

= 100 f(t)D(t,Odt.

Therefore,

1(0 = :~ 100 f(t)D(t,~) dt

holds for almost every ~ E [0,00). This verifies (6.3.2). To prove (6.3.3) it is enough to show

(6.3.5) 1e f(x)dx = 100 ley) 1e x(x,y)dxdy.

CHAPTER 6

Define a sequence offunctions h on [0,1) by hex) = f(k+x). Consider the integral

100 feu,a)D(u,Odu.

By property 1.5.3 for the function X(x, y), we ha.ve

feu, a) = 1a f(x)xe(x)x[x](u) dx

[a]-1 a

= L ce(h)n(u) + 1 f(x)xe(x) X[a](U) dx k=O [a]

for [u) = R.. For each ° < e ~ a we obtain by (1.5.38) that

Since the system {Xk(U)} is orthogonal on [0,1], we have

~-1 ~

(6.3.7) Io(a, e) = ~ co(fk)+co(f[elHO = 10 !(y)dy + co(f[el)co(g)

where

g(x) =

Similarly,

(6.3.8)

00

= L Ct(f[el )Ct(g). £=1

{ 1,

0, if ° ~ x < {O, if {O ~ x < 1.

151

Since ![el(x) E LP(O, 1) and g(x) E Lpl (0,1), it follows from Parseval's identity that

Identities (6.3.7) through (6.3.9) imply

(6.3.10)

152 CHAPTER 6

Hence we have by Holder's inequality that

1100(i(u) - j(u,a))D(u,Odu 1:5IIj(u) - j(u,a)llp·IID(u,Ollp·

Therefore, we conclude from (6.3.1), (6.3.4), and (6.3.10) that

100 j(u)D(u,Odu = 1e f(x)dx

i.e., (6.3.5) holds .•

Chapter 7


§7.1. Convergence and integrability. Let {cn}, n = 0,1, ... , be a monotone decreasing sequence of real numbers, i.e.,

~cn == Cn - Cn+l ~ 0 for n = 0,1, .... H the sequence also converges to zero, we shall write Cn t O.

We shall frequently use the following easily verified identity, which is called Abel's transformation (see [15], p. 9):

n-l n-2

L akbk = L ~akBk+1 + an-IBn - amBm, k=m k=m

where Bo = 0 and Bk = E;:; bi for k ~ 1.

7.1.1. If Cn t 0 then the series

(7.1.1)

converges on the interval (0,1) and is uniformly convergent on the interval [8,1) for eachO<8<1.

PROOF. Let Sn(x) = E~':; CkWk(X) be the partial sums of (7.1.1). Let 1 :5 m < n and notice by Abel's transformation that

(7.1.2) n-l

Sn(x) - Sm(x) = L CkWk(X) k=m n-2

= L ~ckDk+1 + cn-IDn - cmDm, k=m

where Dk(x) = E;':; Wj(x) is the Dirichlet kernel. Recall from §1.4 that

(7.1.3)

153

154 CHAPTER 7

for ° < x < 1 and k = 1,2, .... Consequently, from (7.1.2) we obtain

(7.1.4) n-2

1 '"' Cn-l Cm 2cm ISn(x) - Sm(x)1 ~ - L.t ~Ck + - + - = -. x x x x k=m

Since Cm --+ ° and n > m, (7.1.4) implies that the series (7.1.1) converges at each point x E (0,1). Moreover, if 8 E (0,1) is fixed, then

sup ISn(x) - Sm(x)1 ~ 2~m, 6:5%<1

n>m.

Hence the partial sums Sn are uniformly Cauchy and it follows that (7.1.1) converges uniformly on (8, 1) .•

We shall now prove that the sum of (7.1.1) need not be Lebesgue integrable on (0,1).

7.1.2. There exists a sequence {cn } which converges monotonically to zero such that the function

ex:>

(7.1.5) f(x) = L cnwn(x) n=O

is not integrable on the interval (0,1).

PROOF. We proved in §2.2 that the Lebesgue constants

n=I,2, ...

satisfy the inequality

(7.1.6) for n ~ 4,

and that there is a subsequence of natural numbers no < nl < ... such that

(7.1.7) k = 0,1, ....

Moreover, the sequence {n,,} satisfies the condition 2k ~ nk < 21:+1 for k = 0, 1, ... (see (2.2.4)).

Choose a subsequence {md of {n,,} which satisfies the inequalities

(7.1.8)

where 2"' ::; ms < 2",+1. Define the sequence {cn} by

00 1 Co = C1 = ... = Cml -1 = ~ .y'log2 mk'···'

(7.1.9) Cmo = Co, Cmk = Cml +1 = ... = Cmk+1 -1, ••• ,

1 cml - Cml+1 = J: ' k = 0,1, ....

log2 mk

155

Obviously, Cn 10. Hence by 7.1.1 the series on the right side of (7.1.5) converges uniformly on the interval (6,1) for each 0 < 6 < 1. Hence the function f(x) is integrable (even in the Riemann sense) on [6,1). We introduce the notation

1(6) = 11\f(x)\ dx,

and will show that 1(6) --t 00 as 6 --t O. Surely, this will verify that the function f(x) is not integrable on (0,1).

Apply Abel's transformation. By (7.1.9) we have

(7.1.10)

1(2-11,) = l~v, I ~ cnwn(x) I dx

= l~v, I ~ .6.cnDn+l(X) I dx

= 11 If:: Dml(x) I dx 2- V , k=1 .y'log2 mk

?.y' 1 ( t IDm, (x)\ dx - t- V

, IDm. (x)\ dX) log2 ms Jo Jo

s-1 L 00 1 11 - L mk L J: IDmk(x)ldx. k=1 yllog2 mk k=s+1 10g2 mk 2- V •

Since the sequence {md is a subsequence of {nk} we have by (7.1.7) that

1 Lml ? 410g2 mk, k = 0,1, ....

Using this observation together with inequalities (7.1.3), (7.1.7), (7.1.8) and the estimate \Dm.(x)\ ::; m s, we see by (7.1.10) that

156 CHAPTER 7

as S -+ 00. Since the function 1{ 6) is monotone, it follows that 1{ 6) -+ 00 as 6 -+ O .•

Thus we see that in order to insure that the sum of a Walsh series with monotone decreasing coefficients is integrable, it is necessary to impose some further restriction on the coefficients. Such a restriction is contained in the following theorem:

7.1.3. Let {cn ~ O} be a monotone decreasing sequence which satisfies

00

(7.1.11) '"' Cn L...J - < 00. n

n=1

Then the series (7.1.1) converges on the interval (O, 1) to an integrable function and is its Walsh-Fourier series.

PROOF. First notice since {cn} is monotone decreasing that (7.1.11) implies

(7.1.12) k -+ 00,

so in particular, Ck -+ O. Consequently, we have by Theorem 7.1.1 that (7.1.1) converges on the interval (0,1) to some function f{x). We shall prove that f E L[O, 1).

Let v be any natural number and let 1/{v + 1) < x :5 l/v. By (7.1.4) we have

n-l m+II-l n-l

sup I L CkWk{X) 1:5 L Ck + sup I L CkWk(X) I n>m k=m k=m n>m+II k=m+II

m+II-l m+II-l

< L Ck +4vcm+ II :5 5 L k=m k=m

Consequently,

1 n-l

1m == 1 :~e.1 k~ CkWk{X) I dx

00 II/II n-l

= L sup I L CkWk{X) I dx 11=1 1/(11+1) n>m k=m

WALSH SERIES WITH MONOTONE DECREASING COEFFICIENTS 157

But by Abel's transformation, N 1 m+/I-1 N-1 1

~ v{v + 1) k~ Ck:::; ~ ~Cm+/I-1. Therefore,

(7.1.13) 00 1 00

Im :::; 5 L -Cm+/I-1 = 5 L Cn 1 v n-m+ /1=1 n=m

00

:::; C(cm log2 m + L ~). n=2m

Since (7.1.11) converges and satisfies condition (7.1.12), the right side of (7.1.13) converges to zero as m -+ 00. Thus the inequality

11 ISn(x) - sm{x)1 dx :::; Im for n > m,

implies that the partial sums sn{x) of the series (7.1.1) are Cauchy in the L(O,l) norm. Since this space is complete (see A2.1, A5.1, and A5.2), it follows that (7.1.1) converges in the L(O, 1) norm to some integrable function s(x). But earlier we noted that (7.1.1) converges everywhere on (0,1) to the function f(x). Consequently (see A5.2.4), sex) = f{x) almost everywhere. In particular, the function f(x) must be integrable.

To see that (7.1.1) is the Walsh-Fourier series of f(x), notice that

lim [1 ISn(x) - f(x)1 dx = 0 n-+oo 10

implies

(7.1.14) lim t Wm(X)(Sn(X) - f(x»dx = 0 n--+oo Jo

for m = 0,1, .... But for each n ;::: m we have by orthogonality that

11 Wm(X)Sn(X) dx = Cm.

We conclude by (7.1.14) that

Cm = 1 1f(x)Wm(X)dX, m=O,I, ....•

An examination of the proof of Theorem 7.1.3 shows that we have in fact established the following theorem:

158 CHAPTER 7

7.1.4. Under the hypotheses of Theorem 7.1.3, the sequence Sn{x) of partial sums of the series (7.1.1) converges in L{O, 1) and has integrable majorant, i.e.,

sup ISn{x)1 E L(O, 1). n~l

If Cn L 0 then by Theorem 7.1.1 the series (7.1.1) converges uniformly on (8,1) for each 0 < 8 < 1. Thus its sum f{x) is Riemann integrable on [8,1) for each 8 E (0,1). It turns out that under these conditions the function f(x) is improperly Riemann integrable on all of [0,1], i.e., that the following limit exists and is finite:

lim [If(x)dx. 6-+0+ J6

In fact, the following theorem is true.

7.1.5. IT cn L 0 then the function f{x) defined by (7.1.5) is improperly Riemann integrable on (O, 1]. Moreover, the series on the right side of (7. 1.5) is its "improper" Walsh-Fourier series, i.e.,

(7.1.15) cn = lim t f(x)wn(x)dx, 6-0+ J6

n =0,1, ....

PROOF. For the case n = 0, it is clear that (7.1.15) implies that f(x) is improperly Riemann integrable. Hence it is enough to prove identity (7.1.15).

Since the hypotheses imply that the series on the right side of (7.1.5) converges uniformly on [8,1), 0 < 8 < 1, we have for any N = 0,1, ... that

Choose 8 > 0 so small that WN(X) = 1 on [0,6]. Then the previous identity can be written in the form

1006 1 f(x)wN(x)dx=cN -L: cn1 wn{x)dx. 6 n=O 0

Thus the theorem will be proved if we show

(7.1.16) 00 16 1(8) == ~ Cn 0 wn{x) dx -+ 0

as S -+ 0+. Write the number S in the form S = E~l 2-P~ where 1 $ Pk < PH1 and each

Pk is a natural number. We shall use the inequality

(7.1.17) 2P1 -1 00 6

IJ(S)I $ S L Cn+ 1 L Cn 1 wn(x)dx I· n=O n=2P l 0

By 1.1.4

Thus

(7.1.18)

If n = 2q2p1 - 1 + s where 0 $ s < 2P1 - 1 then

(7.1.19)

If n = (2q + 1 )2P1 - 1 + s where 0 $ s < 2P1 -1 then

160 CHAPTER 7

Observe for each x E [2-Pl,2-Pl+1) that w.(x) = 1 but W2Pl-1(X) = T p1 -1(X) =-1. Consequently,

(7.1.20)

Combine (7.1.18) through (7.1.20) to obtain

(7.1.21) 00 (2q+l)2 P1 - 1-1 2(q+l)2'1- 1-1

111(0)15 0LI L cn - L cnl q=1 n=2q·2'1- 1

00

== 0 L IA2q ,PI - A 2q+1,PII· q=1

n=(2q+l)2' 1 - 1

Since the sequence {Cn} is monotone decreasing, so is the sequence {A.,pJ~2' Consequently,

00 00

L IA2q ,Pl - A 2q+1,Pll = L( -1)0 A.,Pl 5 A 2,Pl q=1 0=2

3.2'1-1_1

= L Cn 5 2Pl-1c2Pl. n=2'1

Thus it follows from (7.1.12) that

Ill(0)1502Pl-1c2P1.

But 052· 2-P1 implies 11(0)1 5 C2 Pl. Consequently, it follows from the definition of It(o) (see (7.1.18» and (7.1.17) that

2"'-1 11(0)1 52-P1 +1 L Cn + C2' == 20'2'1 + C2 Pl

n=O

where O'n represents the (C, 1) means of the sequence {cn }. If 0 --+ 0 then PI --+ 00.

Since Cn --+ 0 and the method (C, 1) is a regular method of summability (see 4.1.3), it follows that 0'2Pl --+ O. Thus 1(0) --+ 0 as 0 --+ 0 and the theorem is proved .•

When studying the problem of integrability of Walsh series with monotone decreasing coefficients, it is sometimes useful to replace the improper Riemann integral with a concept called the A- integral. This notion is defined in the following way.

A measurable function f( x) defined on a measurable set E is called A-integrable if the following two conditions are satisfied:

1) limN ...... oomes{x: x E E, If(x)1 > N} = 0; 2) the limit

lim { [f(x)]N dx N ...... oolE

exists and is finite, where

[f(X)]N = { ~(x) for If(x)1 :::; N,

for If(x)1 > N.

This limit is called the A-integral of the function f(x) on the 3et E and will be denoted by

(A) Lf(X)dX.

If the function f(x) is Lebesgue integrable on the set E, i.e., if f(x) belongs to the class L(E), then it satisfies conditions 1) and 2). Hence any Lebesgue integrable function is A-integrable. However, the converse of this statement is not true. This can be verified by comparing 7.1.2 with the next theorem.

7.1.6. If Cn ! 0 then the function (7.1.5) is A-integrable on the interval (0,1) and satisfies

(7.1.22) Cn = (A) ( f(x)wn(x) dx, leo,!)

n = 0,1, ... ,

i.e., (7.1.1) is the Walsh-Fourier series of the function f in the sense ofthe A-integral.

PROOF. We may suppose that Co > 0 since otherwise all the coefficients Cn = 0 and the result is trivial. Let N > 0 be given and set no = no(N) = [N/(2co)] where [a] represents the greatest integer in the number a. Replace in (7.1.4) the index m by no and let n tend to 00. We obtain

(7.1.23) O<x<1.

If x E (0,1) satisfies If(x)1 > N then

00

I L CkWk(X)1 > N/2 k=no

162 CHAPTER 7

since I E;!~1 CkWk(X)1 :5 N/2. Consequently, N/2 < 2cno /x, i.e., x < 4cno /N. Since Cno -+ 0 as N -+ 00 it follows that

1 mes{x : x E (0,1), If(x)1 > N} = o( N)' N -+ 00.

Thus condition 1) (from the definition of A-integrability) is satisfied by the set E = (0,1).

To verify condition 2) set eN = sup{ x : If(x)1 > N}. Since x < 4cno /N holds for all x's which satisfy If(x)1 > N, it is evident that eN :5 4cno /N. Consequently,

(7.1.24)

But by Theorem 7.1.5,

lim 11 f(x)wn(x)dx = Cn N-oo t:N

for n = 0,1, .... Thus it follows from (7.1.24) and the hypothesis limN-+oo Cno = 0 that

n = 0,1, ....

Hence by definition the function f(x) is A-integrable and satisfies (7.1.22). I We close this section with the following theorem:

7.1.7. lfcn ! 0 then the function f(x) defined by identity (7.1.5) belongs to LP(O, 1) for every 0 < p < 1 and

(7.1.25) lim t If(x) - Sn(x)IP dx = 0, n-+oo Jo

where Sn(x) = E:':~ CkWk(X).

PROOF. Fix 0 < p < 1. By (7.1.23) we have

o < x < 1, n = 1,2, ....

Integrate this inequality to obtain

11 11~ If(x) - Sn(X)IP dx :5 (2cn)P -. o 0 xP

The right side of this last inequality tends to zero as n -+ 00. Hence (7.1.25) is true. But the conditions

imply

11 I/(xW dx ::;11 ISn(x)IP dx + 11 I/(x) - Sn(x)IP dx < 00.

Thus the proof of 7.1.7 is complete .•

§7.2. Series with quasiconvex coefficients. Let {cn }, n = 0,1, ... , be a sequence of real numbers. Set .6.cn = Cn - Cn+b and

.6.2 cn = .6.(.6.cn ) = Cn - 2cn +l + Cn +2 for n = 0,1, .... If .6.cn ~ 0 for n = 0,1, ... , then the sequence {c n } is called monotone decreasing, and if .6.2 Cn ~ 0 for n = 0,1, ... , then the sequence {cn} is called convex.

7.2.1. 1) If a sequence {Cn} is convex and bounded above, then it is monotone decreasing.

2) If a sequence {cn } is convex and bounded, then

(7.2.1 ) n -+ 00,

and

co (7.2.2) ~)n + 1).6.2cn < 00.

n=O

PROOF. 1) Let {cn } be convex and bounded above. We shall show that it is monotone decreasing, i.e., that .6.cn ~ 0 for n = 0,1, .... Suppose to the contrary that there is a non-negative integer m such that .6.cm < O. Since .6.2cn ~ 0 for all integers n ~ 0, it follows that .6.cn < 0 and l.6.cn l ~ l.6.cm I for all n > m. Consequently,

n-1 n-1

Cn - Cm = - L .6.ck = L l.6.ckl ~ (n - m)l.6.cm l -+ 00, n -+ 00,

k=m k=m

which contradicts the fact that {cn } is bounded. 2) Let {cn} be convex and bounded. By 1), it is also monotone decreasing, i.e.,

.6.cn ~ 0 for n = 0,1, .... Since {cn } is bounded below it has a finite limit, say C = limn_co Cn' Consequently,

(7.2.3) 00

Co - C = L .6.cn < 00.

n=O

164 CHAP1ER 7

But the terms of (7.2.3) are monotone decreasing because {cn } is convex. Therefore, convergence implies (7.2.1).

By Abel's transformation we have

n n-l

(7.2.4) E~Ck = E(k + 1)~2ck + (n + 1)~cn' k=O k=O

But we already have verified that (n + 1 )~Cn --+ 0 as n --+ 00 and n

~~Ck = Co - Cn+l --+ Co - C,

k=O n --+ 00.

Thus if we let n --+ 00 in (7.2.4), we obtain

n-l

~(k + 1)~2ck = Co - c.

k=o

This proves (7.2.2) .•

7.2.2. If the sequence {cn} is convex and Cn --+ 0 then {cn} is monotone decreasing and satisfies conditions (7.2.1) and (7.2.2).

PROOF. This result follows directly from Proposition 7.2.1 since the condition Cn --+ 0 implies that the sequence {cn } is bounded .•

If the sequence {cn } satisfies 00

(7.2.5) ~(n + 1)~2cn < 00

n=O

then it is called quasiconvex. Notice by Proposition 7.2.1 that every bounded convex sequence is quasi convex.

7.2.3. Any quasiconvex sequence {cn} satisfies (7.2.1)'

PROOF. Indeed,

00 00 1 I~cnl =1 ~ ~2Ck I:::; ~ k + 1 (k + 1)1~2ckl

k=n k=n

1 00 ( 1 ) :::; -- ~(k + 1)1~2ckl = 0 --n + 1 k=n n + 1

n --+ 00 ••

The following theorem gives a sufficient condition for a series

(7.2.6)

to be the Walsh-Fourier series of some function from the space L(O, 1).

7.2.4. If Cn -t 0 and the sequence {cn} is quasiconvex, then (7.2.6) is the WalshFourier series of some function from the space L(O, 1).

The proof of this theorem requires the following lemma:

7.2.5. If Cn ! 0 and the sum f(x) of the series (7.2.6) belongs to the space L(O, 1), then (7.2.6) is the Walsh-Fourier series of f.

PROOF. The estimate

(7.2.7)

is obvious. Since f is integrable, we also have

(7.2.8) lim 16 If(x)1 dx = O.

6-+0+ 0

Consequently, it follows from (7.1.15), (7.2.7), and (7.2.8) that

n = 0,1, ....

This completes the proof of 7.2.5 .• PROOF OF 7.2.4. Let

n-l

sn(x) = L: CkWk(X) k=O

represent the partial sums of (7.2.6). Two applications of Abel's transformation yield

n-3

(7.2.9) sn(x) = L:(k + 1)~2CkKk+l(X) k=O

where Kn(x) = (lin) E~=l Dk(X). Since IDn(x)1 < l/x and consequently IKn(x)1 < 1/x for 0 < x < 1, we have by Proposition 7.2.3 and the hypothesis Cn -t 0 that the last two terms on the right side of (7.2.9) converge to zero, as n -t 00, for all 0 < x < 1. Furthermore, the inequality IKn(x)1 < l/x implies

00 1 00

L:(k + 1)1~2CkKk+l(X)1 :s -; L:(k + 1)1~2ckl· k=O k=O

166 CHAPTER 7

But the series on the right side of this latest inequality must converge since the sequence {cn } is quasiconvex. Thus the series

00

L:(k + 1)~2CkKk+1(X) k=O

converges for all ° < x < 1. Consequently, the limit as n --+ 00 of the right side of inequality (7.2.9) exists and is finite for 0 < x < 1. Passing to the limit in (7.2.9), as n --+ 00, we obtain

00

f(x) == lim sn(x) = "(k + 1)~2CkKk+1(X). n-looo ~

k=O

Thus 00

If(x)1 :5 L:(k + 1)1~2ckIIKk+l(X)I· k=O

But 4.2.2 implies

'E(k + 1)1~2Ckll1 IKk+l(X)1 dx :5 2 'E(k + 1)1~2ckl < 00.

k=O 0 k=O

Therefore, f(x) E L(O, 1) with

Since f(x) E L(O, 1), we conclude from Lemma 7.2.5 that (7.2.6) is the WalshFourier series of f. •

§7.3. Fourier series of functions in LP. In this section we shall identify necessary and sufficient conditions for the sum of

a series

(7.3.1)

with monotone coefficients to belong to LP[O, 1] for some 1 < p < 00.

First of all we establish a lemma.

7.3.1. Let 1 (x) ~ 0 be a function such that ¢>P(x)xp- 2 is integrable on (1,00), and set

(7.3.2) q,(x) = 1% ¢>(t)dt.

Then q,P(x)x-2 is also integrable on (1,00) and

where Ap is an absolute constant which depends only on p.

PROOF. We begin by using Holder's inequality to verify

(7.3.4)

1% ¢>(t) dt = 1% ¢>(t)t(p-2)/p r(p-2)/p dt

(1% ) l/p (1% ) (p-l)/p ~ a ¢>P(t)tp- 2 dt a r(p-2)/(p-l) dt

for any real numbers 1 ~ a < x. We shall use this inequality to show that

(7.3.5) x ---+ 00.

Indeed, the second factor on the right side of (7.3.4) is of order x l / p and by hypothesis, one can choose a sufficiently large so that the first factor is arbitrarily small. Consequently, given c > 0 it is possible to choose an a such that

1% ¢>(t) dt < cx l / p

holds for all x > a. Fix such an a. We obtain the estimate

x> a.

But if x > a is sufficiently large then q,(a) < cx l / p , and for such x's we have q,(x) < 2ex l / p • Since c was arbitrary, (7.3.5) follows at once.

Integrate by parts to obtain

(7.3.6)

168 CHAPTER 7

By (7.3.5), (~( x))P / x -+ 0 as x -+ 00. Hence it remains to estimate the integral on the right side of (7.3.6). For this we shall use the identity

Applying Holder's inequality, we see that

Consequently, it follows from (7.3.6) that

as x -+ 00. Dividing both sides of this inequality by the first factor of this last term, we obtain

Finally, (7.3.3) follows directly from this relationship .•

7.3.2. If a sequence {cn ;::: O} is monotone decreasing and

(7.3.7) 00

~ cP nP- 2 < 00 ~ n ,

n=l

1 < p < 00,

x -+ 00.

then (7.3.1) is the Walsh-Fourier series of some function f E LP[O, 1].

PROOF. For each natural number k set

(7.3.8) 1 (m )P h = 1 sup I L cnwn(x) I dx.

o m?.k n=k

We shall prove the inequalities

(7.3.9) for 1 < p < 2,

for p;::: 2

where Cp is a constant which depends only on p.

By the proof of Theorem 7.1.3 in §7.1, if II is any natural number which satisfies 1/(11 + 1) < x ~ 1/11, then

m k+v-l

sup I L cmwm(x) I~ 9 L Cm· n~k m=k m=k

Thus

h ~ 9P ~ :2 eEl Cm ) P

Define a function <p( x) on (1, 00) by <p( x) = CnH-l for n ~ x < n + 1. It is easy to see by hypothesis (7.3.7) that the function <pP(x)xP- 2 is integrable on (1,00). Thus 7.3.1 leads to the estimate

~ :2 (,t Cn+k-l ) P ~ Ap ~ C~H-l n P- 2 •

Consequently,

00

(7.3.10) h ~ 9P Ap L c~(n - k + 1)P-2. n=k

For the case p :2: 2 we obtain the second estimate in (7.3.9) with Cp = 9P Ap. For the case 1 < p < 2, we have

Hence the first estimate in (7.3.9) follows from this inequality and (7.3.10). We have verified (7.3.9), and thus by (7.3.7) and (7.3.9), that h -+ 0 as k -+ 00.

Hence it follows from the definition of h (see (7.3.8)) that the sequence of partial sums of the series (7.3.1) is Cauchy in the space U[O, 1]. Since LP[O, 1] is complete, there exists a function f(x) E U[O, 1] such that

lim r ISn(x)-f(x)IPdx =0. ( 1 )lh

n---+oo io In particular,

lim t ISn(x) - f(x)1 dx = O. n-+oo Jo

From this, as we did in the proof of Theorem 7.1.3, we conclude that (7.3.1) is the Walsh-Fourier series of the function f .•

In fact, we have proved the following somewhat stronger theorem:

170 CHAPTER 7

7.3.3. If the sequence Cn ~ 0 is monotone decreasing and satisfies condition (7.3.7) for some p E (1,00), then (7.3.1) converges in the LP[O, 1] norm to some function f and is its Walsh-Fourier series. Moreover, the partial sums sn(x) of the series (7.3.1) have a majorant sUPn~O ISn(x)1 which belongs to the space LP[O, 1].

The converse of Theorem 7.3.2 is also true. To prove this we need the following lemma:

7.3.4. If ¢>(x) E LP[O, 1] for some 1 (t)1 dt.

PROOF. We begin by fixing c > 0 and integrating by parts:

I(c) == 11 (<li~X) r dx

= _ 1<li(x)IP 11 +_1_11 (<liP(x))'x1-P dx (p-1)xP- 1 e p-1 e

= _1<li(1)IP + 1<li(c)IP + -P-l1 (<li(x))P-ll¢>(x)lx1-p dx (p - 1) (p -1)cp - 1 p - 1 e '

i.e.,

(7.3.13) I(c) :::; 1<li(c)IP + -P-l1 (<li(x))P-11¢>(x)lx1- p dx. (p-1)cP- 1 p-1 e

Apply Holder's inequality to (7.3.12) to obtain

<li(x) :::; (l r 1¢>(t)IP dt) lip (l r dt) (p-l)/p = O(X(P-1)/p),

Consequently, l<li( c)IP --+ 0 cp - 1 '

By Holder's inequality again, we also have

(7.3.14)

x--+O+.

11 (<li(X))P-1 (11 (<li(x))P ) (p-l)/p (11 )l/P - 1¢>(x)ldx:5 - dx 1¢>(x)IPdx

t: X e X e

( 1 ) l/p

= (I(c))(P-1)/P l'¢>(x)IP dx

We find from (7.3.13) and (7.3.14) that

( 1 ) l/p

(I(C»l/P<O(l)+P;l 114>(XW dX ,

Let c ---+ 0+ to obtain

( l( »)P )l/P ( 1 )l/P 1 Cf!~ dx ~ P; 1 114>(xW dx

This verifies (7.3.11). I

7.3.5. If Cn ! 0 and the sum J(x) of the series (7.3.1) is a function which belongs to LP[O, 1] for some p > 1 then condition (7.3.7) holds.

PROOF. Set

F(x) = 1x J(t)dt,

We shall use the identity

Fl(X) = 1x IJ(t)1 dt.

for 0 ~ x < 2-n ,

for 2-n ~ x < 1.

(see §1.4). Since the sequence {cn} is monotone, we have for any n ~ 1 that

1 2n_l

F(2-n) = 2-n 1 J(x)D2n(x) dx = 2-n L Ck ~ C2 n-l' o k=O

A straightforward estimate establishes

00 00

L~n_l2n(p-l) ~ LFP(Tn)2n(p-l) n=l n=l

Hence it follows from 7.3.4 that

fc~n_l2n(p-l) ~ ( ~l)P t IJ(xWdx. n=l p Jo

172 CHAPTER 7

Bearing in mind that the sequence {cn } is monotone decreasing, we conclude

00

< max(2P - 2 22- P ) " <! 2k(p-l) - 'L....J 2k_l k=l

~ Cp 11 If(x)IP dx.

In particular, condition (7.3.7) follows from the fact that f E U[O, 1] .• Combining Theorems 7.3.2 and 7.3.5, we see that the following is true:

7.3.6. A series (7.3.1) with Cn ! ° is the Walsh-Fourier series of some function in UfO, 1],1 < p < 00, if and only if condition (7.3.7) is satisfied.

Chapter 8

LACUNARY SUBSYSTEMS OF THE WALSH SYSTEM

§8.1. The Rademacher system. The Rademacher system {Tk(X)} = {W2'(X)}, k = 0,1, ... , which was used to

define the Walsh system (see §1.1), is a typical example of what is called a lacunary subsystem of the Walsh system. We shall study these systems in the next several sections.

The Rademacher system and Rademacher series satisfy some interesting properties. The first such property we isolate is the multiplicative property which is expressed in the following proposition:

8.1.1. If nl ~ n2 ~ ... ~ nk, k = 1,2, ... is any increasing collection of nonnegative integers, then the integral

11 Tn,(X)Tn2(X) •.. Tn. (X) dx

is either 0 or 1. In fact, this integral is always 0 unless the integrand consists only of products of identical pairs of factors.

PROOF. This result follows easily from 1.1.1 and orthogonality of the Walsh system .•

8.1.2. If the series E:=o a~ converges then the Rademacher series

(8.1.1) n=O

converges almost everywhere on [0,1).

PROOF. Since Tn(X) = W2n(X), n = 0,1, ... , (8.1.1) can be written in the form

(8.1.2) n=O

By the Riesz-Fischer Theorem (see A5.4.3), (8.1.2) is the Walsh-Fourier series of some function f(x) E L2[0, 1]. Moreover, we also have

k-1

(8.1.3) S2k(X, f) = I>nW2n(X). n=O

By Theorem 2.5.12 the subsequence of partial sums (8.1.3) of the Walsh-Fourier series of the function f(x) E L2[0, 1] C L[O, 1] converges almost everywhere on [0,1). But S2k(X,f) coincides with the partial sums of order k of the series (8.1.1). Thus (8.1.1) converges almost everywhere. •

173

174 CHAPTERS

8.1.3. 1f E:"=o a~ = 00 then (8.1.1) diverges almost everywhere on [0,1).

PROOF. Set Sn{x) = E:':~ a"r,,{x). Suppose to the contrary that there exist coefficients which satisfy E:"=o a~ = 00 and a set Eo C [0, 1) of positive Lebesgue measure such that the series (8.1.1) converges on Eo. By Egoroff's Theorem this series converges uniformly on some set E C Eo also of positive Lebesgue measure. In particular, there is a constant M > 0 such that

xEE,

for any natural numbers n and p. Consequently,

Ie (sn+p{x) - Sn{X»2 dx ~ M2IEI,

i.e.,

(8.1.4)

By 8.1.1 the system of functions {r,,{x)ri{x)}, 0 ~ k < i, is orthonormal on [0,1). Hence it follows from Bessel's inequality that

(8.1.5) L (f r,,{x)ri{x) dx)2 ~ 11 X~{X) dx = lEI, o~"<;<oo iE 0

where XE{X} is the characteristic function of the set E, i.e., XE{X) = 1 when x E E and XE{X) = 0 when x fI. E. The expression on the left side of (8.1.5) is a series of squares of Fourier coefficients of the function XE{X) with respect to the system {r,,{x}ri{x)}, 0 ~ k < i. Since lEI ~ 1, this series must converge. Hence there is a natural number N{E) such that

n9<~+p-l (Ie r,,{x)r;{x) dX) 2 < 1~12 for all n ~ N(E) and p = 1,2,.... Continuing this inequality by the CauchySchwarz inequality, we obtain

LACUNARY SUBSYSTEMS OF THE WALSH SYSTEM 175

Consequently, by (8.1.4) we have

n ~ N(E), p = 1,2, .... Ie= ..

In particular, the series 2::=0 a; converges contrary to hypothesis. This contradiction proves 8.1.3 .•

8.1.4. H p > 0 and n is any natural number then

(8.1.6)

PROOF. We first prove (8.1.6) in the special case when p = 2r for some natural number r. It is evident that the identity

holds, where the second sum is taken over all non-negative integers nl, n2, ... , n j, ai, a2,"" aj which satisfy nle < n and al + ... + aj = 2r. Integrate this identity with respect to x over the unit interval [0,1] and use the multiplicative property 8.1.1 to obtain

(8.1.7)

where the second sum is taken over all non-negative integers /311 /32,'" ,/3j which satisfy /31 + ... + /3j = r. We shall use the identity

(8.1.8)

Indeed, (8.1.7) and (8.1.8) imply

where C is a constant which maximizes the ratios

(2/31+ ... 2/3j)!: (/31+ ... /3j)! = (2r)!/31! ... /3j! < (2r)! <rT. (2/31)!. .. (2/3j)! r3t!. .. /3j! r!(2/3d!. .. (2/3j)! - r!2T -

176 CHAPTER 8

This proves inequality (8.1.6) for the case p = 2r. ( )

l/p It is easy to see that the integral 101 I/( x)IP dx is an increasing function of

the parameter p E (0,00). Indeed, if 0 < p < p' < 00 then by Holder's inequality (see A5.2.2) we have

t ( t )P/P' ( t )l-P/P' Jo I/(x)IP dx ~ Jo I/(x)IP'dx Jo IP'/(p'-p) dx

( t )P/P' = Jo I/(x)IP'dx

Consequently by the case already proved,

for any p which satisfies 2r - 2 < p ~ 2r .• One easy consequence of 8.1.4 is the following result.

8.1.5. If aO,a1,'" ,an -1 is any finite collection of real numbers then

(8.1.9)

PROOF. Apply Holder's inequality for p = 3/2 and q inequality

where n-1

Sn(x) = I:>krk(X). k=O

3. We obtain the

However, the following inequality was proved in 8.1.4:

Combining these two inequalities, we see that

which is equivalent to (8.1.9) .•

§8.2. Other lacunary subsystems. We shall call a sequence of natural numbers {nk}k:l lacunary in the Hadamard

sense, or simply lacunary, if there exists a number q > 1 such that nk+1/nk > q for k = 1,2, .. " A system of functions {wnk } will be called a lacunary subsystem of the Walsh system if {nd is a lacunary sequence. Since rk(x) = W2k(X), k = 0,1, ... , the Rademacher system is a lacunary subsystem of the Walsh system.

A series will be called a lacunary Walsh series if it has the form

(8.2.1)

where {nd is a lacunary sequence. Many properties of Rademacher series actually hold for any lacunary Walsh series. For example, both 8.1.2 and 8.1.3 remain valid if "Rademacher series" is replaced by "lacunary Walsh series" .

We shall prove the following theorem:

8.2.1. If sm(x) represents the partial sum of order m of the lacunary Walsh series (8.2.1), and if

(8.2.2) limsupsm(x) < 00 m-oo

holds at each point x in some interval I C [0,1), then

00

(8.2.3) L lakl < 00.

k=l

We shall need several lemmas for the proof of this theorem.

178 CHAPTER 8

8.2.2. Under the hypotheses of 8.2.1 there is an interval [a, b] C I on which the partial sums sm(x) of the lacunary Walsh series (8.2.1) are uniformly bounded above, i.e ..

(8.2.4) m = 1,2, ....

PROOF. To prove this result, fix a dyadic interval l' C I. Let N be the subset of the group G which corresponds to I' under the transformati,," >. (see §1.2). We shall use the notation

• •• Emk = {x EN: ISm(x)1 ~ k}, m,k = 1,2, ... ,

where

(8.2.5)

Since the partial sums (8.2.5) are all continuous on G, each set Emk is closed in G. Thus the set Ek = n:=l Emk is closed and the set N' = U:=l Ek is an :F" set (recall that an :F" set is any finite or countable union of closed sets). By

construction, the set N' is simply the set of all sequences ; E N which do not terminate in 1 'so Hence for each such element ; there is a point x E I' such that >.(x) = ; and

In particular, since

(8.2.6) limsupsm(x) < 00 m->oo

we have that; E n:=l Emk for large k. Therefore, N = Z U (U:=l Ek)' where Z is at most countable.

On the other hand, the set N is a neighborhood in the group G and is not of the first category, i.e., it canlll)t be expressed as a countable union of nowhere dense subsets of G. Hence one of the sets Ek is not nowhere dense. Let EM be such a set. Thus it contains some neighborhood N of the group G. By the definition of EM it follows that

(8.2.7) • • sup ISm(x)1 ~ M, m= 1,2, .... ~EN

Thus (8.2.4) is satisfied by any dyadic interval [a, b) which is contained in I'. Since I' is contained in the image under the transformation >. of the neighborhood N, such dyadic intervals exist .•

8.2.3. Let nk+I/nk > q > 3 for k = 1,2, ... , and suppose

N

(8.2.8) PN(X) = II (1 + e:kWn.(X)), k=l

Then for any interval ~ = ~~) = [m2- 8 , (711 + 1)2-8 ), m = 0,1, ... 28 - 1, s = 0,1, ... the identity

(8.2.9)

holds for nl sufficiently large.

PROOF. Since wm(x)wn(x) = WmE9n(X) (see (1.2.17)), the product (8.2.8) can be written in the form

(8.2.10)

where the sum on the right is a finite sum over indices II. Indices II over which the sum in (8.2.10) is taken satisfy the inequality II ~ (q - 2)nI/(q -1). Consequently, if (q - 2)nI/(q - 1) ~ 28 then for each such index II we have by 1.1.4 that

L w,,(x) dx = O.

Thus if we integrate (8.2.10) over ~ we obtain (8.2.9) for (q - 2)nI/(q - 1) ~ 28 ••

PROOF OF 8.2.1. Let {nd be a lacunary sequence with nk+t/nk > q > 1, k = 1,2, .... Suppose q = 1 + 2e: and choose e:' > 0 such that the following three conditions are satisfied:

(8.2.11) {

a) (1 + 2e:)( 1 - e:') > a > 1, 1 +2e:

b) -1--' > a > 1, +e:

l-e:' c) 3--, > a > 1.

1+e:

Find a number Q = Q(e:') > 0 such that p> Q implies (p - 2)/(p - 1) > 1- e:' and pl(p -1) < 1 + e:'. Fix a natural number r such that qr > max(3, Q) and divide the sequence {nd into r subsequences {nkr+p}k:,O, where p = 1,2, ... , r. Set

N-l

PN,p(X) = L(1 +e:kr+pWnkr+p(x)), k=O

ISO

(8.2.12) N-l

p~~!(X) = II (1 + ck'r+pWn"'r+p(X)), k'=o k'oFP

where ckr+p = ±1, p = 1,2, ... , r, and let

Nr SNr(X) = LakWn,,(X)

k=1

CHAPTERS

represent the partial sums of the series (8.2.1). Then for any interval fl C [0,1) we have

(8.2.13) N-l

f SNr(X)PN,p(x) dx = L akr+p f (1 + ckr+pWn"r+p(x))w%+p(x)P~!(x) dx ~ k=O ~

+ p~ (~ akr+p' i wnkr+p , (x)PN,p(x) dX)

P'oFP

N-l

= L akr+p f p~~!(X)Wnkr+p(X) dx k=O J~

N-l

+ L akr+pckr+pl P~!(x)dx k=O 6

r N-l

+ L L akr+p' 1 PN,p (x )wnkr+p ' (x) dx p'=1 k=O 6

P'oFP

== II + 12 + 13 •

Let fl = fl~) = [m2-', (m + 1)2-') c [0,1), where S ~ 0 and m ~ 0 are integers. By 8.2.3 we have

1 p(k) (x) dx = Ifll N,p 6

if nl is large enough. Consequently,

(8.2.14)

for nl sufficiently large.

N-l

12 = Ifll L akr+pckr+p k=O

To estimate the integrals I;}. which appear in the terms of II, notice that

(8.2.15)

II;}. I =1 i P;:'~(X)Wnkr+p(X) dx I N-l II

::;11 Wnkr+p(X) dx 1+ :E 11 Wnkr+p(x):E Gjll(x) dx I A 11=0 A j=O

=Ii+I;,

where for each j the term G jll( x) is the sum of all products of j + 1 Walsh functions whose indices nir+p do not exceed nllr+p" Since nkr+p ~ nl, it is clear by 1.1.4 that

(8.2.16)

when nl ~ 2S •

To estimate the sum Ii notice that

(8.2.17)

where ct are the binomial coefficients. Since n sr+p/n(s-l)r+p > qr > max(3, Q), each index, which satisfies w-y(x) E Gjll necessarily belongs to the interval [(1 -e')nllr+p , (1 + e')nllr+p] where 1/ =I- k. But if snvr+p = [log2,] then snvr +p =I[log2 nkr+pJ. Consequently, R ==, EB nkr+p =I- 0, and by 2.7.5 we have

(8.2.18)

where XA(X) is the characteristic function of the interval 6. If k < 1/ then

since it follows from (8.2.11) that e' < 1/2. On the other hand, if 1/ < k then

182 CHAPTER 8

Consequently, (8.2.18) implies

Substituting this estimate into (8.2.17) we obtain

12 < ~ ~ ~ C j = 12 ~ ~ < 12qr • 1 - L...J L...J" L...J - (r 2)

,,=0 n"r+p j=O 1'=0 n"r+p q - nl

Combine this inequalit~· with (8.2.15) and (8.2.16). We have

Let 8 > 0 and choose nl so large that 12qr I( (qr - 2)nl) < 8. Then II;~ I < 8. In particular, the sum II in (8.2.13) can be estimated by

N-l

(8.2.19) 1111 < 8 L lakr+pl· k=O

Finally, we estimate the integrals which appear in the terms of 13:

(8.2.20)

IJ;~I =/ L W nkr+p ' (x)PN,p(x) dx /

:s:/ L W nkr+P' (x) dx I N-I v

+ ~ ~ct max I r W nkr+p , (x)w-y(x) dx I L...J L...J w~EGJ" lAo 1'=0 )=0

= Ji + J;, where pi f- P but here v may equal ~'. If nl is sufficiently large then

(8.2.21) J; = O.

This follows from 1.1.4. As above, any index 'Y which satisfies w-y E Gjv must belong to the interval [(1 - cl)nvr+p, (1 + cl)nvr+p]. This interval can always be located near the point 'nkr+p' on the real axis by the following choices of the pairs v and p: a) /, = k, p = pi ± 1, b) v = k -1, pi = 1, P = r, c) v = k + 1, pi = r, p = l. In all th .. oc cases, n llr+p and nkr+p' are successive terms of the lacunary sequence

{nk}. The expressions on the right side of (8.2.20) are greatest when'Y and nkr+p' are nearest each other, i.e., in the following cases;

1) 'Y E [(1 - e')nk, (1 + e')nk] and nkr+p' plays the role of nk+1, 2) 'Y E [(1 - e')nJ., (1 + e')nk] and nkr+p' plays the role of nk-l' In case 1), we have by (8.2.11) that

'Y (1 e')n -- > - k > q(l- 10') = (1 + 210)(1- 10') > a > 1. nk-l nk-l

It is not difficult to verify that there exist constants A and B, depending only on a, such that m > n > A and min> a> 1 imply mEEln ~ 2Bm - B , where m = 28m +m' and 0 ~ m' < 2Bm. Thus we have by estimate (8.2.18) that

Since 2B., ~ 'Y < 28 .,+1 and'Y E [(1 - e')nvr+p, (1 + e')nvr+p] it follows that

2B+2 C=--.

1- 10'

Combining this estimate with (8.2.20) and (8.2.21), we can dominate J;rv as follows:

(8.2.22)

for nl sufficiently large. We shall suppose that nl is so large that the right side of (8.2.22) is less than the number 6 > 0 chosen earlier. Thus the sum 13 in (8.2.13) satisfies the inequality

(8.2.23) r N-l

1131 ~ 6L L lankT+P,I. p'=1 k=O p'#p

Hence it follows from (8.2.13), (8.2.14), (8.2.19), and (8.2.23) that

(8.2.24)

i SNr(X) PN,p(X) dx ~ 1121-1111-1131 N-l N-l r N-l

> L akr+pekr+pl.6.l- 6 L lakr+pl- oL L lakr+p,l· k=O k=O p'=1 k=O

p'#P

184 CHAPTER 8

This inequality holds for any sequence ck = ±1. Let Ck = ±1 be the sequence which satisfies Ckak = lakl, for k = 1,2, .... Then summing both sides of inequality (8.2.24) as p runs from 1 to r and using the fact that 6 < 161/(2r) we arrive at

(8.2.25) r Nr Nr L 1 SNr(X)PN.p(x)dx ~ (161- 6) L lajl- (r -1)6L lajl p=1 ~ j=1 j=1

~ I~I Elajl. j=1

Suppose now that the hypotheses of Theorem 8.2.1 hold. Choose by 8.2.2 a dyadic interval 6 = [m2-·, (m + 1 )2-·) and a constant M > 0 such that Sk (x) :::; M for k = 1,2, ... , and in particular, SNr(X) :::; M for all N, r, and x E 6. Using the last inequality and the fact that PN.p(X) ~ 0 (see (8.2.12)), we obtain from (8.2.25) and (8.2.9) that

i.e., 2:7:1 lajl :::; 2Mr for N = 1,2, .... By letting N --+ 00, we see that the series (8.2.3) converges.

Notice that the entire proof to this point is predicated on the fact that the first element nl of the lacunary sequence {nd is sufficiently large. But by eliminating the first few terms of the sequence {nk} we may suppose this is true. Since the convergence of the series (8.2.3) does not depend on any finite number of terms, it follows that the proof of Theorem 8.2.1 is complete .•

Evidently, we can replace hypothesis (8.2.2) in Theorem 8.2.1 with the condition

liminf sm(x) > -00, m-oo

since it can be obtained from (8.2.2) by multiplying each coefficient of the series (8.2.1) by -l.

If we combine this remark with Theorem 8.2.1, it is not difficult to prove the following result:

8.2.4. If a lacunary series (8.2.1) is the Walsh- Fourier series of some bounded function then it converges absolutely, i.e., it satisfies condition (8.2.3).

PROOF. Since {nd is lacunary we can choose a number q > 1 such that nHJ/nk > q, for k = 1,2, .... Suppose that (8.2.1) is the Walsh-Fourier series of some bounded, measurable function f(x), i.e., If(x)1 :::; M, x E [0,1]' where

M ~ 0 is some fixed constant, and j(n) = ak for n = nk and j(n) = 0 for n i:- nk, k = 1,2 ... , Denote the arithmetic means of the series

00

(8.2.26) L j(n)wn(x) n=O

by O"m(x). Then

N=2,3, ....

Let nN-l < m ~ nN. Then the m-th partial sum sm(x, f) of the series (8.2.26) satisfies

(8.2.27) N-l

ISm (x, f) - O"nN(x)1 =1 L ~akWnk(X) 1 k=l nN

N-l N-l

~ L ~Iakl ~ L q-(N-k)lakl == dN. k=l nN k=l

Since Walsh-Fourier coefficients always converge to zero (see 2.7.3), we know that ak -t 0 as k -t 00. Moreover, since I/(x)1 ~ M we have lakl ~ M for k = 1,2, .... Let c > 0 and choose Nl so large that if k > Nl then M Nlq-N1 < c/2 and lakl < (q - I)c/2. Then for N > Nl we obtain

We see then that dN -t 0 as N -t 00. But it is easy to see by (4.2.2) and 4.2.2 that 100nN(x)1 ~ 2M for N = 1,2, ... and x E [0,1) since I/(x)1 ~ M. Consequently, we have by (8.2.27) that ISm(x,f)l ~ B for m = 1,2, ... and x E [0,1), where B = 2M + maxdN. Hence the partial sums of the larl1nary series (8.2.26) are uniformly bounded and we conclude by Theorem 8.2.1 that this series converges absolutely .•

§8.3. The Central Limit Theorem for lacunary Walsh series. Our main goal in this section is to prove the following theorem:

186 CHAPTER 8

8.3.1. Let {ad be a sequence of real numbers which satisfies the conditions

(8.3.1) A - (2 2 2 )1/2 N = a1 + a2 + ... + aN -+ 00, N -t 00,

and

(8.3.2) N -+ 00.

If {nk} is any lacunary sequence of natural numbers and E C [0, 1] is any Lebesgue measurable set of positive measure lEI, then

. 1 I{ 1 ~ }I 1 jY _)..2/2 (8.3.3) J~ lEI x: x E E, AN ~akWnk(x) S Y = ~ -00 e d)",

for all -00 < y < 00.

To prove this theorem we shall derive a preliminary result concerning lacunary sequences of natural munbers.

8.3.2. Let {nd be a lacunary sequence of nat ural numbers, i.e, there exists a q > 1 such that nk+t!nk > q for k = 1,2, .... Then there is a number T = T(q) > 0 such that for any r > T and any choice of'Y = 0, 1, . .. the equation

(8.3.4)

has only finitely many solutions when nk1 /nka > qr . Moreover, given any'Y there is a constant C = C('Y,q) ~ 1 such that equation (8.3.4) has no more that CP solutions.

PROOF. We shall write 'Y in the form 'Y = 2:;~o ci2i where Ci = 0 or 1. Let (nk 1 , nk2 ,.'" nkp) denote a solution to equation (8.3.4) and write

lj

nkj = LCji2i, i=O

Cji = 0 or 1, Cjij = 1.

Let 9 = max1<j<p €j and notice that s"( S g. By definition of the operation ill (see §1.2), equatio~ (8.3.4) can be interpreted in the following way:

B1 B2 nk 1 C10 C11 CIs.., C1s..,+1 C1g

nk2 C20 C21 C2s.., C2s..,+1 c2g

(8.3.5) nka C30 C31 c3s.., C3s..,+1 C3g

nkp CpO Cp1 Cps.., cp • ..,+l cpg

'Y co C1 1 0 0

LACUNARY SUBSYSTEMS OF THE WALSH SYSTEM IS7

The elements of the matrix B1 satisfy the equations

p

(8.3.6) ~:::>ji = ei (mod 2), i = 0, ... ,s'1' j=1

and the elements of the matrix B2 satisfy the equations

p

(8.3.7) L:>ji = 0 (mod 2), i = s'1 + 1, ... ,g. j=1

Since q > 1 we can choose a natural number j such that 1 + 2- j ~ q. For each natural number a let Sa represent the largest integer n which satisfies the inequality 2n ~ a, i.e., set Sa = [log2 a]. If for some 'Y the number of solutions of (8.3.4) is infinite then we can find solutions which satisfy

nk Ink < 2g+1-,8 1 3 - ,

Hence it follows from (8.3.7) that nk, and nk2 must have g+l-fj identical coefficients in their dyadic expansions, i.e.,

(8.3.8) i = fj, fj + 1, ... ,g.

Suppose that 9 - fj + 1 ~ 9 - s'1' i.e., s'1 ~ fj - 1. Then by (8.3.8) we have

(8.3.9)

If nk)nks > qr for r 2: T(q), then the inequality nkl/nka ~ 2g+1-,8 implies that 9 - fj 2: j for T(q) chosen sufficiently large. But then we would have by (8.3.9) that nkJnk2 ~ 1 + 2- j ~ q which is impossible. Consequently, the assumption s'1 ~ f3 - 1 leads to a contradiction. Thus S'1 2: f3 if 9 - f3 2: j. We see then that all dyadic digits of rank greater than s'1 in the dyadic expansion of the number nka must be 0, i.e., e3i = 0 for i = s'1 + 1, ... ,g. Hence

nk, < (2g + eg_12g-1 + ... + e",+128"1+1) + (2""1 + ... + 2°) nk2 - 2g + eg_12g-1 + ... + e8"1+128"1+1

2""1+1 < 1 + -- = 1 + 2-(g-(8"1+1)). - 2g

188 CHAPTER 8

If 9 - (S,,! + 1) ~ j then nkJnk2 :5 1 + 2-; :5 q which again contradicts the fact that {nk} is lacunary. Hence 9 - s"! :5 j. Therefore, all solutions of (8.3.4) necessarily satisfy nk1 :5 2q+1 :5 26 .,+;+1 and the number of these solutions cannot exceed 2(6.,+;+1)p == CP(-y,p) .•

To prove Theorem 8.3.1 we introduce the notation

(8.3.10)

Thus equation (8.3.3) takes on the form

(8.3.11) 1 j" 2 lim FN(y,E)=F(y)== tiC e->'/2d)". N~oo v2~ -00

The function FN(y, E) is the distribution function of the sum l/AN Ef=l akWn~ (x) on the set E. Consequently, according to the inverse limit theorem for characteristic functions (see, for example, [51, p. 241) formula (8.3.1) will be established if we prove that the characteristic functions (8.3.12)

iPN()..,E)=IEI-ljOO ei>'YdFN(y,E) = IEI-l {exp(i)..tak:n~(X)) dx -00 JE k=l N

of the distribution functions (8.3.10) converge uniformly to the function e->.2/2, as N -+ 00, on any finite subinterval of the real axis.

We shall divide the sum Ef=l akWn~(X) into blocks of r terms plus a certain remainder term in the following way. Suppose N > r. Set N = Tr + N' where 0:5 N' < r, and write

(8.3.13)

T-l == L ZN,k(X) + ZN,T(X),

k=O

Since

and

we have by (8.3.2) that

(8.3.14) N -+ 00.

Since eZ = (1 + z)e z2 / 2 eO(lzl\ as z -+ 0, it follows from (8.3.12) and (8.3.13) that

where

lXN(X) = exp (t,O(IZN,k(XW)) .

Let £ > 0. Choose by (8.3.14) a natural number No = No(£) such that N ~ No implies

(8.3.16) T

IlXN(X)1 ::; £ L IZN,k(X)1 2

k=O

,; >.',AN' (~(t,ai'+;) + ~~' aJ+T ')

+ 2>.2£ I AiV2( L akr+iakr+iWnkr+iEIlnkr+j(x)

l~i<i~r

+ L ai+Trai+TrWni+TrEIlnHTr(x)) I

It is clear that IPN(x)1 ::; r(r - 1) and thus that lXN(X) converges, as N -+ 00,

uniformly to zero with respect to both x E [0,1) and >. E [a, b], for any interval [a, b]. Consequently,

(8.3.17)

where (3N(X, >.) -+ 0, as N -+ 00, uniformly in x and >..

190 CHAPTER 8

We shall now estimate the middle factor of the integrand in (8.3.15). We obtain by (8.3.13) and (8.3.16) that

T

(8.3.18) LZ~,k(X) = _,\2(1 + 2PN(X». k=O

Suppose for a moment that we have proved that PN(X) converges in measure to ° (see A4.1), i.e., that

(8.3.19) I{X : x E [0,1), IPN(x)1 ~ 8}1 ~ 0, N ~ 00,8> 0.

Then it would follow from (8.3.18) that

(8.3.20) exp (~~Z~'k(X») ~ exp (_ ~2), N~oo

in measure, uniformly with respect to ,\ E [a, bj, for any interval [a, bj. Notice that

(8.3.21 )

I fJ (1 + ZN,,( x)) I $ (fJ (1 + IZN,,(x )1')) 'f'

~ exp (~ ~ IZN,k(XW) ~ M('\), x E [0,1).

In particular, it would follow from (8.3.15), (8.3.17), (8.3.20) and a version of the Lebesgue Dominated Convergence Theorem (see A4.4.1), that

(8.3.22) _>.2/2 f T

<PN('\, E) - lEi- JE Do (1 + ZN,k(X» dx ~ 0, N -+ 00,

uniformly with respect to ,\ E [a, bj, for any interval [a, bj. We see, then, that it suffices to show (8.3.19) and the following proposition:

8.3.3. If -00 < a < b < 00, and E is any measurable set then

T Ie Do (1 + ZN,k(X)) dx ~ lEI, (8.3.23) N~oo,

uniformly with respect to ,\ E [a, bj.

PROOF OF (8.3.19). We shall use Chebyshev's inequality (see (5.1. 7)), namely, that

(8.3.24) I{X : x E [0,1), IPN(X) I ~ 8}1 ~ 11 0-2 Pj.(x) dx.

Hence it suffices to show that the right side of (8.3.24) converges to 0 as N -t 00.

By construction (see (8.3.16)), we have

(8.3.25) 1 1 1 T-l 1 P~(x)dx = 1 A4 1 L) L.: akr+iakr+jWntr+i(Bntr+i(X))+

o 0 N k=O l:::;i<j:::;r

where

+ L.: ai+Traj+TrWni+Tr(BnHTr(X) 12 dx l:::;i<i:::;N'

= A: (~( L.: a~r+iair+j) + L.: a~+Tra]+Tr) N k=o l:::;i<i:::;r l:::;i<i:::;N'

+ :t 11 E(x) dx

=Siv+S~,

and this sum is taken over indices i, j, u, v ranging from 1 to r and over all indices Q, (3 which satisfy 0 ~ Q :5 (3 :5 T with the further restriction that i 1: u, j 1: v when Q = (3.

It is simple to estimate Siv:

(8.3.26)

, r(r -1) TL.:-1 4 N'(N' -1) 4 SN ~ 2A4 m!lX lakr+il + 2A4 ~ax laTr+il N k=O l:::;.:::;r N l:::;.:::;N'

r r - 1) 4 r(r - 1) 0 ( N ()2 < a' < max a' -t - 2At ~I.I - 2A~ l:::;;:::;N I •1 , N -t 00.

To~stimate S'Jy we shall break the sum E(x) in (8.3.25) into two pieces SN(X)

and SN(X) in the following way. _Let the sum SN(X) represent all terms whose

indices satisfy (3 - Q ~ 2, and let SN(X) represent the remaining terms, i.e., those terms for which 0 :5 (3 - Q :5 1.

We shall estimate the number of terms in the sum SN(X). If 0 :5 Q ~ T and o ~ (3 :5 T then there are precisely T pairs (Q, (3) which satisfy (3 - Q = -.! and

precisely T + 1 pairs (Q, (3) which satisfy (3 = Q. The number of terms of S N( x) which correspond to pairs (Q,(3) with (3 = Q cannot exceed (r(r -1)/2)2. Thus the

192 CHAPTER 8

total number of terms in SN(X) cannot exceed 2(T + 1)(r(r - 1)/2)2. Combining this observation with (8.3.2), we obtain

N -+ 00.

A typical term from the sum S N( x) looks like

(8.3.28)

where f3 - a ~ 2, 1 ::; i < j ::; r, 1 ::; u < v ::; r. Since ar + j ::; (f3r + u) - r, there are at least r members of the given sequence {ntJ which lie between n",r+i and npr+u. Consequently, npr+u/n",r+i > qr. But the integral over the interval [O,IJ of the expression (8.3.28) is different from zero only when the index of the Walsh function there equals zero. Hence to estimate the integral Jo1 SN(X) dx we need only consider the terms in (8.3.28) for which

(8.3.29)

But by Lemma 8.3.2 (applied to the case p = 4 and I = 0) we can choose an r such that the number of solutions of (8.3.29) does not exceed C4 (0, q). Therefore,

(8.3.30) N -+ 00,

where this maximum is taken over all nk, for which (8.3.29) has a solution. Finally, observe that

Therefore, we conclude from (8.3.25), (8.3.26), (8.3.27), and (8.3.30) that

lim t pJ.(x) dx = o. • N ..... oo}o

PROOF OF 8.3.3. By definition of the quantities ZN,k(X) (see (8.3.13)) we have

T

(8.3.32) flu + ZN,k(X)) = 1 + 2:a~N)w'Y(x), k=O 'Y~O

where

(8.3.33)

and this sum is taken over all indices (nk"nk 2 ••• nkp ) which solve (8.3.4) for p = 1,2, ... , N. Notice that the factors ankl ' a n • 2 , a nka in (8.3.33) are taken from three different sums ZN,k(X), k = 0,1, ... , N. Consequently, there are at least r terms of the sequence {nd which lie between nkl and nka. Hence nkl/nka > qT and we have by 8.3.2 that

N ( Ia-I)P la~1 ::;"" 1>'ICh,q) m.ax -A' , ~l l~.~N N P=

1=1,2, ...

But if a ::; >. ::; b then given any c > ° there is an No such that

la·1 1>'ICh,q) max -A' < c l~.~N N

for N ~ No. Consequently, a~ -t ° as N -t 00 for any I ~ 1. The same kind of estimate is also true for ar;' - 1, i.e., ar;' -t 1 as N -t 00.

We have proved that

(8.3.34) N -t 00, 1=1,2, ...

uniformly for>. E [a, b]. By integrating identity (8.3.32) over the set E, and applying (8.3.34), it is easy to verify that (8.3.23) holds uniformly for ,\ E [a, b], where -00 < a < b < 00. This completes the proof of 8.3.3 and thus the fundamental result 8.3.1 is established .•

Chapter 9

DIVERGENT WALSH-FOURIER SERIES. ALMOST EVERYWHERE CONVERGENCE

OF WALSH-FOURIER SERIES OF L2 FUNCTIONS

In Chapter 2 we saw that even for a continuous function it is necessary to impose additional conditions to insure that its Walsh- Fourier series converges at every point. Without such conditions, as we remarked in §2.3, the Fourier series of a continuous function may diverge at some points.

Given a particular continuous function, or a particular Lebesgue integrable function f, how large can the set of points be on which the Walsh-Fourier series of f diverges? We shall investigate this extremely complicated question in this chapter.

In §9.1 we shall establish that the Walsh-Fourier series of a Lebesgue integrable function need not converge at even one point.

It turns out that the Walsh-Fourier series of a continuous function can diverge only on a set of measure zero. This will follow as a special case of the general theorem proved in §9.2 which states that the Walsh-Fourier series of any L2[O, 1) function must converge almost everywhere.

§9.1. Everywhere divergent Walsh-Fourier series. We shall prove an analogue of a theorem of Kolmogorov about the existence of

functions whose Fourier series diverge everywhere. We notice that for the Walsh system, the construction is somewhat simpler in comparison to the classical case for the trigonometric system.

We shall prove the following lemma which plays a key role in this section:

9.1.1. For any natural number n there is a polynomial of the form

2,,+2" -1

(9.1.1) Pn{x) = 1 + L aiwi{x), i=2"

which satisfies the properties: 1) Pn{x) ~ 0 everywhere; 2) for each x there are integers lz and m., with 2n < l., < m., < 2n+2" such that

the corresponding partial sums of the polynomial Pn satisfy

194

DIVERGENT WALSH-FOURIER SERIES 195

PROOF. Fix n and choose a natural number m of the form (2.2.4) which satisfies 2n - 1 < m < 2n. From inequality (2.2.5) and Theorem 2.2.2 we obtain

(9.1.2)

Since m < 2n , the kernel Dm(t) is constant on each of the intervals A~n), for j = 0,1, ... , 2n -1 (see 1.4.3). By (1.2.15) and definition (1.4.8), it is evident that the kernel Dm(x EB t) is constant for x E A~n) and t E A)n). Denote this constant value by Dj,i. Set

i.e.,

(9.1.3)

for sgnDj,i = 1,

for sgnDj,i = -1,

(-l)'Yi.; = sgnDj,i'

From each A)n) select the dyadic interval of rank n + 2n of the form

By construction it is clear that if t E dj then the coefficients of 2-(n+i+1), for o ::; i ::; 2n - 1, in the dyadic expansion of t are precisely "Ij,i' Thus by (1.2.12) we have

W2n+i(t) = (-l)'Yi.i.

In particular, (9.1.3) implies that

(9.1.4)

for each 0 ::; i ::; 2n - 1. We also notice that the interval dj is of length

(9.1.5) Idjl = 2-(n+2").

We shall show that the polynomial

(9.1.6)

satisfies the required conditions, where Xdi(x) is the characteristic function of the interval dj .

196 CHAPTER 9

Since the function Pn(x) is constant on any dyadic interval ~(n+2"), we have by Theorem 1.3.2 that it is a Walsh polynomial of order 2n +2 " - 1, i.e., the upper limit on the sum (9.1.1) is correct. To show that the lower limit in (9.1.1) is correct, we shall examine the Walsh-Fourier series of Pn(x) and show that all its Fourier coefficients P n ( i) are zero for 1 ::; i ::; 2 n - 1.

Notice first that

for all ii, h, and thus that

(9.1.7)

for all 1 ::; k ::; n and all j1, h. Fix 1 < i < 2n - 1 and choose k < n with 2k- 1 < i < 2k such that the function

Wi(t) is ~onstant on each interval ~k), and if ~~k-:'l) = ~~~) U ~~~~1 then Wi(t)

changes signs from ~~~) to ~~~~1. Using (9.1.7) we obtain

for aliI::; i ::; 2n - 1 and 1 ::; k ::; n. In particular,

That Pn(O) = 1 is easy to verify by (9.1.5). Thus we have proved that (9.1.6) and (9.1.1) are two ways of representing the same function, namely, the polynomial Pn(x).

Property 1) is obvious by (9.1.6). It remains to verify property 2).

Fix x and choose a natural number i = i(x) such that x E ~}n). Set ex = 2n +i

and mx = m + 2n+i , where m is chosen to satisfy (9.1.2). Since 0 ::; i ::; 2n - 1 and 2n- 1 < m < 2n, it is clear that 2n < ex < mx < 2n+2"-1 + 2n < 2n+2n.

Use (2.1.10), (1.4.11) and apply (9.1.6), (9.1.4), and (9.1.2). We obtain for any

DIVERGENT WALSH-FOURIER SERIES

x E ~~n) that •

ISm" (x, Pn) - Sl,,(X, Pn)1 =111 Pn(t)(Dm,,(x E9 t) - D2n+i(X E9 t» dt 1

=111 Pn(t)W2 n+i(X E9 t)Dm(x E9 t) dt 1

=1 W2n+.(X) 11 Pn(t)W2n+i(t)Dm(x E9 t) dt 1 2"

=1 ~22"1; W2,,+i(t)Dm(x E9 t) dt 1 2n

= L 22" l1Dm(X E9 t)1 dt ;=0 d;

2"

= L [ IDm(X E9 t)1 dt JLl.(") ;=0 ;

= 11 IDm(t)1 dt > ~. o 4

This verifies property 2) and the proof of the lemma is complete .•

197

We are now prepared to state and prove the fundamental result of this section.

9.1.2. There is a function f E L[O,I) whose Walsh-Fourier series diverges everywhere on [0,1).

PROOF. Let n; be a sequence of natural numbers which satisfy

(9.1.8)

and choose corresponding polynomials Pn;(x) by Lemma 9.1.1. We notice by (9.1.8) and (9.1.1) that the polynomials Pn;(x) do not have common terms for different j's, except 1.

Consider the function defined by

(9.1.9) 00 1

f(x) = "-Pn.(x). ~n. J j=1 J

Notice by (9.1.8) that nj > 2;-1 and thus that the series 2:~1 lin; converges. Since

the polynomials Pn; (x) are non-negative and satisfy Jo1 Pn; (t) dt = 1, it follows from

198 CHAPTER 9

a theorem of Levy (see A4.4.2) that the series (9.1.9) converges to an f E 1[0,1).

Moreover, since for each i the partial sums ofthe series 2::':1 ~Pn- (x )Wi(X) satisfy } nj J

k 1 k 1 I " -Pn·(x)Wj(x) I~" -Pn·(x) ~ f(x), L...Jn° ) L...J n · ) j=l } j=l J

it follows from the Lebesgue Dominated Convergence Theorem (see A4.4.1) that this series can be integrated term by term. Hence we obtain from (9.1.8) that

for 2nj ~ i < 2nj +2"j, where a~j) are the coefficients of the polynomial Pnj(x) defined by (9.1.1) for n = nj. Consequently, if 2nj < C < m < 2nj+2"j then the partial sums of f of order m and C satisfy

In particular, this inequality holds for the numbers m~j) and C~i) (see 9.1.1). Therefore, we obtain the following estimate for all x E [0, 1):

Since for each point x this inequality holds for all j, we conclude that the WalshFourier series of the function f(x) diverges everywhere on [0,1) .•

§9.2. Almost everywhere convergence of Walsh- Fourier series of L2 [0, 1) functions.

We shall prove the Walsh analogue of Carleson's Theorem which established that the trigonometric Fourier series of an L2 [0, 21l"] function converges to it almost everywhere.

Indeed, the following theorem is true:

9.2.1. For any function f E L2 [0, 1), the partial sums Sn(x, J) of its Walsh-Fourier series converges to f(x) almost everywhere on [0,1).

This theorem, as we shall show, follows from the fact that the operator

f ---+ Mf(x) == sup ISn(.r,!)1 n~l

is of weak type (2,2), i.e., from the following result:

9.2.2. There is an absolute constant C such that

C 11 mes{x: sup ISn(x,J)1 > y} ~ 2 If(t)12 dt n~1 Y 0

holds for all functions f E L2[0, 1) and for all real numbers y > 0.

In turn, 9.2.2 follows from a lemma which is fundamental for this section:

9.2.3. Given a function f E L2[0, 1), a number y > 0, and a natural number N, there is a set E = E(J, y, N) and some absolute constant C such that

1) mes E ~ ~ t 1!(tW dt; Y Jo

2) ISn(x,J)1 ~ Cy for x E [0,1) \ E and 1 ~ n < 2N.

We first show that 9.2.2 follows from 9.2.3. Fix y > 0. By condition 2),

and thus by condition 1) we have

mes eN = mes {x: sup ISn(x, J)I > CY} 1~n<2N

~ mes E ~ ~ t If(t)12 dt. y Jo

Now, this inequality is true for any N, and the sets eN satisfy eN+! :::) eN, and

{ x:sup1Sn(x,J)I>Cy}= U eN· n~1 N=1

Hence it follows that

mes {x: sup ISn(x, 1)1> CY} = lim meseN ~ ~ I 11f(tW dt. n~1 N--+oo Y 0

If we replace y by y / C in this last inequality, we obtain the inequality in 9.2.2 with absolute constant C3 •

Notice that in Lemma 9.2.3 we may exclude the case when the identity Sn( x) = S2N (x) holds for some n < 2N. Indeed, in this situation we can estimate the sums in 9.2.2 directly, using 5.2.4 and 5.1.1, to see that

200 CHAPTER 9

To prove that 9.2.1 follows directly from 9.2.2 we need only apply Theorems 5.1.3, 5.4.1, and the fact that for each Walsh polynomial Theorem 9.2.1 is trivial.

Thus to complete the proof of Theorem 9.2.1 it remains to establish Lemma 9.2.3.

PROOF OF 9.2.3. We shall need several new concepts and some additional notation.

For each dyadic interval b. of rank m (m > 0), denote the unique dyadic interval of rank m -1 which contains b. by b.*. Notice that 1b.1 = 2-m and 1b.*1 = 21b.1 = 2-m +1 .

Corresponding to each number n with dyadic expansion n = L:f=~l ej2j and each interval b. of rank m define a natural number by

N-l

(9.2.1) n(b.) = L ej2j, j=m

where the empty sum is defined to be identically zero. Then n = n(b.) + s where

(9.2.2) m-l

S = L ej2j < 2m = lb.r 1.

j=O

By definition of the operation EEl on the set of natural numbers (see §1.2), it is clear that n = n(b.) EEl S and thus by (1.2.17) that

(9.2.3)

Hence we see by (9.2.2) and 1.1.3 that the function w.(t) is constant on each interval b. of rank m, taking on the value +1 or -1 on each of them.

We introduce "local" Fourier coefficients, which are defined by the formula

(9.2.4)

Notice for b. = [0,1) that these local coefficients are the usual Walsh-Fourier coefficients, namely, an(J) = an([O, 1),1). By (9.2.3), we also have

(9.2.5)

If b. is an interval of rank m and b.' is another interval of the same rank such that b.* = b. Ub.', then substituting s = 2m - 1 = 1/(21b.1) = 1/1b.*1 into (9.2.3) and using the definition of W2m-1(t), we see that the function wn(Ll)+ILl.I-1 takes on the value Wn(Ll)(t) on one of the intervals b., b.' and takes on the value -Wn(Ll)(t) on the other interval. Consequently,


Since

, we obtain

(9.2.6)

Notice that 2m divides n(~) and recall (see 2.6.6) that {2m/2wi2m(t)}~O is a complete orthonormal system on each interval ~ of rank m. Moreover, notice that the Fourier coefficients of f with respect to this system are precisely the weighted local coefficients 2- m/2 ai2m (~, f). Consequently, it follows from Parseval's identity (see A5.4.2) that

(9.2.7) 1 [ 2 ~ 2 ~ le:.1f(t)1 dt = ~lai2m(~,f)1 .

We introduce one more bit of notation:

(9.2.8) ~=~/U~"·

(9.2.9)

The main technical step of this proof is the construction of a set Q* which contains special pairs (n(~), ~). The set Q* will be composed of subsets Q'k by means of Q* = U~l Q'k. For each pair (n(~,~) E Q'k we shall define a partition of the interval ~, which we shall denote by n = n(n(~),~, k). By using the partition n we shall identify intervals ~ C ~ and corresponding sums

(9.2.10)

which will be used to estimate the partial sum Sn(x, f). The partition n will help estimate the sums (9.2.10). These estimates will be obtained for all x except those belonging to some sets of small measure. The choice of the pairs Q*, the partition n, and the corresponding estimates will be carried out by using the numbers An(~) (see (9.2.8)).

We begin to carry out this plan. The numbers An (~) will be controlled by the set

(9.2.11) S* = U { ~ * : I~I L If(t)12 dt ~ y2} ,

202 CHAPTER 9

where this union is taken over the collection of intervals ~ which satisfy the following property: if ~ =~' U~" then the corresponding inequality in (9.2.11) holds for at least one of the intervals ~' or ~". Hence if ~ is not a subset of S* then both of its halves ~' and ~" satisfy the opposite inequality, so we have by Parseval's identity (9.2.7) that

and

lan(<l")(~"W ::; ,~",l" If(tW dt < y2,

for any n. By (9.2.5) the same kind of inequality holds for lan(~')1 and lan(~")I. In particular (see 9.2.8),

(9.2.12) if ~ is not a subset of S· then An(~) < y for n = 1,2, ....

It is clear by construction that the set S* can be written as a union of nonoverlapping intervals ~; such that for each j, one of the halves ~i of ~; satisfies the inequality

j = 1,2, ....

Consequently,

(9.2.13) 1 t mesS· = L I~;I = 2 L I~il ::; 22" Jo If(t)12 dt.

j j y 0

The set S* is part of the set E for which we search. Consider an interval which is not contained in S·, i.e., an interval which satisfies

(9.2.12). For each natural number k define a set Qk of pairs (n(~),~) which satisfy two conditions:

(9.2.14)

and

(9.2.15)

for (n(~),~) E Qk,

lan(<l)(~)1 ::; 2k for I~I = 2"; {

2y 1

if 1/2N ::; I~I < 1/2 then lan(~~ < y/2k ~or all n and ~ which satisfy n(~) = n(~) and ~ :::) ~, I~I ::; 1/2.

"#

By definition of the pairs in Qk it is evident that

(9.2.16) { if (n(~),~) E Qk then (n(~), ~) ~ Qk for all

~ :::) ~ and all n which satisfy n(~) = n(~). "#

DIVERGENT WALSH-FOURIER SERIES

Consider the polynomials

(9.2.17)

Notice that

(9.2.18)

Pk(Ll) == Pk(x,Ll) == L an(~">(Ll)wn(a)(X). (n(a),a)Eq,

a::>a

(n(a),a)Eq"

203

If (n(Ll), Ll) E Qk then by (9.2.16) the polynomial Pk(X, Ll*) cannot contain terms amwm(x) whose indices satisfy m(Ll) = n(Ll). Consequently, the polynomial Pk(X, Ll *) is orthogonal to Wn(a)(x) on the interval Ll, i.e.,

Thus for each (n(Ll), Ll) E Qk we have

This formula, (9.2.18), and Parseval's identity imply

(9.2.19) L If(x) - Pk(X, Ll)12 dx

= 11 f(x) - Pk(x, Ll*) - L an(a)(Ll)wn(a)(x) 12 dx a (n(a),a)Eq"

= llf(X)-Pk(x,Ll*Wdx- L lan(a)(LlWILlI· a (n(a),a)Eq"

Notice that if Lll U Ll2 = Ll, then

Consequently,

204 CHAPTER 9

Repeated applications of (9.2.19) and (9.2.20) eventuate in

o~ L llf(X)-Pk(X,fl.WdX 161=2-N 6

= L llf(X) - Pk(X, fl.*)12 dx - L lan(6)(fl.WIfl.1 161=2-N 6 (n(6),6)EQ.

161=2-N

= L llf(X) - Pk(X, fl.W dx - L lan(6)(fl.WIfl.1 161=2-N+1 6 (n(6),6)EQ.

161=2-N

= .. , = 11 If(xW dx - L lan(6)(fl.WIfl.I· o (n(6),6)EQ.


In particular,

(9.2.21 ) 22k t L 1fl.1 ~ -2 Jo If(xW dx.

(n(6),6)EQ. Y 0

Let Qk represent the set of pairs (n, fl. *) such that (n( fl.), fl.) E Q k and n = n(fl.*). Thus each pair (n(fl.), fl.) E Qk generates two pairs in Qk: (n(fl.), fl.') and

(n(fl.) + I~ * I' fl. *). Since Ifl. * I = 21fl.1, we have by (9.2.21) that

(9.2.22) 22k t L 1fl.1 ~ 4-2 Jo If(xW dx.

(n(6),6)EQ: Y 0

Consider the set 00

Notice that

(9.2.23) {if fl. = [0,1) and Y.. ~ An(6)(fl.) < -JL,

2k 2k- 1

then (n(fl.), fl.) E Qt.

This implication follows directly from definition (9.2.8), inequality (9.2.14) and the first inequality in (9.2.15).

Moreover, we shall prove that

(9.2.24)

if ~ is not a subset of S* and ;k ::::; An(a)(~) then one can choose

a triple (n,~,k) such that (n(~),~) E Q~, ~:J~,

1 ::::; k ::::; k, and such that if ii is any interval of rank one greater

than the rank of ~,i.e., 21iil = I~I), then n(&) = n(&) .

Indeed, if ~ = [0,1) then choose by (9.2.12) an integer k which satisfies (9.2.23) and 1::::; k ::::; k. Thus (n(~),~) E Q~ and the triple we want is given by (n(~),~, k).

If ~ is a proper subset of [0,1) and (n(~),~) ¢ Qi; then use definition (9.2.8) to see that at least one interval & C ~ of rank one greater than the rank of ~ must satisfy

A A Y lan(a)(~)1 = lan(A)(~)1 ~ 2k '

i.e., the pair (n(ii),ii) satisfies (9.2.14). On the other hand, from the assumption (n(~),~) ¢ Qi; and the definition of Qi; it follows that (n(ii), ii) ¢ Qk. Moreover, we have 1&1 < I~I ::::; 1/2. Thus (n(ii), &) fails to satisfy (9.2.15) and there must be an n and a~' such that n(&) = n(&), ~' ~ &, I~'I ::::; 1/2, and lan-(~')I ~ y/2k.

In particular, IAn{~)1 > y/2k where ~ =~' U~", ~:J~. - ~

If (n(~), ~) E Qi; then the triple for which we search is given by (n(~), ~, k). If (n(~), ~) ¢ Qi; then repeat the argument above, perhaps several times, until we either arrive at a pair (n(~),~) E Qk or until ~ = [0,1). But this last situation is considered above, because the fact that ~ is not a subset of S* implies that ~ is also not a subset of S*. Therefore, the proof of (9.2.24) is complete.

We can stipulate that the k chosen in (9.2.24) is minimal. Under this stipulation, it is easy to see that

(9.2.25) { - y -

A-(A)(~) < -_- for all ~ which satisfy n.... 2k - 1

~ C ~ C ~ and all n(~) such that n(~) = n(~).

Indeed, suppose to the contrary that (9.2.25) is false, i.e., for some ~ which satisfies ~ C ~ C ~ we have Aii(~)(~) ~ y/2k- 1 • Then the triple (n(~), ~, k -1) satisfies

(9.2.24), which contradicts the minimality of k. We shall show that

(9.2.26) if (n(~),~) E Qk then An(a)(~) < 2;-1.

206 CHAPTER 9

Suppose first that IL\I < 1. We have by the assumption (n(L\), L\) E Qi that one of the halves L\' of L\ must satisfy (n(L\'), L\') E Qk. Hence by (9.2.15)

Let L\" be the other half of L\. Then it follows from L\' U L\" = L\, (9.2.6), and L\ = (L\')* = (L\")* that

lan(A,)(L\')1 = lan(A'HIAI- 1 (L\')1 < 2;-1 and

lan(A,)(L\")1 = lan(A'HIAI- 1 (L\")1 < 2;-1· Since n(L\) equals either n(L\') or n(L\') + IL\I-l, we conclude by definition (9.2.8) that (9.2.26) holds in the case IL\I < 1.

Notice by (9.2.26) and (9.2.14) that if IL\I < 1 then the pair (n(L\), L\) can belong to only one Qi. In sharp contrast, the pair (n, [0, 1» can belong to two different sets Qt and Qi 2 , kl < k2 • Exclude (n, [0, 1» from Qi2 • Then we see by the definition of Qi and by the first inequality in (9.2.15) that (9.2.26) holds in the case L\ = [0,1).

We now show that for each interval L\ which appears in a pair (n( L\), L\) E Qi, it is possible to construct a partition n = n((n(L\),L\,k) whose elements are dyadic intervals L\' E n which satisfy the following three conditions:

(9.2.27) L\' c L\ IL\'I > ~. to' - 2N '

(9.2.28)

and

(9.2.29) either IL\'I > 2~ and An(A)(L\') 2:: 2k~1 ' or IL\'I = 2~·

We begin to construct the partition n by dividing each interval L\ into two halves (which are necessarily of rank one greater that the rank of L\), and placing into n those halves .& which satisfy An(A)(.&) 2:: Y /2 k - 1 • (Notice by the definition of Q'k that IL\I > 1/2N .) For those halves which did not get placed into n (it is possible to retain both halves) continue this process of dividing and placing. At each stage, the newly divided intervals L\ are placed into n if IL\I = 1/2N

or if An(A)(L\) 2:: y/2k - 1 . On the other hand, those intervals L\ which satisfy

An(A)(Li) < y/2 k - 1 and IL\I > 1/2N are retained for further division. It is easy to

verify that the partition of ~ obtained by this process generates non-overlapping dyadic intervals ~' which satisfy conditions (9.2.27) through (9.2.29).

For each pair (n(~),~) E Qk we shall call sums of the form

(9.2.30) x E ~, ~'C~ C~, ~'E n

diJtinguiJh ed. Notice for each interval ~" E n that the intersection ~" n~ is either empty or

is a subset of ~". Let us examine one of the distinguished sums (9.2.30). Let

N-l N-l

n(~) = L c j 2j , n(~) = L cj2i , j==m j=m

where m is the rank of the interval ~ and m is the rank of the interval ~. Notice that m > m and thus n(~) < n(~). The sum (9.2.30) can be evaluated by means of the modified Dirichlet kernel using (5.3.1) in the form

Indeed, by (5.3.4) we have

(9.2.31 ) Sn(e.)(x, f) - Sn(:6)Cx, f)

= 11 J(t)wn(e.)(X ffi t)D~(e.)(x ffi t) dt -11 J(t)Wn(,C;)(x EB t)D:(Eix ffi t) dt

N-l 1

= LCj 1 J(t)Wn(e.)(xEBt)D;i(xEBt)dt j=m 0

N-l 1

- LCj 1 J(t)Wn("E:) (x EBt)D;i(X EBt)dt j=m 0

for all x E ~. Notice for all j ? m that the function D;i (t) = W2i (t)D2i (t) is different from zero only on the intervals ~~ C ~lF but for j < m the functions W2i(t) are all constant on these same intervals, with value 1. Notice also that

m-l

Wn(e.)(t) = II (W2j (t»e j wn(:6)(t) = Wn("E:)(t), t E ~~. j=m

Consequently,

j ? m, t E [0,1).

208 CHAPTER 9

Combining this identity with (9.2.31), we see that the following expression is valid for all x E .6.:

m-l 1

(9.2.32) Sn(A)(X, f) - Sn(7S:/ x , f) = j~ ej 1 f(t)Wn(A)(X EB t)D;j (x EB t) dt.

We introduce an auxiliary function

(9.2.33) get) == g(t,n) == { ~n(A)(.6.') for t E .6.' E n, for t tJ. .6.'.

By (9.2.28) it is clear that

(9.2.34) y

Ig(t)1 < 2k - 1 ' tE[a,1).

Fix x E .6.. Choose an interval .6.' E n such that x E .6.' C .6.. The kernels D;j (x EB t) are constant on the intervals of rank j + 1 and thus constant for j ::; m - 1 on all intervals of rank m, in particular for t E .6., in fact for t E .6.' C .6.. These kernels D;j (x EB t) are also constant in t on any interval .6./1 E n, since in this case x tJ. .6./1 and we can apply 5.3.1. Finally, for j ~ m and t tJ. .6. we have D;j (xEBt) = a since x E .6. implies x EB t tJ. .6.~m), i.e., the point x EB t does not belong to the support of the kernel D;j (x EB t). If we combine these observations with (9.2.33) and (9.2.4), we see that the identity (9.2.32) can be continued in the following way:

(9.2.35) m-l

Sn(A)(X, f) - Sn(t>.) (x, f) = Wn(A)(X) L ej L 1 f(t)Wn(A)(t)D;j (x EB t) dt j=m A'Ef! A' m-l

=Wn(A)(X) Lej L 1 g(t)D;j(xEBt)dt j=m A'Ef! A' m-l 1

=Wn(A)(X) Lej 1 g(t)D;j(xEBt)dt j=m 0

Use the notation (5.3.2) and formula (5.3.4). Since n(.6.) = L:;:-~ ej 2j , it follows that

This last sum can be viewed as a partial sum of the Walsh-Fourier series of the function 8~(A)(x,g). Thus, we can interpret the sum on the right side of (9.2.35) as a partial sum of a Fourier series, i.e.,

m-l 1 m-l 21+1_1 1

Lej 1 g(t)D;j(xEfH)dt= Lej L Wj(X) 1 g(t)Wj(t)dt j=m 0 j=m i=2 j 0

= 82m-(x, 8~(A)(g))·

Consequently, using the notation (5.2.7) and (9.2.35), we obtain

(9.2.36) x E~.

We emphasize that the right side of this last estimate does not depend on ~. We now introduce a set on which the distinguished sums (9.2.30) can be relatively

large. Set

(9.2.37) U* == U*(n(~),~, k) == {x E [0,1) : H(S~(A)(g))(x) > 2;/2}'

Recall from 5.2.4 that the operator H is of strong type (p,p), hence (see 5.1.1) of weak type (p,p) for alII < p < 00. Use this fact for p = 6. We obtain

Since the operator S~(A) is also of type (6,6) (see 5.3.2), we can continue this estimate using (9.2.34) and (9.2.33) as follows:

(9.2.38) mes U* < G,23k IIgl16 < G,,23k • L I~I = G" ~

- y6 6 - y6 26k 23k .

On the other hand, we have by (9.2.36) and (9.2.37) that

(9.2.39)

for any ~, C D. C ~ and ~' E n. We have completed construction of the set E, namely,

X d U*, x E A v:. U,

(9.2.40) E == S* U U ( U u*(n(~)'~'k)) . k=1 (n(A),A)EQZ

210 CHAPTER 9

By (9.2.13), (9.2.22), and (9.2.38), we have

(9.2.41 ) ( XlI) 1 11 mes E ~ 2 + C"4 E 21< 2" If(t)12 dt, 1<=1 Y 0

i.e., this set E satisfies condition 1) from the conclusion of Lemma 9.2.3 with 2+4C" as the absolute constant.

It remains to verify condition 2), i.e., to estimate Sn(x, I) for any x f/. E and 0< n < 2N.

We may suppose that An(~o) > 0, where ~o == [0,1). Indeed, if this were not true then by (9.2.9) an(~O) = an (f) = 0 and either there is a natural number m, n < m < 2N , such that Sn(x,1) = Sm(x,1) for all x E [0,1) and Am(.~o) > 0, or Sn(x, I) = S2N(X, I), and as we remarked at the beginning of this section (after the statement of Lemma 9.2.3), we can exclude this case from consideration.

Fix x f/. E. Since x f/. S· we have ~o f/. S· and (9.2.12) holds for ~o. Thus we can choose ko ~ 0 such that

and by (9.2.23) such that (n,~o) E Qio' Hence the partition no = n(n,~o,ko) is defined.

Let ~1 E no satisfy x E ~1' By (9.2.39) we have

By (9.2.27), ~1 C ~o. If n(~d = 0 then estimate 2) is obtained for this point x. ¢

If n(~d "I- 0 then we continue this process further. Notice that if n(~d"l- 0 then I~d > 1/2N (see (9.2.1)). By (9.2.29)

Y An(atl(~d = An(ao)(~d ~ ---.

21<0- 1

Since x f/. S· we see by (9.2.12) that An(a,)(~l) < y. Thus we can find a kI,

1 ~ k1 < ko, such that

(9.2.42) y y

21<, ~ An(atl(~l) < 21<,-1

The inequality on the left side of (9.2.42) allows us to apply (9.2.24) and (9.2.25). The result of this is we can choose a triple (n}, li 1 , kt) such that the halves Li1 C ~I, 21Li11 = I~d, satisfy

(9.2.43)


such that .6.1 :::> 61, k1 ~ k1' and k1 is minimal such that (iiI (.6.1 ), .6.1 ) E Qt. Thus by (9.2.25) we see that all intervals 6 which satisfy 6 1 C 6 C .6.1 also satisfy

(9.2.44)

Thus the partition 0 1 = 0(ii1 (.6.1 ),.6.1 ,kt} is defined. Choose 6 2 E 0 1 such that x E 6 2 and by (9.2.29) such that

A. . (6) > -Y-. nl(Ad 2 - 2k1 - 1

The interval 6 2 cannot be one of the intervals 6, 6 1 C 6 C .6.1 , since such a 6 satisfies the opposite inequality (9.2.44). Consequently, 6 2 C 6 1 and 6 2 C '&1'

'¢

Thus we see by (9.2.43) that n1(62 ) = n(62 ). We shall estimate

ISn(Ad(X, I) - Sn(A2)(x, 1)1

~ ISn(A1)(X, f) - Sn(ad(x, 1)1 + ISn1(at}(x, I) - Sn(ad(x, 1)1

+ ISnl(~d(x, I) - Sn(A2)(X, 1)1

== 81 +82 +83.

To estimate 81 notice that n(61) equals either n(a1 ) or n(a1) + 161 1-1 (see (9.2.1)). In the first case 81 = O. In the second case 161 1 = 2-m , 6 1 = a1 ua~, and we obtain

81 = ISn(ad+2m(x, I) - Sn(al)(x, 1)1

=111 f(t) C~1 Wn(al)+i(X (f) t)) dt 1

=111 f(t)Wn(ad(t)D2m(x (f) t) dt 1

=1 2m (!al f(t)wn(ad(t)dt+ !a~ f(t)Wn(a1)(t)dt) 1 1 . • ,

~ 2(lan(a1)(6dl + lan(ad(61 )1) 1 • • ,

= 2(lan(A1)(61 )1 + lan(Ad(61 )1) ~ An(Ad(61 ).

We used (9.2.5) and (9.2.8) to obtain these last estimates. The terms 82 and 83 are distinguished sums and can be estimated by using (9.2.39)

for k = kl' where .6.1 plays the role of 6 and '&1 and 6 2 play the role of 6.

212 CHAPTER 9

Combining these estimates for SI, S2, and S3, using (9.2.42) and bearing in mind that kl ~ k1 , we obtain

This change from ~1 to ~2 and the corresponding estimate obtained above are the first step in an inductive argument which we repeat if n(~2) f O. This argument can be repeated only finitely many times. The result is that we obtain nested intervals

[0,1) = L\o :::> ~1 :::> ••• :::> ~.+1 # # #

and integers ko > kl > ... > k. ~ 1 such that n(L\j) f 0 for j ~ S, n(~.+I) = 0 and

Summing these estimates over j we have . ~

ISn(x,J)1 =/ LSn(aj)(X,J) - Sn(ai+,j(x,J) I~ 4y LTk/2. j=o k=1

Adjusting the constants in this inequality and in (9.2.41) we see that condition 2) of Lemma 9.2.3 holds. This completes the proof of the lemma, and thus of Theorems 9.2.1 and 9.2.2 .•

Chapter 10


§10.1. Approximation in uniform norm. Let J(t) be a function continuous on the interval [0,1). Consider the quantity

n-l

(10.1.1) En(J) = inf IIJ(t) - L akwk(t)lI, {all k=O

n = 1,2, ... ,

where the infimum is taken over all real coefficients {ak} and the norm is the uniform one, i.e.,

IIJII = sup IJ(t)l· 09<1

We shall call the quantity En(J) the best uniJorm approximation oj the function J by Walsh polynomials of order n. By definition it is clear that

En(J) :5 IIJ(t) - Sn(t, f)1I,

where sn(t, f) = E~':~ j(k)Wk(t) is the partial sum of order n of the Walsh-Fourier series of J. As we showed in 2.3.1,

IIJ(t) - S2n(t,J)II-+ 0, n -+ 00

for all continuous functions f. Consequently, E2n(J) -+ ° as n -+ 00. Moreover, the sequence {En(J)}~=o is obviously non-increasing, i.e., En(J) ~ EnH(J). Therefore, En(J) -+ ° as n -+ 00 for every function continuous on the interval [0,1).

Below, we shall obtain two-sided estimates for the best approximation En(J) to a function f in terms of its modulus of continuity. Moreover, we shall establish estimates for the best approximation of a continuous function by Haar polynomials as well.

We begin by recalling the definition of the Haar system and its connection with the Walsh system (see §1.3). The Haar functions hn(x) are defined as follows: ho(x) = 1 for x E [0,1); if n = 2k + m, k = 0,1, ... , m = 0,1, ... 2k - 1, then

(10.1.2)

for

for

for

213

E A (HI) X L.l.2m ,

(HI) x E 6 2m+1 ,

[ \ (k) xE 0,1) ~m.

214 CHAPTER 10

Here A~) = [m/2k, (m + 1)/2k), k = 0,1,00', m = 0,1,00' 2k - 1. As we showed in §1.3, the identities

2'-1

(10.1.3) W2'+m(X) = Tk/2 E W~~nh2.+n(X), n=O

and

2'-1

(10.1.4) h2.+n(x) = 2k/2 E W~~~W2.+m(X), m=O

hold for all x E [0,1), where (w~~~) is a symmetric orthogonal matrix of order 2k. Since wo(x) = ho(x), we have by (10.1.3) and (10.1.4) that every Haar polynomial

2:!~~1 anhn(x) of order 2k is also a Walsh polynomial 2:!~~1 bnwn(x) of order 2k and conversely. Consequently, we have by (10.1.1) that

k = 0,1, ... ,

where

n-l

(10.1.5) E2• (f)h = inf Ilf(t) - E akhk(t)1I {ad k=O

is the best uniform approximation to the function f by Haar polynomials of order n. (Here, the infimum is taken over all choices of n real numbers ao, aI, ... ,an-I')

Notice that each Haar polynomial 2:~':~ akhk(t) is a step function of a special

type. Namely, if n = 2m +1:'+ 1, 1:' = 0,1, ... , 2m -1, and A~m) = [i/2 m , (i + 1)/2m ),

i = 0,1, ... 2m -1, m = 0,1, ... , then this Haar polynomial is a step function which is constant on the intervals A~m+l), 0::; i::; 21:'+ 1, and .6.;m), 1:' < j ::; 2m -1. This follows immediately from the definition of the Haar functions (see (10.1.2)). The converse of this statement is also true:

10.1.1. Ifn = 2m +1:'+ 1, e = 0,I,oo.,2m -1, m = 0,1, ... , then any step function 'Pn(x) which is constant on the intervals A~m+1), 0 ::; i :::; 21:' + 1, and AJm),

1:' < j ::; 2m - 1 can be represented as a Haar polynomial 2:~;:~ akhk(t) of order n.

PROOF. Let 'P n (x) be a step function which satisfies the hypotheses of Theorem 10.1.1. We need to show that the equation

n-l

(10.1.6) E akhk(x) = 'Pn(x), o ::; x < 1, k=O

APPROXIMATIONS BY WALSH AND HAAR POLYNOMIALS 215

can be solved for the coefficients ak. We suppose at first that n - 1 = 2m for some m = 0,1, .... In this case we need to prove that if 'P2m+I(X) a step function which takes the value c~m) on the interval D.~m) for i = 0,1, ... , 2m - 1, then there is

a Haar polynomial of the form I:!:;;-l akhk(x) which coincides with this function on the interval [0,1). But we have already noticed that a Haar polynomial of the form I:!:;;-l akhk(x) is also a Walsh polynomial of the form I:~:;;-l bkWk(X), and conversely. Consequently, that equation (10.1.6) can be solved in the case n = 2m +l follows immediately from 1.3.2.

Suppose now that n = 2m + £ + 1, where 0 5 £ < 2m -1, for some m = 1,2, .... Then we can write the left side of equation (10.1.6) in the form

n-l 2m_l n-l

(10.1. 7) L>khk(X) = E akhk(x) + E akhk(x), k=O k=O

By what we just proved, we can choose coefficients for the first polynomial on the right side of this identity so that it coincides with any step function which is constant on intervals of the form D.im) , s = 0,1, ... 2m - 1. To deal with the second polynomial on the right side of (10.1.7), notice by the definition of the Haar functions (10.1.2) that this second polynomial is a step function which vanishes on the intervals D.}m), £ < j 5 2m - 1, and takes on constant values b~m+l) on

the intervals D.~m+I) which satisfy the additional condition b~::+l) = -b~:::/), s = 0,1, ... ,P. Consequently, if the step function 'Pn( x) on the right side of equation (10.1.6) takes the values c;m+l) on the intervals D.;mH), j = 0,1, ... ,2£ + 1 and

the values C}m) on the intervals D.jm), j = £ + 1, ... , 2m - 1, then equation (10.1.6) is equivalent to the following system of linear equations:

b~m) + b~::H) = c~::H),

b(m) _ b(mH) _ (m+l) 8 28 - C28+1 ,

b(.m) = /m) ] ]'

S = 0, ... , £, j = £ + 1, ... ,2m - 1.

This system has a unique solution for any choice of the numbers on the right side. Therefore, (10.1.6) can be solved and the proof of Proposition 10.1.1 is complete. I

Given any function f defined on the interval [0,1), we shall use the notation

(10.1.8)

(10.1.9)

o:~m)(f) = sup f(x), xE~~m)

for s = 0,1, ... ,2m - 1, m = 0,1, ....

216 CHAPTER 10

10.1.2. If a function f(x) is continuous on [0,1] and w( 0, I) is its modulus of continuity, then

(10.1.10) n = 1,2, ... ,

where En(J)h is defined by (10.1.5).

PROOF. The inequality is obvious for n = 1 since

Suppose now that n ~ 2. Write n in the form n = 2m +£+ 1, £ = 0,1, ... ,2m -1, m = 0,1, .... We shall prove that

(10.1.11) En(J)h = -21 max { max w~m+l)(J), 0~i9l+1

Toward this, let Pn(x) = L:;';;-~ akhk(x) be a Haar polynomial of order n which takes

on the value ~(a~m+l)(1) + .aim+1)(J» on each interval boim+1), s = 0,1, ... ,2£ + 1,

and the value ~(ajm) (J) + .a;m) (J» on each interval bo;m), j = £ + 1, ... ,2m -

1. Such a polynomial exists by 10.1.1. This polynomial gives the best uniform approximation by constants to the function f (x) on each of the intervals boi m+ 1) , S =

0,1, ... 2£+ 1, and on each of the intervals boJm), j = £+ 1, ... ,2m -1. Consequently, on the entire interval [0,1), this polynomial gives the best uniform approximation to the function f(x) by Haar polynomials of order n. By the construction of this polynomial, we see that (10.1.11) holds. Using this identity together with the notation (10.1.8) and (10.1.9), we see that

1 En(J)h :5 2 max{w(Tm - 1 , l),w(2-m, I)}

= ~w(Tm, I) :5 w(2-m- 1 , I) :5 w(~, I).

In particular, the left hand inequality in (10.1.10) is proved. But, the definition of the modulus of continuity together with (10.1.9) and (10.1.11) imply

1 w(-,I):5 w(Tm,1)

n :5 max{2 max w(m+l)(J) 2 max w~m)(J)

0<0<2l+1 0 , l<j~2m) ,

w~~~ (I) + w~;'+l) (J) + w~;'+il) (J)}

< max{3 max w(m+l)(J) 3 max w)(m)(J)} = 6En(J)h. - 0~0~2l+1 0 , l<j~2m

This proves the right hand inequality in (10.1.10) .•

10.1.3. If f is continuous on the interval [0,1] then

(10.1.12) 1

En(J) :s 2w( -,1) :s 24En(J), n

n = 1,2, ....

PROOF. Let n = 2m , m = 0,1, ... , and notice by (10.1.3) and (10.1.4) that

(10.1.13)

Thus in this case (10.1.12) follows immediately from 10.1.2. Suppose now that 2m < n < 2m +! and write n in the form n = 2m + £ where

o < £ < 2m. Since the sequence {En(J)} is monotone, non-increasing, we see by identity (10.1.13) and 10.1.2 that

i.e., the left hand inequality in (10.1.12) is true. Similarly,

w(l/n, 1) :s w(2-m, 1) :s 2w(2-m- l , f)

:s 12E2 m+l (J)h = 12E2 m+l (J) :s 12En(J).

This proves the right hand inequality in (10.1.12) .• As a corollary of 10.1.2 and 10.1.2 we obtain

10.1.4. A function f, continuous on the interval [0,1] belongs to the class Lip a for some a E (0,1] if and only if either of the following two conditions are met:

In the theory of approximation, an upper estimate of the best approximation to a given function (in a fixed norm) by polynomials of a certain order in some system of functions, by means of the modulus of continuity of this function (in the same norm) is called a direct theorem. Results in which the best approximations are estimated below by the modulus of continuity are called inverse theorems. Direct theorems are represented by Jackson's Theorem, which uses the modulus of continuity on the interval [0,27l'] of a continuous 27l'-periodic function f to estimate the best uniform approximation En (J)T by trigonometric polynomials

of order n as follows

n-l

tn(x) = ao + 2)akcoskx + bksinkx) k=I

1 En(J)T:S 12w( -,1).

n

218 CHAPTER \0

(More recently, the exact inequality

7r En(f)T < w( -, f),

n If:. const, n = 1,2, ... ,

was obtained by N. P. Kornelcukom, who proved that the constant which precedes the modulus of continuity on the right side of this inequality cannot be replaced by any constant smaller than 1.)

The left hand inequalities in (10.1.12) and (10.1.10) can be viewed as analogues of Jackson's Theorem for the best approximation by Walsh and Haar polynomials. However, the right hand inequalities of (10.1.12) (respectively, (10.1.10)) are peculiar to the Walsh case (respectively, the Haar case) in the sense that there does not exist a constant C such that

n = 1,2, ...

holds for all 27r-periodic, continuous functions for any constant C. It is easy to use 10.1.2 and 10.1.3 to obtain the following result:

10.1.5. If I is continuous on the interval [0, I] and either En(f)h = o(l/n) or En(f) = o(l/n), as n -+ 00 then 1== const.

PROOF. If a continuous function I satisfies one of these conditions then we have by 10.1.2 or 10.1.3 that w(l/n, f) -+ 0 as n -+ 00. Since the modulus of continuity is monotone, it follows that

(10.1.14) w(a,f) = o(a), a-+O+.

But I/( x ± a) - I( x)1 :5 w( a, f) for any x E (0,1) and x ± a E (0,1). Thus we obtain from (10.1.14) that

limsupl/(x±a~-/(x)1 <limsupw(a,f) = lim w(a,f) =0. 6--0+ - 6--0+ a 6--0+ a

Since the left side of this display is always non-negative, it must be identically zero, i.e., I'(x) = O. Since x E (0,1) was arbitrary, we see that the function I is constant on (0,1). Since it is continuous on the entire interval [0,1], it follows that I is constant on [0,1] .•

Thus we see that a non-constant, continuous function cannot be uniformly approximated by a Haar or Walsh polynomial of order n with a rate as fast as a(l/n), as n -+ 00. This fact can be explained by the discontinuous nature of the Haar and Walsh functions. This is not a serious drawback. For a continuous but non-smooth function, i.e., a continuous function which does not have a continuous derivative, the Walsh and Haar polynomials provide sufficiently good uniform approximations on the interval [0,1]. Moreover, we should keep in mind the fact that for the best

uniform approximation by Haar polynomials there is the simple formula (10.1.11) which in the case n = 2m , m = 0,1, ... , remains true for uniform approximation by Walsh polynomials as well (see (10.1.13). We notice also that the proof of Proposition 10.1.2 contains a simple and effective means to construct a Haar polynomial which attains the best uniform approximation to a given continuous function be Haar polynomials of order n. (Effective means for constructing trigonometric polynomials which are best uniform approximations to a given continuous, 211'-periodic functions are not yet known.)

§10.2. Approximation in the V' norm. We shall look at direct and inverse theorems for approximation by Walsh and

Haar polynomials of functions in L'[O, 1) norm, 1 :::; p < 00.

Let J be a function which belongs to L'[O, 1), for some 1 :::; p < 00, i.e., J is Lebesgue measurable on [0,1) and the norm

(10.2.1) ( 1 ) 1/,

IIJllp = lIJ(x)I'dx

is finite. The quantity

n-1

(10.2.2) E~)(1) = inf IIJ(t) - L akWk(t)lIp, {II.} k=O

where the infimum is taken over all choices of the n real numbers ao, all ... , an-I, will be called the best approximation to the function J in the norm oj L' [0, 1) by Walsh polynomials oj order n. It is clear by definition that

(10.2.3)

where

n-1

(10.2.4) Sn(X,j) = L!<k)Wk(t) k=O

represents the partial sums of order n of the Walsh-Fourier series of the function J.

10.2.1. If J E V[O, 1), 1 :::; p < 00, tben

(10.2.5) n=O,l, ....

PROOF. Recall from §2.1 that these partial sums satisfy

(10.2.6) S2,,(1) = 2n ( J(t) dt, Ja(")

I

for x E d~n), i = 0, ... ,2n -1,

220 CHAPTER 10

where A~n) = [i/2n,(i + 1)/2n). Let ~ +! = 1. We shall use Holder's inequality p q

(10.2.7) b ( I ) lip ( I ) llg 11 <p(t)t/J(t) dt 1$ 11<p(x)IP dx l lt/J(xW dx

for the case [a, b] = A~n), <p(t) = f(t), and t/J(t) = 1. Thus it follows from (10.2.6) that

For the next several pages we shall assume that the function f and the Walsh functions are periodic of period 1. The quantity

(10.2.8) wp(b,f)= sup (If(x+h)-f(x)IPdx , ( I ) lip

099 10

is called the modulus of continuity of the function f in the LP norm , or simply, the LP-modulus of continuity.

10.2.2. Iff E LP[O, 1), 1 $ p < 00, then

(10.2.9) IIf - 52" (x, f)lIp $ 21IPwp(2-", f), n = 0, 1, ....

PROOF. By identity (10.2.6), we see by the definition (10.2.1) of the LP norm that

IIf - ,,"(f)II, ~ Ct.' ii") I f(x) - 2" ii"/(t) <it I' <Ix) 'I,

~ Ct.' ii') 12" ii",u(X) - f(t»<it I' <Ix) 'I,

Hence using Holder's inequality (10.2.7) to estimate the inner integral, we obtain

IIf - ,,·(f)II, :5 Ct.' 2" ii") (ii',If(X) - f(t»I' dt) dX) 'I,

= (2I:I 2" 1 .. (1(;+1)/2"-Z If(x) _ f(x + t»)IP dt) dX) IIp

;=0 .o.l) '/2"-z

:5 Ct.' 2" ii') (f".lf(X) - f(x + t»I' <it) dX) 'I,

APPROXIMA TlONS BY WALSH AND HAAR POLYNOMIALS 221

By Fubini's Theorem (see A4.4.4), change the order of integration in this last expression and use the definition of the LP-modulus of continuity. Thus obtain

IIf - ". (1)11, <; (2. L~ ('t: ii.,If(x) - f( x + '))1' dX) dt )'"

= (2. L~ ([ If(x) - f(x + '))1' dX) dt r ~ 21/ pwp (2- n , f) .•

10.2.3. If IE LP[O, 1) for some 1 ~ p < 00 then

(10.2.10) n = 1,2, ...

PROOF. Write n in the form n = 2m + e where e = 0, ... ,2m -1, m = 0,1, .... By (10.2.3) and (10.2.9) we have

E~)(f) ~ E~lj(f) ~ 21/pwp (Tm, f)

~ 21+1/pwp(Tm-l , f)

~ 21+1/pwp(l/n, f) .•

Inequality (10.2.10) can be viewed as an analogue of the well-known estimate

where E~)(f)T = inf{tn} III - tnllp is the best approximation to a function I E LP[0,27l') in the LP norm by trigonometric polynomials tn(x) of order n.

For 0 < a ~ 1, and 1 ~ p < 00, denote the class of functions I E LP[O, 1) which satisfy wp(o,f) = OW") by Lip (a,p). The following result is a corollary of 10.2.3:

10.2.4. If I E Lip (a,p), for some 0 < a ~ 1, 1 ~ p < 00, then

In order to prove a reverse theorem, that is one which estimates the LP-modulus of continuity from above by the best approximation in LP[O, 1) norm, we need three preliminary results.

10.2.5. If IE LP[O, 1) for some 1 ~ p < 00 then

n = 0,1, ....

222 CHAPTER \0

PROOF. The left hand inequality is trivial by the definition of E~~)(f). To prove the right hand inequality let 'P2 n (x) denote a Walsh polynomial of order 2n

which gives the best approximation to the function I in the LP[O,l) norm, i.e., E~~)(f) = III - 'P2n llp· This polynomial exists since for I E LP[O,l) the quantity III - L::':~ akwk(x )llp is a continuous function of the coefficients ao, ... , an-I' Apply the triangle inequality for the LP norm

(10.2.11)

which is frequently called Minkowski's inequality, and Proposition 10.2.1. We obtain

III - 82n(f)llp = 111- 'P2n - 82n(f - 'P2n )lIp ~ III - 'P2 n lip + 11 8 2n (f - 'P2 n )lIp ~ 2111 - 'P2 n lip = 2E~~\f). I

Let hn(x) be the Haar functions as defined in equation (10.1.2).

10.2.6. For any p > 0 and any real numbers ak, 2n ~ k < 2m+!, m = 0,1, ... , the following identity holds:

(10.2.12)

PROOF. If k = 2m + i, 0 ~ i < 2m, then Ihk(x)1 = 2m/2 for x E ~~m) =

[i/2m,(i + 1)/2m) and hk(x) = 0 for x E [0,1) \ ~~m). From this it is easy to see that (10.2.12) holds. I

Let I E L[O, 1) be fixed. Let 'Pn(x) == 'Pn(x,f), n = 1,2, ... denote the partial sums of order n of the Haar-Fourier series of I, i.e., let

n-l

(10.2.13) 'Pn(x) = L akhk(x), k=O

where

(10.2.14)

are the Haar-Fourier coefficients of the function I.

10.2.7. If n - 1 = 2m + f for some £ = 0,1, ... ,2m - 1, m = 0,1, ... , then

for x E ~~m+l), 0 $ i $ 2£+ 1,

for x E A (m) £ < J' < 2m J ' ,

where A~8) = [i/2 B , (i + 1)/28 ), i = 0,1, ... ,28 - 1.

223

PROOF. Substitute the expression (10.2.14) for the Haar- Fourier coefficients of f into the right side of (10.2.13) to obtain

(10.2.15) 'Pn(x) = 11 f(y)Kn(x, y) dy,

where

n-l

(10.2.16) Kn(x, y) = L hk(x)hk(Y) i=O

is the Haar kernel of order n. We shall isolate several properties of this kernel. Recall that ho(x) = 1 for 0 $ x < 1. Thus K 1 (x, y) = 1 for 0 $ x, y < 1. Notice by (10.1.2) that if k ~ 1 with k = 2m + £ for some 0 $ f $ 2m - 1 and

some m = 0,1, ... , then the function h}m)(x) = hk(X) satisfies

for x E ~~;+l), £ A (m+l) or x E uU+l ,

for x E [0,1) \ A}m+l).

Thus using the notation Q~i) = A~m) X ~~m}, we see that the function hl(X)hl(Y)

equals 1 on the squares Q~~) and Qg}, and equals -Ion the squares QW and Q~~ . Thus the kernel K2(X, y) = K 1(x, y) + hl(X )hl(Y) equals 2 for (x, y) E Q~~} U Qi;), and equals 0 for (x,y) E Q~;) UQW. In fact, it is easy to prove by induction that

K2m(X, y) = 2m on the squares Q~;") for i = 0, 1, ... , 2m - 1 and K2m(X, y) = 0 on

the squares Q~i} for i :f:. j, 0 $ i,j $ 2m - 1, m = 0,1, .... Suppose now that n = 2m + £ + 1 for some 0 :::; £ :::; 2m - 1 and m = 1,2, ....

Then by (10.2.16) we have

t

Kn(x,y) = K2m(X,y) + Lh~m>Cx)h~m}(y). k=O

Th f . h(m}( )h(m)() al 2m h Q(m+l} d Q(m+l} e unctIOn k x k y equ s on t e squares 2k,2k, an 2k+l,2k+l' I 2m th Q(m+l} d Q(m+l} d al 0 th .. equa s - on e squares 2k,2k+I' an 2k+I,2k, an equ s on e remammg

224 CHAPTER 10

f h Qo [0 1)2 S' h ~ Q(m+1) Q(m+1) parts 0 t e square 00 = , . 10ce t e lOur squares 2",2'" 2"+1,2"+1' Q(m+1) d Q(m+1) d"d h Q(m) . t ~ al' d . 2",2"+1' an 2"+1,2" lVl e t e square "," 10 0 lOur equ pleces, an smce

K 2",(x, y) = 2m for (x, y) E Q~m~, it follows that Kn{x, y) = 2m+1 on the squares (m+1) (m+1) , ( ) (m+1) (m+1)

Q2k,2" and Q2"+1,2"+1' and Kn x, y = 0 on the squares Q2",2"+1 and Q2"+1,2'" Consequently,

~ ( ) Q(m+1) UQ(m+1) lOr x, y E 2",2" 2"+1,2"+1'

for (x, y) E Q~j+1)

for i :f:. j, 0 ~ i,j ~ 21 + 1, and k = 0,1, .. . 1. Thus the kernel Kn(x, y) is determined by its values on a union U~,~!'~ Q~j+1) of

squares of rank m + 1. Moreover, since

t L him)(x)him)(y) = 0, (x,y) E Q~r;) "=0

for I < j < 2m - 1 or I < r < 2m - 1, we see that

(x,y) E Q~r;).

It is now easy to prove 10.2.7 using (10.2.15), (10.2.16) and the properties of the kernel Kn(x, y) just established .•

10.2.8. The inequality

(1O.2.17) wp{1/n, f) ~ n~ip t k(1/p)-1 Elp)(f),

"=1 n = 1,2, ... ,

holds for any function 1 E LP[O, 1), 1 ~ p < 00. Moreover,

PROOF. As we noted in §10.1, every Walsh polynomial of order 2m is also a Haar polynomial of order 2m , and conversely. Let S2'" (x, f) denote the partial sums of order 2m of the Walsh-Fourier series of the function I. From the definition (10.2.8) of the LP-modulus of continuity and Minkowski's inequality (10.2.11), we obtain

(10.2.18) wp(Tm, f) ~ wp(Tm ,1 - S2'" ) + wp{Tm , S2'" )

~ 21\1 - s2",l\p + wp(Tm, S2'" )

~ 4E~~ (f) + wp(2-m , S2 m ),

where S2 m = S2 m (I).

Notice by 10.2.7 and (10.2.6) that the partial sums of order 2m of a WalshFourier series are identical with the partial sums of order 2m of the corresponding Haar-Fourier series. Hence

2m_l

S2m(X) = L akhk(x), k=O

where ak = ak(l) are the Haar-Fourier coefficients (10.2.14) of the function f. Let o < h ~ 2-m , m = 1,2, .... Since ho(x) = 1 it follows that

m-l 2,+1_1

(10.2.19) II S2 m (x + h) - S2m(X)lIp ~ L II L ak(hk(x + h) - hk(x))llp· 8=0 k=2'

If k = 28 + £ for some £ = 0,1, ... ,28 - 1, and s = 0,1, ... , then the term hk(X + h) - hk(X) differs from zero on the set

A (h) = (~ - h ~) U (2£ + 1 _ h 2£ + 1) U (£ + 1 _ h £ + 1) k 29' 29 29+1 '29+1 29' 28 '

and is identically zero on the complement of the closure of the set Ak(h), where this complement is taken with respect to any interval of length 1 which contains the set Ak(h). Notice that the intervals which were used to define the set Ak(h) are non-overlapping since 0 < h ~ 2-m , 0 ~ s < m, m = 1,2, .... Moreover, it is obvious that if Ak(h) == A~s)(h) then A~8)(h) nA~12(h) = 0 for 0 ~ £ < 2m - 2 and Ihk(x + h) - hk(X)1 ~ 2J28 on the middle interval which defines the set Ak(h) but Ihk(x + h) - hk(x)1 ~ J28 on the other two intervals which define the set Ak(h). Using the inequality

1 ~ p < 00,

226 CHAPTER 10

we obtain

2'+'_1

II 2: ak(hk(x + h) - hk(x))IIp k=2'

~ 2'-'/p ([ 't,la,".,IP Ih,".,(x + h) - h,".,(x)IP dX) '/p

" 2'-'/p Ct,I",".,IP(h'i' + 2Ph'i' + h'i')h) '/p

Consequently, it follows from 10.2.6 and 10.2.5 that

2,+1_1

II 2: ak(hk(x + h) - hk(x))llp $ 2(1 + 2P- 1 )1/P2'/Pll82,+l(1) - 82' (!)IIph1/p

k=2' $ 4· 2'/P(II82,+l(1) - flip + 1182,(1) - fllp)h 1/ p

$ 16· 2'/Ph1/ p E~~)(!).

Using this inequality in conjunction with (10.2.19) we see that

m-1

1182m(x + h) - 82m (x)lIp $ 16h1/p 2: 2'/p E~~)(!) . • =0

Since h E (0,2-m ] was arbitrary, we see by the definition of the LP-modulus of continuity (10.2.8) that

m-1

wp(2- m, 82m) $ 16· 2-m/p 2: 2'/p EW(!)· .=0

Combining this estimate with (10.2.18) and the inequality E~Ij(!) $ E;P,!-, (I), we obtain

m-1

wp(T m ,!) $ 4(21/ P + 4)2-m/p 2: 2'/p E~~)(!). ,=0

APPROXIMA nONS BY WALSH AND HAAR POLYNOMIALS 227

Suppose that n ~ 2 is an integer. Write n in the form n = 2m + l, for some l = 0, 1, ... 2m-I, m = 1,2, .... Then since the sequence {E~p)(f)}~o is monotone decreasing, we see that

wp(l/n, f) :5 wp(2- m , f) m-1

:5 C;2-m / p I: 2·/p E~~)(f) .=0

n

:5 2C;Tm /p I: k(1/p)-1 E~p)(f) 1:=1

n

:5 21+1/PC;n-1/P I: k(1/p)-1 E~p)(f), 1:=1

where C~ = 4(21/ P + 4), i.e., we have proved (10.2.17) for the case n ~ 2. If n = 1 then by definition of the V-modulus of continuity and by (10.2.20) for

m = 1, we have

We notice since these norms are continuous as p ---t 00 that inequality (10.2.17) also holds for p = 00 with a constant of 96.

10.2.9. 1) If 0 < a < l/p:5 1 then the conditions E!!')(f) = O(n- cr ), as n ---t 00

and f E Lip (a,p) are equivalent; 2) If 0 < l/p < a :5 1 then the condition E!!')(f) = O(n-cr ), as n ---t 00, implies

that f E Lip (l/p,p); 3) If 0 < a = l/p:5 1 then the condition E!!')(f) = 0(n-1/P), as n ---t 00, implies

that wp(6,j) = 0(61/ P I1n61); 4) If the series

co

(10.2.21) I: k1/p- 1 E~p)(f) k=1

converges for some 1:5 p < 00, then f E Lip (l/p,p).

PROOF. Part 1) follows from 10.2.4 and 10.2.8. Indeed, if E!!')(f) = O(n-cr ), n ---t 00, where 0 < a :5 1, 1 :5 p < 00, then by 10.2.8 we have

(10.2.22) wp(l/n, f) = 0 (n -lIp t k(1/P)-cr-1) . 1:=1

228 CHAPTER 10

If 0 < a < lip then I:~=l k(1fp)-Of-l = O(n(1lp)-Of), n -t 00. Consequently, wp(lln,J) = O(n-Of), n -t 00, i.e., wp(lln,J) ~ Cn-a, n = 1,2, ... , where C > 0 is some constant. Let 6 be any number which belongs to the interval (0,1]. Choose a natural number n = n(8) which satisfies 1/(n + 1) < 6 ~ lin. Then

Thus f E Lip (a,p). The fact that any f E Lip (a,p) must satisfy E~p)(J) = O(n-a), as n -t 00,

follows directly from 10.2.4. Thus the proof of 1) is complete. If E!!)(J) = O(n-a), as n -t 00, for some 0 < lip < a ~ 1 then as we showed

above, (10.2.22) holds. Since a> lip we have I:~=I k(1lp)-a-1 = 0(1), so it follows from (10.2.22) that wp(lln, J) = O(n-I/p), n -t 00. As above, this estimate implies that f E Lip (llp,p) and part 2) is established.

Suppose now that E~p)(J) = O(n-Ilp), as n -t 00. Use (10.2.22) for a = lip. Since I:~= I 1 I k = O(ln n) we see that

n -t 00.

As we did in the proof of 1), we can use this relationship to verify that wp( 8, J) = 0(81IP lln81), as 6 -t 0+. This completes the proof of 3).

Finally, if the series (10.2.21) converges for some 1 ~ P < 00 then 10.2.8 implies that wp(lln,J) = O(n-Ilp), n -t 00. From this it follows easily that wp(8,J) = 0(81/p ), i.e., f E Lip (llp,p). I

A consequence of 4) is that if the series I:~=l E~I)(J) converges for some f in L[O, 1) then f satisfies the condition WI (8, J) = 0(6). Thus the function f is equivalent to a function of bounded variation on [0,1), i.e., it is possible to change the values of f on a set of Lebesgue measure zero to obtain a function of bounded variation on [0,1).

Implications 2) through 4) of 10.2.9 are exact in the following sense:

10.2.10. 1)!f0 < lip < a ~ 1 then the condition E!!)(J) = O(n-a), as n -t 00,

does not necessarily imply that wp( 8, J) = o( 81/p ) as 8 -t 0+; 2) The condition E~p)(J) = O(n-Ilp), as n -t 00, does not necessarily imply that

wp(8,J) = o(81/p lln81) as 8 -t 0+; 4) Convergence of the series (10.2.21) does not necessarily imply that wp( 8, J) =

o(81/p ) as 8 -t 0+.

PROOF. Consider the function fo(x) = WI (x). Clearly, E!!)(Jo) = 0 for n 2:: 2, 1 ~ p < 00, and thus E!!>Cfo) = O(n-a), n -t 00, for any a > 0, in particular, for

O! > l/p. But since Wl(X) = 1 for 0 ~ x < 1/2, Wl(X) = -1 for 1/2 ~ x < 1, and Wl(X + 1) = Wl(X), we have for any 0 < h < 1/2 that

{ -2

W1(X + h) - W1(X) = 0 for x E [1/2 - h, 1/2), for x fj. [1/2 - h, 1/2] n[O, 1].

Thus

(11 IWl(x + h) _ W1(X)IP dX) lip = 2hl /p •

Consequently, wp(li,/o) = 2P/p for 0 < Ii < 1/2, i.e., 10 satisfies the condition wp( Ii,/o) = O( lil7p) but fails to satisfy the condition wp( li,/o) = o( li1/p), as li ....... 0+. This completes the proof of 1).

Since E~p)(fo) = 0 for n ~ 2, it is obvious that the series 2::::"=1 n(l/p)-l E~)(fo) converges. Thus the function lo(x) = W1(X) provides an example which verifies 3).

To prove 2), consider the function

00

hex) = LTn/2 h2n(x). n=O

This series obviously converges pointwise on the interval (0,1). We shall show that it converges in LP[O, 1) norm for all p ~ 1. Indeed, since

00 00

L 2-n/2 I1h2 n lip = LTn/p < 00, 1 ~ p < 00,

n=O n=O

we see that hELP [0, 1) for p E [1,00). Moreover,

m-l

E~Ij(fd ~ 11/1 - L 2-n/2h2nllp n=O

00

= II L Tn/2 h2n lip n=m 00

n=m 00

= L 2-n / p = O(2- m / p ), m ....... 00.

n=m

If n = 2m + i!, i! = 0,1, ... 2m - 1, m = 0,1, ... , then

m ....... 00.

230 CHAPTER 10

Therefore, we have by 10.2.5 and the definition of the LP-modulus of continuity that

wp(2-m, 11) ;:::: wp(2-m, S2m (h» - wp(2- m ,11 - S2 m (h»

;:::: wp(2-m, s2m(h» - 211h - s2m(h )lIp ;:::: wp(2-m, s2m(h» + O(2-m/p), m -+ 00.

It remains to prove that

m -+ 00.

But this follows from the inequality

§10.3. Connections between best approximations and integrability conditions.

We search for conditions on the sequence {E~)(f)} of best approximations to a given function I E LP[O,I), 1 ~ p < 00, which will guarantee that the function f belongs to the space Lq[O, 1) for some q > p. We shall also derive estimates of best approximations in the Lq[O, 1) norm in this situation. Recall that the space Loo[O,l) consists of measurable functions which are essentially bounded on [0,1). A norm for this space is given by the eJJential Jupremum 01 the function I on the interval [0,1),

11/1100 = ess sup If(x)1 < 00, O~z<1

which equals the smallest constant C;:::: ° for which the inequality I/(x)1 ~ C holds almost everywhere on [0,1).

For each function IE L[O, 1), let Pn(x,f) == Pn(x), n = 1,2, ... , represent the partial sum of order n of the Haar-Fourier series of I. 10.3.1. Ifn = 2m + f. for some f. = 0,1, ... 2m -1, and some m = 0,1, ... , then

holds for all functions f E LP[O, 1), and alII ~ p < 00.

PROOF. This result has already been proved in the case n = 2m (see 10.2.2). For arbitrary n, the proof can be based on 10.2.7 exactly as 10.2.2 was based on (10.2.6). The details are left to the reader .•

10.3.2. IT f E LP[O, 1) for some 1 ::; p < 00 then

PROOF. In the case n = 2m this result reduces to 10.2.1, since 'P2 m (f) = 82m (f), where 82m (f) represents the partial sum of order 2m of the Walsh-Fourier series of the function f. For arbitrary fl., the proof can be based on 10.2.7 exactly as 10.2.1 was based on (10.2.6). The details are left to the reader .•

Proposition 10.3.2 (and 10.3.4 below) can also be obtained from the following facts. The sequence fPn(f)} is a martingale (see [20]), {IIPn(f)lIp} is increasing for any p ~ 1, and limn_oo lI1'n(f)lIp = Ilfllp since the Haar system is a basis in the space LP[O, 1).

10.3.3. IT f E LP[O, 1) for some 1 ::; p < 00 then

n = 1,2, ....

PROOF. In the case n = 2m this result reduces to 10.2.5. For arbitrary n, the proof can be based on 10.3.2 exactly as 10.2.5 was based on 10.2.1. •

10.3.4. The inequality

n+m n+m (10.3.1) II L akhkllp::; 211 L akhkllp

k=O

holds for any real numbers ak and any 1 ::; p ::; 00.

PROOF. Set f(x) = L:;~: akhk(x) and notice that the numbers ak are the Haar-Fourier coefficients of the function f. Thus 10.3.2 implies

n n+m II Lakhkllp::; II L akhkllp, 1::; p < 00.

k=O k=O

We continue this estimate by Minkowski's inequality (10.2.11) and obtain

n+m n+m n n+m n+m II L akhkllp ~ II L akhkllp -II Lakhkllp ~ II L akhkllp -II L akhkllp'

k=O k=n+l k=O k=n+l k=O

This proves inequality (10.3.1) in the case when p f. 00. But Ilflloo = limp_ oo IIfllp, so the case p = 00 can be obtained from (10.3.1) by taking the limit as p -+ 00 ••

232

10.3.5. If 1 ~ p < 00 and m = 0,1, ... , then

holds for any choice of real numbers an'

PROOF. From

and 10.2.6 we obtain

2m +'_1

II L anhnll p

CHAPTER 10

1 ~ p < 00,

,; 2'-' (1.'1 'i' a.h.Cx) I' dx + l' I :I: a.h.Cx) I' dX) 'I,

(

2m+l_1 2m+2_1 ) lip

= 21-; 2m(f- 1) n~m lanlP + 2(m+1)(f- l ) n=~+l lanlP

,; 2'-' v'2 . 2m (;-,) C~-' la.I' f' Since 2- l/p < 1 for any 1 ~ p < 00, the proof of the right hand inequality in (10.3.2) is established.

To prove the right hand inequality in (10.3.2), let

2m +1 _l

L'm(x) = L anhn. n=2m

By 10.3.4 we have

2 m +2 _l

(10.3.3) max {11L'm(x)lIp, II L'm+1 (x) lip} ~ 211 L anhnllp •

n=2m

Use Proposition 10.2.6 and then inequality (10.3.2). Thus verify

C~>nl}" ~ = (2m(1-;) 11 I Em(x) IP dx + 2(m+I)(1-~) 11 I Em+1(X) IP dX) IIp

2 m +2 _1

::; max{1,2;-!}. 2m(;-!). 21+;11 L anhnllp" n=2m

Since 1 ::; p < 00, this completes the proof of the left hand inequality of (10.3.2) .•

10.3.6. Suppose I E LP[O, 1) for some 1 ::; p < 00 and n = 2m + k for some k = 0,1, ... , 2m -1, m = 0,1, .... Then

lan(f)1 ::; 4· 2m(;-!)E~)(J)h'

where an(f) are the Haar-Fourier coeflicients of the function I.

PROOF. Indeed, by 10.3.4 and 10.3.3 we have

lan(f)1 IIhnil p = IIanhnil p = IIPn+I(f) - Pn(f) lip

::; IIPn(f) - Ilip + IIPn +1(f) - Ilip ::; 4E~)(fk

Since IIhnilp = 2m(;-~), the promised inequality is established .• We come now the main result of this section:

10.3.7. Let I E LP[O, 1), 1::; p < q ::; 00 and suppose the series

00

(10.3.4) L E~)(J)hn;-~-l n=l

converges. Then I E Lq[O, 1) and

E~q)(fh ::; 128 (E~)(f)hn;-! + f EkP)(f)h k!-!-l) k=n+1

(10.3.5)

for n = 1,2, ... , where E~)(fh is the best approximation in the LP norm to the function I by Haar polynomials of order no greater than n.

PROOF. Since I E LP[O,l), we have by 10.3.1 that the Haar-Fourier series of the function I converges in LP norm, in fact, III - Pn(f)IIp ~ ° as n ~ 00.

Consequently, for all n the inequality

00

I ~ Pn(f) + L(Pn2k+l(f) - Pn2k(f)) k=O

234 CHAPTER 10

holds, where equality is understood in the sense that the series on the right side converges to I in the LP[O, 1) norm.

Let n = 2m +r, r = 0,1, ... ,2m -1, m = 0,1, .... Then 2k+m :::; n2k, and n2k+l < 2Hm+2. Since 1 :::; p < q < 00, it follows from 10.3.5, 10.3.6, and 10.3.3 that

:::; 2v2· 2(Hm)(t-!) [4v2. 2(Hm)(;-t) II 1'n2H 1 (f) -1'n2~(f)lIpr/q x

x [max (1,2;-t). 2(Hm)(;-H. 4E~)~(f)hr-~

:::; 2v2 . 2(Hm)(;-!) [16v'2E~~)~(fh r/q [4v'2E~~)~ (f)h] l-~ :::; 64(n2k );-! E~~)~(f)h.

Let q --+ 00 and verify that this estimate also holds for 1 :::; p < q = 00. Thus we obtain by (10.3.6) and (10.3.7) that

E~q)(f)h :::; III -1'n(f)lIq 00

:::; L l11'n2~+1(f) -1'n2~(f)lIq k=O

00

:::; 64 L(n2k );-! E~j~(fk k=O

holds for any 1 :::; p < q :::; 00. Since

00 00

L(n2k );-! E~)~(f)h :::; n;-! E~)(f)h + 2 L ElP)(f)hk;-!-l, k=O k=n+l

we conclude by the previous inequality that (10.3.5) holds as promised .• The following result l is a corollary of the previous one.

1 This result is an analogue of a result of A.A. Konjuskov and S.B. Steckin for approximation by trigonometric polynomials.

10.3.S. Let f E LP[O, 1), 1 ~ P < q ~ 00 and suppose the series

00

(10.3.8) LEip)(f)k;-;-l k=l

converges. Then f E Lq[O, 1) and

(10.3.9) E~a)(f) ~ 128 (E~~)(f)2n(;-;) + f EiP)(f)k;-;-l) k=2"+1

for n = 1,2, ... , where E~)(f) is the best approximation in the LP norm to the function f by Walsh polynomials of order no greater than n.

PROOF. It is easy to see that the series (10.3.8) is equiconvergent with the series

00

L EW(f)2n(;-~) n=O

and thus with the series 00

LEW (f)h 2n(;- ~), n=O

since E~~)(f) = EW(fh for n = 0,1,.... But this last series is equiconvergent with the series (10.3.4). Consequently, if (10.3.8) converges then so does (10.3.4). Thus it follows from 10.3.7 that f E Lq[O, 1) and (10.3.5) holds. If we let n = 2m

and use the fact that the sequence Eip)(f)h is monotone, we obtain

E~'!2(f)h ~ 128 ( E~~(f)h2m(;-;) + s~ 2s(}-;) E~~)(fh) .

Since E~~(f)h = E~~(f), the proof of (10.3.9) is complete .• We notice for f E LP[O, 1), 1 ~ P < q < 00, that the inequalities

E~q)(f)h ~ C1(p, q) (E~)(fhn;-~ + f EiP)(fhk;-~-l) k=n+l

and

E~q)(f) ~ C1(p, q) (E~)(f)n;-; + f EiP)(f)k}-~-l) k=n+l

where the constants Ci(p, q) depend on P and q, will be derived by another method in the following section as corollaries of a more precise theorem.

A corollary of 10.3.7 deserves special mention:

236

10.3.9. Iff E ufo, 1), 1 ~ p < 00 and the series

00

2: nt-1 E~)(f)h n=l

converges, then f E Loo[O, 1) and

This result together with 10.1.2 imply

10.3.10. If a function f is continuous on [0,1] then

for all 1 ~ P < 00 and n = 1,2, ....

The following result is a special case of 10.3.8:

10.3.11. If a function f is continuous on [0,1] then

for all 1 ~ p < 00 and m = 0,1, ....

CHAPTER 10

§10.4. Connections between best approximations and integrability conditions (continued).

In this section we shall prove more precise results of the type illustrated in 10.3.7 and 10.3.8. Under the hypotheses 1 ~ p < q < 00, these results will contain Theorems 10.3.7 and 10.3.8 as corollaries, but the constants on the right sides of the corresponding inequalities will depend on p and q. For the proofs of these results we shall need several preliminary observations.

The modulus of continuity of a function f E LP[O, 1), 1 ~ P < 00 will be defined somewhat differently than in the previous sections, namely, we shall set

(10.4.1) w;(8,f)= sup r If(x+h)-f(x)JPdx , ( I-h ) lip

0~h~6 10

Notice that this LP-modulus of continuity depends only on the values f takes on the interval [0,1), in contrast to the LP-modulus of continuity given by (10.2.8) which was defined for periodic functions of period 1.

10.4.1. If f E U[O, 1), 1 :::; p < 00, then

(10.4.2)

for any 0 :::; a < b :::; 1.

PROOF. By changing variables, reversing the order of integration using Fubini's Theorem, and applying definition (10.4.1), we obtain

Ip = lb (lb If(x) - f(y)IP dX) dy

= lb (l~~Y If(u + y) - f(y)IP dU) dy

= l b-

a (l b-

U If(u + y) - f(y)IP dY) du

+ l~b (l~u If(u + y) - J(y)IP dY) du

= l b-

a (l b-

U If(u + y) - f(yW dY) du

+ lb-

a (l:v lf(y-V)-J(y)IPdY) dv

= l b-

a (l b-

U If(u + y) - f(y)IP dY) du

+ l b-

a (l b-

V If(t) - f(t + vW dt) dv

= 21b-

a (l b-

t If(t + y) - f(yW dY) dt

:::; 21b-

a Iw;(t, J)IP dt .•

10.4.2. Iff E LP[O, 1), 1 :::; p < 00, and

t+ h

<h(t) = it f(x)dx o :::; t :::; 1 - h, 0:::; h < 1,

238 CHAPTER 10

then

(10.4.3)

PROOF. First suppose that 1 < p < 00. By Holder's inequality (10.2.7) we have

{l-h t-h 1 (h Jo IJ(t) - 1/1h(t)IP dt ~ Jo I h Jo [J(t) - J(t + u)] du IP dt

~ h-P 11-

h (l h IJ(t) - J(t + u)IP dU) dt· h(l-;)p

1 t ( (l-h ) = h Jo Jo IJ(t) - J(t + u)IP dt du

1 {h ( t-u ) ~hJo Jo IJ(t)-J(t+u)IPdt du

~ X l h(W;(U,J))P du ~ (w;(h,J))P.

This verifies inequality (10.4.3) for 1 < p < 00. If p = 1 then the proof is similar, but Holder's inequality is not needed. I

10.4.3. If J E prO, 1), 1 ~ p < 00, and

l (k+l)/n tPh(t) = n J(u)du,

kin k/n~t~(k+1)/n, k=0, ... ,n-1,

then

(10.4.4) t- 1/ n

I~) == Jo ItPl/n(t) - tPn(t)IP dt ~ (w;(1/n,J)?

for n = 1,2, ....

PROOF. For the case n = 1, (10.4.4) is obvious since in this case the left side reduces to zero and the right side is evidently non-negative. Suppose now that

n 2:: 2 and 1 < p < 00. By Holder's inequality we obtain

n-21

(k+l)/n It+1 l n l(k+l)/n I~p) = nP L I f(u)du - f(u)du IP dt

k=O kin t kin

n-21(H1)/n jt+1/n it = nP L I f(u)du - f(u)du IP dt

k=O kin (k+l)ln kin

n-21

(H1)ln (it 1 )P ~nPL If(u+;;-)-f(u)ldu dt

k=O kin kin

n-21(k+l)ln (1(H1)ln 1 )P ~ nP L If(u + -) - f(u)1 du dt· n(;-l)p

k=O kin kin n

11-.1. 1 = n If(u+-)-f(uWdu~(w;(1/n,f))p,

o n

i.e., (10.4.4) holds for 1 < p < 00. If p = 1 then the proof is similar, but Holder's inequality is not needed .•

10.4.4. If f E LP[O, 1), 1 ~ p < 00, then

(10.4.5) ( 1 ) lip

J~p)== l lf(t)-vJn(tWdt ~4w;(~,f), n = 1,2, ....

PROOF. Using the inequality la + Wlp ~ lal 1/p + IWlp, 1 ~ p < 00, we see that

( 1-.1. )l/P ( 1 )l/P

J~p) ~ 1 n If(t) - vJn(tW dt + 1-;. If(t) - vJn(t)IP dt

Applying the triangle inequality (10.2.11) to the first term on the right side of this last inequality, we obtain

Estimate the first two terms on the right side by using (10.4.3) and (10.4.4):

.r,,') ,; 2w;(~,f) +n (L. 1 L. (f(t) - f(x))dx I' dt) 'I,

== 2w;(!, f) + I~p). n

240

But by Holder's inequality (10.2.7) and estimate (10.4.2), we obtain

This together with the preceding inequality imply (10.4.5) .•

10.4.5. If IE LP[O, 1), 1 ~ p < 00, then

(10.4.6) 0~8~1.

CHAPTER 10

PROOF. This inequality follows directly from the definition (10.4.1) of the modulus of continuity and the inequality Iial - Ibll ~ la - bl .• 10.4.6. Let IE LP[O, 1) and 1 ~ p < 00.

1) Ifp = 1 then for n = 1,2, ... we have

(10.4.7) w;(-!., I) ~ ~ (sup [1/(t)1 dt - inf [1/(t)1 dt) j n 17 EC[0,lj J E EC[O,lj J E

IEI9/n IEI~I/n

2) If 1 < p < 00 then for n = 2,3, ... we have

(10.4.8)

1 n l-

l/p (1 1) w;( -,1) ~ -7- sup I/(t)1 dt - sup I/(t)1 dt

n 1 EC[O,lj E E, C[O,lj\E E, IEI=I/n IEd=l/n

PROOF. Suppose first that p = 1. Inequality (10.4.7) is obvious for n = 1 since the right side is identically zero and the left side is non-negative. For n ~ 2 set

(10.4.9) l (k+I)/n 1jJ(t) == 1jJn(t) = n I/(u)1 du = q,

kin

k/n~t«k+l)/n, k=O,I, ... n-1.

It is clear that

11-.1. 1 w;(I/n,t/J) "2 n It/J(t+-)-t/J(t)I dt

o n n-21(k+l)/n 1

=L 1t/J(t+-)-t/J(t)ldt k=o kin n

1 n-2 1 ( ) = - L !ck+I - Ckl "2 - ~ax Ci - rpin Ci •

n n 0<.<n-1 O<.<n-l k=O - - - -

But since

~ax Ci = n sup r 1t/J(t)1 dt, O~.~n-l EC[O,l] 1E

IEI~l/n

and

min Ci = n inf r 1t/J(t)1 dt, 0~'~n-1 EC[O,I] 1E

IEI~l/n

it follows that

(10.4.10) wr(.!.,t/J)"2 sup r t/J(t)dt- inf r t/J(t)dt. n IEI9/n 1E IEI~I/n 1E

Consequently, we have by 10.4.4, 10.4.5, and (10.4.10) that

w; (.!., J) "2 w;(.!., III) n n

= sup r1-

h III(t + h)I-II(t)11 dt 0~h9/n 10

= sup r1-

h IIJ(t + h)l- t/J(t + h) + t/J(t + h) -1j;(t) + 1j;(t) -IJ(t)11 dt 0~h9/n 10

"2 sup t-h 11j;(t + h) - t/J(t) I dt - 2 t III(t)I-1j;(t)1 dt 0~h9/n 10 10

"2 sup r t/J(t)dt- inf r 1j;(t)dt-8w;(.!.,IJI) IEI9/n 1E IEI~l/n 1E n

241

242

= SUp ( [t/J(t) - II(t)1 + II(t)1l dt IEI9/nJE

- inf ( [t/J(t) -11(t)1 + II(t)l] dt - 8w;(~, III) IEI?l/n JE n

~ SUp { IJ(t)1 dt - inf ( II(t)1 dt IEI9/n JE IEI?,l/n JE

11 1 - 2 111(t)l- t/J(t)1 dt - 8w;(-, III)

o n

~ SUp ( IJ(t)1 dt - inf [11(t)1 dt - 16w~(~, IJI). IEI9/n JE IEI?l/n JE n

This proves (10.4.7).

CHAPTER 10

Suppose now that 1 < p < 00 and t/J is defined by (10.4.9). Let 0 :::; do :::; dl :::;

... :::; dn - l represent the sequence {Ck}, k = 0,1, ... , n - 1 arranged in increasing order. Then

Two cases are possible: a) dn- l = dn- 2 ; b) dn- l > dn- 2 • In case a), the function t/J achieves its maximum value on a set of points A C [0,1] of measure IAI ~ 2/n, and thus the right side of (10.4.8) is identically zero, i.e., we have derived the inequality:

(10.4.11) w;(l/n, t/J) ~ n l - l / p sup (f t/J(t) dt - sup ( t/J( t) dt) . EC[O,l] JE E,C[O,l]\E JE, IEI=l/n IEd=l/n

In case b), set B = {t : t E [0,1], t/J(t) = dn-d. Evidently,

(10.4.13) w;(l/n,t/J) ~ n-l/P(dn_l - dn- 2 )

= n l - l / p (l t/J(t) dt - ;dn- 2 ) == n l - l /p D,

APPROXIMA nONS BY WALSH AND HAAR POLYNOMIALS 243

where D ~ o. But

(10.4.13) D~ r 1/!(t)dt- inf r 1/!(t)dt=Dl JE E1C[O,1]\E JE1

IE11=1/n

holds for any set E C [0, 1J of Lebesgue measure lEI = l/n. Indeed, if IE nBI = 0 then Dl < 0 and thus D > Dl . If IE \ BI + IB \ EI = 0 then D = Dl . And if IEnBI > 0, IE\BI + IB\EI > 0 then D > Dl . Thus (10.4.13) holds as promised. In particular, we see by (10.4.12) that (10.4.11) also holds in case b).

We have shown that given any function 1/! of the form (10.4.9), inequality (10.4.11) holds. Consequently, proceeding as we did for the case p = 1, using the triangle inequality for LP norms and Holder's inequality together with (10.4.6), (10.4.11), and (10.4.5), we obtain

w;(1/n,J) ~ w;(1/n, III) = sup (r l - h Ilf(t + h)I-lf(t)IIP dt)1/P O~h9/n Jo

~ sup (t- h 111/!(t + h)1 _ 11/!(t)IIP dt) l/p O~h9/n Jo

_ 2 (11 111(t)I-I1/!(t)IIP dt) lip

~ n l -; sup (r 1/!(t) dt - sup r 1/!(t) dt) EC[O,l] JE E1C[O,I]\E JE1 IEI=l/n IEt/=l/n

- 8w;(1/n, III)

~ nl-} sup (r I/(t)1 dt - sup r I/(t)1 dt) EC[0,I] J E El C[O,I]\E J El IEI=I/n IE11=I/n

- 2n l -; sup r 11/(t)I-I1/!(t)11 dt - 8w;(1/n, III) AC[O,I] JA IAI=I/n

~ nl-;Fn - 2nl-} (1111/(t)'-'1/!(t)"P dt) lip (l/n)I-} - 8w;(1/n, If I)

~ n l -! Fn - 16w;(1/n, J).

We conclude that (10.4.8) holds .• We recall the definition of equidistributed functions. Let I( x) be defined and

measurable on the interval (0,1). For a fixed y denote the set of points x E (0,1)

244 CHAPTER \0

which satisfy f(x) > y by EU > y). The function m(y) = lEU> y)1 is called the distribution function of f. Two functions f and g, measurable on (0,1), are called equidistributed if they have the same distribution function. It is clear that if one of two equidistributed functions is integrable on (0,1), then the other one is also integrable, and their integrals are the same.

10.4.7. Given any function f, measurable on (0,1) there is a non-increasing function F on (0,1) which is equidistributed with f.

PROOF. It is obvious that the function m(y) = lEU> y)1 is non-increasing on (-00,00), continuous from the right, and satisfies m( -00) = 1, m( 00) = 0. If this function is strictly decreasing and continuous on ( -00,00) then its inverse function F(x) is also strictly decreasing and continuous on (0,1). It is also clear that F is equidistributed with f on (0,1).

Suppose now that the function x = m(y) is not continuous. Consider the curve I. : x = m(y), -00 < y < 00, and let Yo be one of its points of discontinuity. Associate with this curve I. and the point Yo a horizontal line segment consisting of all points (x, Yo) where m(yo +) < x ~ m(yo -). Since the function m(y) is continuous from the right, the point (m(yo), yo) = (m(yo+), Yo) must belong to the curve I.. Repeat this procedure for each point of discontinuity of the function x = m(y). Let €* denote the curve generated by combining I. with all these horizontal line segments. It is clear that each line x = Xo, for ° < Xo ~ 1, intersects the curve €* at at least one point. The ordinate of this point will be denoted by F( Xo, J) == F( xo). The function F(x) is uniquely determined for all ° < x ~ 1 except those x's which correspond to an interval where m(y) is constant. But such x's constitute no more than a countable set M. For each x E M let F(x) be any value which does not prevent F from being monotone. From the construction of F it is clear that IE(F> yo)1 = lEU> yo)1 = m(yo) for any Yo· Thus f and Fare equidistributed. Moreover, it is also clear that the function F is non-increasing2 • I

10.4.8. Suppose that ° ~ f(x) E L[O,IJ and that F(x, J) is a non-increasing rearrangement of f. If E C [O,IJ is any Lebesgue measurable set then

(10.4.14) f liE' JE f(x) dx ~ 0 F(x, J) dx.

PROOF. Let g(x) = f(x) for x E E and g(x) = ° for x E [0,1) \ E. Then g(x) ~ f(x) for x E [0,1) and thus F(x,g) ~ F(x,J). Consequently,

/ef(x)dx= /e9(X)dX= l 1F(x,g)dx= lEIF(X'9)dX~ lEIF(x,J)dX. I

2The function F is sometimes called a non-increasing rearrangement of /.

10.4.9. If 0 :::; f( X) E L[O, 1] and a E [0,1], then

fa=: sup r f(x)dx= rF(u,f)du, Me[O,I] 1M 10 IMI=a

La =: inf r f(x)dx = t F(u,f)du. Me[O,I] 1M 11-0

(10.4.15)

IMI=o

PROOF. The identities are easily verified if a = 0 or a = 1. Indeed, if a = 0 then the expressions all are identically zero, and if a = 1 then the expressions reduce to

11 f(x)dx = 11 F(u,f)du,

which surely holds since f and Fare equidistributed. Thus we may suppose that 0 < a < 1. Suppose further that there is a point Yo

such that m(yo) = a. Define a function </J by </J( x) = 0 if f( x) :::; Yo and </J( x) = f( x) if f(x) > Yo. Then it is clear that 0 :::; </J(x) :::; f(x) holds for any x E [0,1]. Moreover, F(u, </J) = F(u, f) for 0 :::; u < a and F(u, </J) = 0 for a yo}. It follows that

(10.4.16)

fa ~ sup r </J(x)dx ~ r </J(x)dx = 11 </J(x)dx IMI=01M 1Mo 0

= 11 F(u, </J) du = 10 F(u,</J)du = 10

F(u,f)du.

But (10.4.14) implies

(10.4.17) fa:::; 1° F(u,f)du.

Hence by (10.4.16) and (10.4.17) we obtain the first identity in (10.4.15) for the case a = m(yo).

Suppose now that a is a number for which a ¥- m(y) for all real y. This means that a belongs to an interval where m(y) jumps (along the X axis). Let Yo be the point where m(y) jumps. Set /3 = limy -+yo + m(y) = m(yo). It is clear that /3 < a:::; m(yo-) and

lEU> Yo)1 =: IBI = m(yo) = /3, 1

lEU> Yo - -)1 =: IAnl ~ a. n

Thus 1 I {x : Yo - - < f (x) :::; yo} I = IAn \ B I ~ ex - f3, n

246 CHAPTER IO

and consequently,

00

IDI ? a - (3, D = I{x : f(x) = yo}1 = n (An \ B). n=l

Choose a measurable set E C D such that lEI = a - {3. If Mo = B U E then we have

JOt? { f(x)dx = { f(x)dx+ ( f(x)dx JMo JB JE

= L f(x) dx + yo(a - (3) = L f(x) dx + i'JI Yo duo

But since m(yo) = {3 = IBI, we can repeat the proof of (10.4.16) to verify

(IBI Lf(X)dx? Jo F(u,f)du.

Consequently,

Thus it follows from (10.4.17) that the first identity in (10.4.15) holds in the case a =f. m(y), -00 < y < 00. This completes the proof of the first identity in (10.4.15). The second identity is proved in exactly the same way .•

In the previous proof we established the fact that the supremum of the numbers JM f(x) dx, as M ranges over the subsets of [0,1] which satisfy IMI = a, is attained by some set Mo of measure IMol = a.

10.4.10. Let 0 ~ f(x) E LlO, 1], a E [0,1/2] and Mo be a set which satisfies

sup {f(x)dx = { f(x)dx. {M:IMI=Ot} JM JMo

Then

sup {f(x)dx = 120t F(u,f)du. EC[O,lJ\Mo J E Ot

IEI=Ot

PROOF. This proof is exactly like the one for 10.4.8 .• By combining 10.4.9 and 10.4.10, we obtain the following result.

10.4.11. Let I E LP[O, 1], 1 ~ p < 00.

1) lip = 1 then

(10.4.18) 1 1 (11 In jl ) w;( -, f) ~ 17 F(x, III) dx - F(x, III) dx n 0 I-lin

for n = 1,2, .... 2) If! < p < 00 then

(10.4.19) 1 1-.1 ( tin j21n ) w;(-;;, f) ~ n 1 / 10 F(x, III) dx - lin F(x, III) dx

for n = 2,3, ....

10.4.12. Let I E LP[O, 1], 1 ~ p < 00.

1) lip = 1 then

(10.4.20)

for n = 1,2, .... 2) If 1 < p < 00 then

(10.4.21 )

for n = 2,3, ....

247

PROOF. 1) Let p = 1. Since the function F(x, III) is non-increasing, we have by (10.4.18) that

w;(;,f)~ l~n (F(;,I/D-F(l-;,I/I))

and (10.4.20) follows at once. 2) Suppose p > 1. Then we have by (10.4.19) that

1 n l -.1 (llln j21n ) w;(-,f)~-7P F(x,l/l)dx- F(x,l/l)dx, n 1 0 lin

and

1 (2n )1-.1 (11/(2n) j11n ) w;( -2 ' f) ~ 7 P F(x, III) dx - F(x, III) dx .

n 1 0 1/(2n)

248 CHAPTER 10

Combining these inequalities we obtain

1 1-1. ( 11/(2n) 12/ n ) 2w;( -, f) ~ n 7P 2 F(x, If I) dx - F(x, If I) dx n 1 0 l/n

> n1-} (F(1/2n,lfl) _ F(l/n, Ifl)) - 17 n n

_1. n P

= - (F(1/2n, If I) - F(1/n, Ifl)), 17

i.e., inequality (10.4.21) holds .• Inequality (10.4.21) remains true if p = 1. Indeed, for n ~ 2 we have by (10.4.20)

that

nwi(.!., f) ~ ! . 2nw;(..!:.-, f) n 2 2n

1 ( 1 1) ~ 34 F(2n' If I) - F(l- 2n' If I)

1 (1 1) ~ 34 F(2n' If I) - F(;;, IfD .

10.4.13. Let 1 ~ p < q < 00 and suppose

(10.4.22) n == fn~-2 [w;(~,f)]q < 00.

n=1

Then

(10.4.23) f E V[O, 1] and IIfllq ~ C(q)(lIflll + n 1/ q).

PROOF. Notice first that

11 11 lIn 1 1 IIflh= If(x)ldx= F(x,f)dx~ F(x,f)dx~-F(-,f), o 0 0 2 2

i.e.,

(10.4.24)

Next, use Holder's inequality

1 1 - + - = 1, p p'


to verify

(10.4.25)

(lIflll + ~2k/PW;(Tk,J)) q = (lIflll + ~2k(;-*)W;(2-k,J)2k/(2q») q

~ (lIflir + ~2k(:-!)[W;(2-k,fW) X

X (t2fq~)q-l k=l

~ C1(q)2n / 2 (lIflir + ~2k(:-!)[W;(2-k'fW) .

Use 10.4.12 to see

Adding and telescoping, we obtain

F(2- n - 1 , IfD - F(~, IfD ~ 34 E2k/PW;(T\ J) k=l

for n = 1,2, .... Hence it follows from (10.4.24) that

00

(10.4.26) FCTn - 1 , IfD ~ 211flll + 34 L 2k/pw;CTk, J). k=1

250 CHAPTER 10

Combining (10.4.15) and (10.4.26) we obtain

1 00 2- n +1 1 (F(x, Ifl))9 dx = ~ 1-n (F(x, Ifl))9 dx

00

$ LTn (F(2- n , Ifl))9 n=l

00 (n )9 $ 349 ~ 2-n Ilfllt + t; 2k/p w;(2-k, f)

$ C2 (q) ~ T n/2 (~2k(!-!)W;(2-k, f) + IIfll!)

$ C3 (q) (~2k(!-!)[W;(2-k'fW ~2-n/2 + IIfll!)

$ C4 (q) (~2k(!-!)[W;(2-k,jW + Ilfll!)

(

00 2· )

$ C5 (q) t; m=~l+1 m!-2[w;(~ ,f)]9 + IIfll!

$ C5(q) (f m:-2[w;(~,fW + IIfll!).

Hence we conclude by (10.4.22) that

IIfll9 = 11F1I9

,; C,(q) (lIflll + [~m:-'[W;(~,f)[·r) = C6(q)(lIflh + D 1/9). I

The next inequality concerning real numbers is a result of Hardy and Littlewood.

10.4.14. Ifl < a, p < 00 and an ;::: 0 for n = 1,2, ... , then

(10.4.27)

PROOF. Let ¢Yn = n-<> + (n + 1)-<> + .... Then ¢Yn $ C(a)nl-<>. Set So = 0 and

m m

L n-as~ = L(<pn - <Pn+t)S~ n=l n=l

m

= L <Pn(S~ - S~_l) - <Pm+1s{;. n=l

m

;:; L <Pn(s~ - S~_l) n=l

m

;:; C(a) L nl-a(S~ - S~_l) n=l

But s~ - S~_l ;:; C(p)s~-lan. Consequently,

m m

Ln-as~;:; C(a,p) Lnl-as~-lan n=l n=l

m

= C(a,p) L [(nan)n-:] [n-fr] s~-l, n=l

where ~ + ~ = 1. Applying Holder's inequality for sums p p

251

where an = (nan)n- a/p and f3n = s~-ln-a/pl, and using the fact that p/(p-1) = p, we obtain

Consequently,

Since 1 - ~ = ~ this inequality implies (10.4.27) .• p' p

252 CHAPTER 10

10.4.15. Let 1 :5 p < q < 00 and suppose I E LP[O, 1). Then

(10.4.28)

and

where n-l

E~')U) = inf 11/(x) - L akwk(x)ll, {ak} k=O

represents the best approximations to the function I, in L' norm, by Walsh polynomials of order n. These inequalities also hold for best approximations by Haar polynomials, namely when E~')U) is replaced by E';!\f)h.

PROOF. By 10.4.13 we have

(10.4.30)

for I E LP[O, 1), 1 :5 p < q < 00 and w;(o, f) is the modulus of continuity defined in (10.4.1). Since w;(o,f):5 w,(o,f), we have by 10.2.8 that

(10.4.31) w;(~, f) :5 n~~' t k;-l EkP)U). k=l

Thus it follows from (10.4.30) and the inequality 1I/IiI :5 1I/IIp that

Substitute estimate (10.4.27) into the right side for ak = k;-l EkP)U), k = 1,2, . ... We obtain

This verifies (10.4.28). Since by 10.2.8 inequality (10.4.31) remains true if we

replace E~>Cf) with E~p)Uh, a similar proof establishes the following inequality:

It remains to prove (10.4.29). For this let Sn(x) = I:Z:~ akwk(x) denote a Walsh polynomial which yields the best approximation to the function I in the LP norm,

i.e., E~)(f) = III - Snllp· It is obvious that

k = 1,2, ... ,

and Eip)U) 2': Eip)u - Sn) for k 2': n. Consequently, applying the inequality (10.4.28) to the function I - Sn we obtain

This verifies (10.4.29). A similar proof establishes the analogous statement for best approximations by Haar polynomials .•

We shall show for 1 :::; p < q < 00 and C(p, q) in place of 128, that Theorems 10.3.7 and 10.3.8 are corollaries of Theorem 10.4.15. To accomplish this we need the following lemma:

10.4.16. Let {an };:"=l be a sequence of numbers wbicb satisfy an 2': o and an 2': an+l for n = 1,2, .... Tben

254

lorn = 1,2, ... , 0' E (-00,00), and v ~ O.

PROOF. Since n2m+1 _1

CHAPTER 10

2:: kOl :5 (n2m )(n2m+1)01 :5 (201 +1 + 1)( n2m )1+01, 0' ~ 0,

and n2m+1 _1

2:: k Clt :5 (n2m)(n2m)0I :5 (201 + 1)(n2m)1+0I, 0' < 0, k=n2 m

we have n2m+l_1

(10.4.33) 2:: kOl :5 (2 01 + 1)( n2m )1+01

for m = 0,1, ... , n = 1,2, ... , and all real 0'. Moreover, it is clear that

(10.4.34) (n2m),8 :5 (1 + 2.8)k,8

for n2m- 1 < k :5 n2m, n, m = 1,2, ... , and all real {3. Since v ~ 1 and the sequence {ad is non-negative and non-increasing, it follows from (10.4.33) and (10.4.34) that

(t k'a~) 'I" ~ (t, n:J;:' k.a;) 'I"

$ (t, a:,. n:J;:' k') 'I" $ (1 + 2')'1" (t, (n2m)"'a:,. ) 'I"

00

m=O

( 00 (2m)(1+ 01 )/v n2m )

:5 (1 + 201 )1/V n(1+01)/van + J;. n n2m-1 k=n~l+1 ak

(

00 n2m )

:5 (1 + 201 )1/V n(1+01)/van + 2 J;. (1 + 2~-1) k=n~l+1 k~-1ak

:5 (1 + 201 )1/V (2 + 2(1+0I )/V) (n(1+ 01 )/Van + f k(1+ OI - V)/vak ) k=n+1

:5 (1 + 201 /V)(2 + 2(1+0I )/V) (n(1+ 01 )/van + f k(1+ OI - V)/1I ak) . k=n+1

Thus (10.4.32) holds .• If 1 ~ P < q < 00, then we have by (10.4.32) that

00

~ 9n;--; E~)U) + 8 L k;--;-l ElP)U). k=n+2

The analogous inequality for best approximations E't)Uh by Haar polynomials is also true. Thus if 128 is replaced by a constant c(p, q) then Theorems 10.3.7 and 10.3.8 are corollaries of Theorem 10.4.15.

Notice also that if

(p) q p ( k J._J. )

Ek Uh = 0 In(k + 1) , (p) q p (

k J._J. )

or Ek U)=O In(k+l)

for some 1 ~ p < q < 00 then by Theorems 10.3.7 and 10.3.8, the function f must belong to Lq[O, 1], since in this case the series on the right side of (10.3.5) and (10.3.9) obviously converges. Furthermore, in this case we have by Theorem 10.4.15 that

n -+ 00,

and a similar estimate for best approximations E';t)Uh by Haar polynomials.

§10.5. Best approximations by means of multiplicative and step functions.

As we saw in §2.3 and §2.8, series with respect to multiplicative orthonormal systems share some approximation properties with the trigonometric system and have some approximation properties quite different from the trigonometric system. In this section, we shall obtain one more property which does not have an analogue for the trigonometric system.

Let {Xn(x)}~=o, x E [0,1), be the multiplicative system defined in §1.5 relative to the sequence P = (Pl,P2, .. ' ,Pn,'.')' For each 1 ~ P ~ 00 and each natural number n, let E~p)U) denote the best approximation in the LP[O,l) norm to the function f E LP [0, 1) by polynomials in the system {X n (x)} ~=o of order no greater than n - 1, i.e.,

n-l

E~)(f) = inf IIf(x) - L CkXk(X) lip· {Ok} k=O

Let QnU, x) = l:~':~ (YkXk(x) be a polynomial which yields the best approximation to f in LP norm, i.e.,

256 CHAPTER 10

Such polynomials exist for each natural number n since IIf(x) - L~~~ CkXk(X)llp is a continuous, non-negative function on the collection of coefficients Ck.

Using the metric p*(x, t), we shall define an LP- modulus of continuity, ;:;(p)( 8, I), analogous to (2.5.1), and for 8 = l/mr, mr = Prmr-l, we shall denote it by w~p)(J), i.e.,

wf,!)(J) = sup Ilf(x EB h) - f(x)llp, O'!,h<l/m r

1 5: P 5: 00.

The following result is true (compare with 10.2.5).

10.5.1. If 1 5: P 5: 00 and f E LP[O, 1), then

E!!:!(I) 5: w~p)(J) ~ 2E!!:!(J).

PROOF. Write the partial sums of the Fourier series of a function f with respect to the system {Xn(x)}~=o in the form (2.8.3):

Sn(J, x) = 11 f(t)Dn(xet)dt.

Make the change of variables u = x e t, i.e., t = x e u, and use the fact that the integral is invariant with respect to translation by EB. Thus obtain

Sn(J, x) = 11 f(x e u)Dn(u) duo

Applying (1.5.21), we find that

(10.5.1) t/mr

Smr(J,X)=mrJo f(xeu)du.

Consider the case 1 ~ P < 00. Apply Minkowski's inequality to (10.5.1) (see A5.2.3) to obtain

E!!:!(J) 5: IIf(x) - Smr(J, x)llp

= (11 I f (x)-m r 11/mr f(xeu)duIP dXY/P

,; (l [m, tm'I/(X) - I(x e U)ldU], dX)'"

~ m, [l (llm'I/(X) - I(x e u)1 dU)' dXl'"

5: mr 11/mr (11 If(x) - f(x e u)IF dx yIP du

5: mr t/mr sup ({I If(x) _ f( x e u)IF dX) l/p du = w~p)(J). Jo O'!,u<l/mr Jo

If p = 00 then

E~OO)(f) < max If(x) - Sm (f, x)1 r - 0:::; x:::; 1 r

Thus the left hand inequality of 10.5.1 is established. To prove the right hand inequality, notice that

n(x ED h) = Xk(X)

for all 0 ::; h < l/mr and k = 0,1, ... , mr - 1. Consequently,

mr-l

Qmr(f,x) = L QkXk(X) = Qmr(f,xEDh). k=O

Thus for each 0 ::; h < l/m r we have

IIf(x) - f(x ED h)llp = IIf(x) - Qmr(f,x) + Qmr(f,x ED h) - f(x ED h) lip

Therefore, we find that

::; IIf(x) - Qmr(f,x)lIp + IIQmr(f,x ED h) - f(x ED h)lIp

= 2E~;(f).

w~p)(f) = sup Ilf(x) - f(x ED h)llp ::; 2E~:(f), O:::;h<l/m r

i.e., the proof of 10.5.1 is complete .•

257

We shall now consider approximation of a function defined on the positive real axis 0 ::; x < 00. When approximating functions on the whole axis, entire functions of exponential type are used (instead of polynomials). Moreover by the PaleyWiener Theorem (see (30), Vol. II., p. 272), there is a 1-1 correspondence between entire functions of exponential type and certain Fourier transforms of functions f(t) E L2(-00,00). Namely, the Fourier transform

of a function f(t) E L2( -00,00) has compact support if and only if the function f( t) can be extended to the complex plane to a function f( z) which is entire and satisfies If(z)1 = O(e(u+e)lzl), c > 0, as Izl-t 00.

258 CHAPTER 10

For multiplicative transformations on the positive, real axis [0,00), we see by 6.2.12 and 6.2.13 that an analogue for the collection of entire functions of exponential type is the class of step functions which are constant on intervals of a certain rank. Therefore, we shall study approximations by this class of functions.

We shall call a function gr(x), defined on [0,00), P- adic entire of order less than or equal to r, if it is constant on each interval of rank r:

f. = 0,1, ....

The class of all P-adic entire functions of order less than or equal to r will be denoted by Rn and we shall denote the best approximation to a function f(t) E LP(O,oo) in the LP(O, 00) norm by P-adic entire functions of order less than or equal to r by

£~:(J), i.e.,

£~:(J) = iEn! IIf - 9rIlV(o,oo), 9r "-r

where Iitfollv(o,oo) = (fooo Itfo(x)iP dx)l /p , P I: 00, and IitfoIlLOO(o,oo) = sup Itfo(x)l. xE[O,oo)

We shall also denote the P-adic LP-modulus of a function f( x) E V(O, 00) by

w~p)(J) = sup IIf(x El7 h) - f(x)lIv(o,oo). O~h<l/m.

The following result is an analogue of Theorem 10.5.1:

10.5.2. If 1 :0:; P :0:; 00 and fELl [0,00) n V[O, 00), then

PROOF. Consider the function

tIm. Jm.(J,x)=m r 10 f(xEl7u)du,

which is constant on each interval t5t (r) = [f./mr,(f.+ l)/m r ), f. = 0,1, .... By definition J m. (J, x) E Rr- Consequently,

£~:(J) = iEn! IIf - grIlLP(O,oo) 9r ""'r

:0:; IIf - Jm.(J, x)lb(o,oo) tIm.

= IIf - mr 10 f(x El7 u) dUlb(o,oo)

tIm. = IImr Jo [J( x) - f(x El7 u)] duIlLP(O,oo).

By repeating the proof of 10.5.1, we continue this estimate, obtaining

t lmr £!!:}U) ::; mr 10 Ilf(x) - f(x EB u)lllv(o,oo) du ::; w~p)U),

i.e., the left hand inequality of 10.5.2 is established. Suppose now that g;(x) = g;U,x) is a step function from Rr which satisfies

£!!:}(n = iEni Ilf - grIlV(o,oo) = Ilf - 9;IILP(0,oo). gr "-r

(Such a function in Rr exists and is unique.) Since 0 ::; h < l/m r implies g;(x) = g;(x EB h), we can write

IIf(x) - f(x EB h) IIv(o,oo) = IIf(x) - g;(x) + g;(x EB h) - f(x EB h) IILP(o,oo)

::; Ilf(x) - g;(x)IILP(o,oo) + IIg;(x EB h) - f(x EB h) IILP(O,oo).

Consequently,

sup Ilf(x) - f(x ffi h) IILP(o,oo) = w~p)U) ::; 2£!!:!U)· • O'5. h <l/mr

Chapter 11

APPLICATIONS OF MULTIPLICATIVE SERIES AND TRANSFORMS

TO DIGITAL INFORMATION PROCESSING

In the last decade interest has increased significantly in applications of the Walsh system and its generalizations, especially applications to digital information processing. This interest stems from a peculiarity of the Walsh functions, namely, that each one of them takes on only two values +1 and -1. A consequence of this peculiarity is that multiplication can be avoided when utilizing high speed computers for certain problems. In such cases a discrete Hadamard transform can be computed almost 10 times faster than the corresponding discrete Fourier transform. Moreover, with the discrete Hadamard transform (DHT) and the discrete transfonn with respect to multiplicative systems (DMT), one can perform parallel calculations obtaining output of simultaneous calculations on single instruction processors better than can be done using of the discrete Fourier transfonn (DFT). This allows one to carry out basic calculations using only addition thereby avoiding the more costly operation of multiplication.

In this chapter we shall discuss discrete multiplicative transforms and practical methods for computing them. In this chapter and the next, we shall restrict our attention to systems X(x, y) which are generated by symmetric sequences P = ( ... ,P_j,'" ,P-I ,PI,··· ,Pj,"')' i.e., sequences where P_j = pj for j = 1,2, ....

§ 11.1. Discrete multiplicative transforms. By the discretization of the multiplicative integral

(11.1.1) j(y) = 100 !(t)x(t,y)dt

we shall mean the procedure of approximating this integral using Riemann sums; specifically, we shall pass from the integral (11.1.1) to the sum

N-I

(11.1.2) j(y) = L !(tk)x(tk, y)D.tk' k=O

It is well-known (see, for example [6], p. 316), that if the spectrum

(11.1.3) cp(v) = 1: rjJ(t)exp(,;2;ivt)dt

260

APPLICATIONS TO DIGITAL INFORMATION PROCESSING 261

of a function <p( t) E L1 ( -00,00) has compact support, then discretization of the integral (11.1.3) leads to a "spreading" of peaks in the spectrum (v) and the appearance offalse peaks, but ifthe spectrum (v) does not have compact support, then discretization leads to a "superposition of frequencies". It turns out that these defects are lacking for discretization of the integral (11.1.1).

Before we state and prove some fundamental theorems we establish a number of preliminary results.

11.1.1. Let n be a natural number and Y E [O,mn). Then

m~1 (k ) {mn if [y] = 0, LmnCy) = t:o X mn'Y = 0 if 1:::; [y] :::; m n -1.

PROOF. If [y] = 0 then the result is obvious, since x(k/mn,y) = x(k/mn,O) = 1 (see 1.5.3 and 1.5.1). Suppose [y] ~ 1. Write the numbers k/mn < 1 and [y] E [1, m n ] in the form

k .e1 .e2 .en -=-+-+ ... +-, mn m1 m2 mn

0:::; .ej :::; Pj - 1, j = 1,2, ... , n,

and q

[y] = LY-jmj-1, o :::; Y_j :::; Pj - 1, j=1

where Y_q =I- 0 for some q = q(y) E [1,n]. Using definition (1.5.33) and the multiplicative property of the function X(x, y), we have

But since Y_q =I- 0, q :::; n, we have

P. -1 2·.e P. -1 ( 2·.e) Y • '"""' 1l"Z q Y _ q '"""' 7rZ q -~ exp = ~ exp -- = 0,

1.=0 Pq 1.=0 Pq

since this is the sum of all pq-th roots of unity raised to a fixed power Y_q which satisfies 1 :::; Y_q :::; Pq - 1. Consequently, Lmn (y) = 0 for 1 :::; [y] :::; mn - 1. •

262 CHAPTER 11

11.1.2. Let JI(x) and h(x) be absolutely integrable on [0,(0), and il(Y), i2(Y) represent their respective multiplicative transforms. Suppose that h (x) is bounded on [0,(0) and that its transform i2(Y) E Ll(O, (0). Then the multiplicative transform of the product f(x) = JI (x )h(x) is the p-adic convolution of the transforms il(Y) and i2(Y), i.e.,

i(y)=F[Jl'h](Y)= 1= il(V)i2(y8v)dv= 1= il(y8v)i2(V)dv.

PROOF. Since h(x) is bounded and fl(X) is absolutely integrable, it is clear that F[JI . h](Y) exists. By 6.2.2 and Fubini's Theorem, we find that

i(y) = 1= JI(x)h(x)x(x, y) dx

= 1= fl(X) 1= i2(V)x(x,v) dvx(x,y)dx

= 1= i2(V) 1= JI(X)X(·T,Y8 v )dxdv

= 1= i2(V)il(y8v)dv.

Using the change of variables v = Y 8 u, we conclude that

11.1.3. Let € be a natural number. Then

t/mt { lime Jo X(v,y)dv= ° for ° ~ y < me,

for me ~ y < 00.

PROOF. Since v E [0, lime] implies v < 1, we see by 1.5.3 and 1.5.1 that

x(v, y) = x(v, [yD·

Let n = [y] and write this munber in the form

= n = L n-kmk-l'

k=l

If n < me then n-k = ° for k ~ €. Thus we have by definition (1.5.38) that

(11.1.4) x(v,n) = Xn(v) = 1, n < mi, v E [0, lime].

APPLICATIONS TO DIGITAL INFORMATION PROCESSING

The first part of 11.1.1 follows directly from this identity. Suppose now that n ~ mi. Then

l v v

n = 2..:: n-kmk-l + 2..:: n-kmk-l = (n)( + 2..:: n-k m k-l,

k=1 k=(+1 k=f+l

where n_v :f. 0, i.e., n-v E [l,pv - 1], v ~ e + 1. We have

But X(n)Jv) = 1 for v E [0, 11m£) by (11.1.4). Thus

111m! 111m! v X(v,y)dv= IT Xn_kmk_,(v)dv

o 0 k=f+l

Since the functions v-I

'l/Jr,v(v) = IT Xn_kmk_'(v) k=l+l

are constant on the intervals 6r (v -1),0:S r :S mv-l -1, and mf

we have

tim! (mv_t/md- l pv-1 2' io xCv, y) dv = 2..:: 'l/Jr,v(6r (v -1)) 2..:: exp 7rm-v V v = O.

o r=O Vv=O PI'

263

(This last sum is the sum of all Pv-th roots of unity raised to the power n_I" where 1:Sn_I':Spv- 1) .•

264 CHAPTER II

11.1.4. Let nand r be natural numbers, N = mnmr and

D(u,() = 1~ X(x,u)dx.

Then the function

(11.1.5) 1 N-I (k )

<Pn,r(t) = - L D - et,mr , mn k=O mr

t E [0,00),

is bounded and its multiplicative transform In,r(Y) belongs to V (0, 00) and is givell by

, { mr for o ~ Y < limn, (11.1.6) <Pn,r(Y) = 0 [or limn ~ Y < 00.

PROOF. By (1.5.38) we have

D (~r et,mr) = {;r for o ~ (klmr) e t < limn

for 1/mr ~ (klmr) e t < 1.

Thus for each fixed t there are only a fixed number of terms in the sum (11.1. 5 ) which are different from zero. Thus the function <Pn,r(t) is bounded for 0 ~ t < 171"

and identically zero for t 2:: mr + 1 (because [( klmr) e t] > 1 for t 2:: m" + 1 and for 0 ~ k ~ N - 1).

We shall compute In,r(Y). First, by definition

1 N-I1°O (k ) = - L D -et,mr X(t,y)dt

mn k=O 0 mr

1 N-I1 -= - L mrx(t,y)dt.

mn k=O O~(k/mr)et<l/mr

Making the change of variables t = (k/mr) 8 v, 0 ~ v < l/mr we obtain

m N-l11/mr (k ) ¢>n,r(Y) = _r L X - 8 v,Y dv

mn k=O 0 mr

mnmr-l (k ) l 1/mr = mr L X -,Y X(v,y)dv

mn k=O mr 0

_ mr m n -l U+l)mr-l (k ) l1/mr - - L L X -,y X(v,y)dv mn j=O k=jm r mr 0

l1/mr mn-l mr-l (k )

= :r X(v,y)dv L X(j,y) L X -:;;;,y n 0 j=O k=O r

m l1/mr = _T Dmn(y)Lmr(Y) X(v,y)dv,

mn 0

where Lmr(Y) was defined in (11.1.1). Using 11.1.3 we find that

Applying 11.1.1 we obtain

J () _ { _1 Dmn(y)m r 'l'n,r Y - mn

o

for 0 ~ y < mr,

for mr ~ Y < 00.

for 0 ~ y < 1,

for 1 ~ y < 00.

265

Substituting the value of Dm n (y) into this expression we conclude that (11.1.6) holds as promised. In particular, integrability of ¢>n,r(Y) is obvious .•

We now come to the discrete integral (11.1.1).

11.1.5. Let f(t) E Ll(O,OO) be P-continuous on [0,00) and fey) have compact support, namely, suppose f(y) = 0 for y 2:: m r • Then for the knot discretization tk = k/mr, k = 0,1, ... , mrmn - 1, the step function

(11.1.7) {I m n m r -l (k) (k ) - - L f - X -,y

fey) = :n k=O mr mr for 0 ~ y < m r ,

for mr ~ y < 00

can be obtained from the transform f(y) by the formula

(11.1.8)

266

i.e., j(y) is an average of iCy) over the intervals DIL(n) f.1 = 0,1, ....

PROOF. Consider the function

fI(t) = <Pn,r(t)f(t),

CHAPTER 11

where <Pn,r(t) is defined by (11.1.5). Since <Pn,r(t) is bounded, it is clear that fl(t) E LI(O,oo) and il(Y) exists. We compute the function i1(Y):

Making the change of variables t = (k / m r ) 8 u and using properties of the function X(x, y), we obtain

(11.1.9)

mnmr-I11/mr (k ) (k ) il(Y) = mr L f -8u X -8u,y du

mn k=O 0 mr mr

m mnmr-I (k ) 1 1/ mr (k ) = _r L X -,y f -8u X(u,y)du.

mn k=O mr 0 mr

By hypothesis iCy) = 0 for y ~ m r. By 6.2.12, f(t) is constant on the intervals [v /m r , (v + 1 )/mr) and consequently, f(( k /mr ) 8 u) = f( k /mr) for 0 S; u < l/m r .

Hence it follows from 11.1.3 that

mnmr-I (k ) (k) 1 1/ mr il(y) = mr L X -,y f - X(u,y)du

mn k=O mr mr 0

{I mnmr-l (k) (k ) - L f - X -,y for OS;y<m r ,

= mn k=O mr mr

o for mr S; y < 00,

i.e., (see (11.1.7» j(y) = il(Y)' On the other hand, by 11.1.2 we have

il (y) = F[<Pn,r . fl = 100 i(y 8 v )In,r( v) dv.

We conclude by (11.1.6) and (11.1.7) that

for 0:::; y < mro

for m r :::; y < 00 .•

11.1.6. Let I(t) E Ll(O,OO) be P-continuous on [0,00). If fey) does not have compact support then the function fey), determined by the knot discretization tk = k/mr' k = 0,1, ... ,mrmn - 1, can be written in the form

J(y) ~ {

1 mnmr-l

f* (~J X (~r'Y) L for 0:::; y < m r , mnmr

k=O (11.1.10) mnmr-l 1

L I; (~J X (~r'y) for mr :::; y < 00 mnmr

k=O

where

( k ) t lmr (k ) 1* mr = mr Jo I mr 8 u dlt,

and

for [y) = I:~~~ y-vmv-l' Moreover, the function ley) also satisfies the formula (11.1.8).

PROOF. As in the proof of 11.1.5, we consider the function !t(t) = cPn,r(t)/(t), and show that (11.1.9) holds for il(Y)' We consider the cases 0 :::; y < mr and mr :::; y < 00 separately.

If 0:::; y < mr then by 11.1.3 we have X(u, y) = 1 for 0 :::; u < l/mr . Thus

t lmr I (~8 u) X(u,y)du = ~mr t lmr I (~8 u) du Jo mr mr Jo mr

= ~j* (~). mr mr

Substituting this identity into (11.1.9), we establish the first part of 11.1.6. If mr :::; y < 00 then we can write

q(y) r q(y)

[y] = L y-vmv-l = LY-vmv-l + L y-vmv-l = [y)* + [Y)r. v=l v=l

268 CHAPTER 11

Since [yJ* < mr, it follows from the multiplicative property of the function X(.'!:, y) that

x(u, y) = X(1l, [yJ* + [yJr) = X(u, [yJ*)x(u, [yJr) = X(u, [yJr)

for 0::; u < l/m r , i.e., for m r ::; y < 00 and [yJ = [y)* + [yJr the function X(x,y) is identical with

Consequently, for mr ::; y < 00 we have

t lmr (k) t lmr (k ) q(y) io J;;; 8u X(u,y)du = io J ~ 8u II Xy_"m v _l(ll)dll OrO r ",=r+l

= _1 J; (~). mr mr

The proof of (11.1.S) in this situation is similar to the proof of 11.1.5 .• A consequence of 6.2.4, (ILLS) and (11.1.10) is that

11.1.7. If J(t) E Ll(O,oo) is bounded andP-continllous on [0,00), then

(11.1.11)

Combining 6.2.6 and (11.1.11) we obtain the estimate

11.1.8. Let mq ::; y < m q+l for some q ;::: r, i.e.,

q(y)

[yJ = Ly-",m v -l, 1 ::; y_q < Pq - 1, q = q(y) ;::: r. ",=1

If J(t) E Ll(O, 00) is bounded and P- continuous on [0,00), then

Theorems 11.1.5 and 11.1.6 give a method for approximating the transform l(y) at any point y. However, in questions of spectral processing, of pattern recognition and in a number of other applications the inverse problem is also significant. By the inverse problem we mean the problem of recapturing the values of J(td from the values of i(yv). Since by (11.1.7), ley) = 0 for y ;::: m r , we see that the knot discretization y", of the function l(y) need only be chosen from the interval

0::; y < m r . Furthermore, it follows from (11.1.7) that ICy) is a step function on [0, mrl which is constant on the intervals ol(n) = [C/mn' (C + l)/mn). Partition the interval [0, m r ) into mn mr subintervals Ol( n) and choose some point from each subinterval, for example Yt = C/mn. Thus we may write

(11.1.12)

for C = 0,1, ... , mnmr - 1 We shall call the transform (11.1.12) the direct discrete multiplicative transform

(DDMT). To obtain the inverse transform of (11.1.12), multiply both sides of (11.1.12) by

x(C/mn, q/m r ) where q = 0,1, ... , mnmr - 1 and sum over C. Thus

But

1 m"mr-l mnmr-l (k) (k C) (C q) L L f - X -, - X -,-mn l=O k=O mr mr mn mn mr

1 m"mr-l (k) mnmr-l (C q k ) L f - L X -,-8-mn k=O mr [=0 mn mr mr

1 m"mr-l (k) mr-l (j+l)mn-l (C q k ) = - L f - L L X -, - 8-

mn k=O mr j=O i=jm n mn mr mr

1 m"mr-l (k) mr-l (. q k ) m,,-l (C q k ) = - L f - LX), - 8 - L X -, - 8 - .

mn k=O mr j=O mr mr (=0 mn mr mr

By 11.1.1

m,,-l (C q k) {m n , LX -,-8- = mn mr mr

(=0 0,

[ q k ] if -8- =0, mr mr

if [~8~] ~ 1. mr mr

270

Moreover,

i.e.,

Therefore,

q k 1 if 0< -8- <-,

mr mr mr

if ~ ~ ..!L 8 ~ < 1,

-8- -( q k) {mn m'r mr 0,

mr mr mr

if k = q,

if k =f. q.

m n mr -l _ ( e) (e q) 1 (q) L f - X -, - = -f - m"m r ·

(=0 mn mn mr mn mr

We conclude that

(11.1.13) ( q ) 1 mn mr -1 _ ( e) (e q) f - -- L f - X --

mr - mr (=0 mn m n ' mr

for q = 0,1, ... mnmr - 1.

CHAPTER 11

We shall call the transfonn (11.1.13) the inverse discrete m1titiplicative transform. (IDMT).

§11.2. Computation of the discrete multiplicative transform. We shall consider the question of efficiently computing the multiplicative trans

forms (11.1.12) and (11.1.13). By an arithmetic calculation we shall mean a complex multiplication followed by complex addition. Clearly, the values j(e/mn ),

e = 0,1, ... , mnmr - 1, can be computed using (m n m r )2 arithmetic calculations. However, analogous to the discrete Fourier transform case, it is possible to use what is called a fast algorithm to compute these values using fewer arithmetic calculations. The idea behind this comes from a special way of representing the quantities j(e/mn ).

11.2.1. Let j and v be integers with j E [0, mn - 1], v E [0, mr - IJ written in the form

n

j = Lj-/lffil'-h /,=1

r

V = L V-l' m l'_l,

1'=1


Then

(11.2.1)

( n) 1 pr- 1 () Pr-l-1 ( ) - .c- _ V-r _ V-r+l f mn = mn ~ X[l/mnl Pr L_ X[l/mnl PrPr-l '"

lI_ r -O JI_ r +l- 0

PROOF. Set k = jmr + v for 0 ~ j ~ mn - 1, 0 ~ v ~ mr - 1 and use the multiplicative property of the function X(x, y) to see that

272 CHAPTER 11

where

f (~J = f (11_1 + ... + lI_ r m r -I + ~:I + ... + j-nmn-I)mr ). •

For each fixed e = O,l, ... ,mnm r -1, the sums in (11.2.1) (beginning with the sum over j _ d take respectively PI. P2, ... Pn, PI. ... ,Pr arithmetic operations. Thus we have proved the following result:

11.2.2. The values j(e/mn), e = 0,1, ... ,mnmr -1 of the discrete multipiica.tivc transform (11.1.12) can be computed in

]{ = (PI + P2 + .. ·Pn + PI + ... + Pr)mnmr ~ (lnmn + In mr)mnmr

aritilmctic operations.

A similar result holds for the inverse discrete multiplicative transform (11.1.13). The transforms (11.1.12) and (11.1.13) are more conveniently written in matrix

form. To see this we introduce the notation

(11.2.2) k, e = 0,1, ... mnmr - 1,

XT = (XO,XI,'" ,Xmnmr-I),

yT = (Yo,YI, ... ,Ymnmr-d,

and W = (We,d, k, e = 0,1, ... mnmr - 1, where Wt,k = x(k/mr, e/mn). We shall call the vector Y the Jpectrum of the input vector X.

Using this notation we can write the transform (11.1.12) in the form

1 Y=-WX,

mn

and the inverse transform (11.1.13) in the form

X = ~W*X, mr

where W* = (W;'l)' W;,l = x(l!/mn, k/mr), q, I! = 0,1, ... mnmr - 1. Writing (11.1.12) in the form (11.2.1) means that the mnmr X mnmr matrix W

is a product of n + r sparse matrices W(v), each of order mnmr x mnm r . Namely,

W = w(n+r)HT(n+r-I) ... w(n+l)w(n) ... W(l),

where for 1 ::; j ::; n the matrices W(j) contain only Pi non-zero elements in each row (each a power of q = exp( -27ri/pj», and for n + 1 ::; j ::; n + r each row contains pj-n non-zero entries (each a power of q = exp( -27ri/Pj_n»'

We shall presently give a detailed account of the DMT (11.1.12) for the case when Pk = P ;::: 2 for all k = 1,2, ....

We begin by establishing the following result:


11.2.3. The identity

holds for any choice of k = 0, 1, .. . pr+n -1 and f = 0,1, .. . pr+n -l.

PROOF. This identity is obvious for k = 0 or f = o. Let

and

n+r k = L k_ vpV-l ,

1'=1

n+r f = L f_ vpv-l ,

1'=1

k-v = 0 for v ~ n + r + 1,

f-v = 0 for v ~ n + r + 1,

Lv == [pVk_l] (mod p), Lv == [pV~I] (mod p).

Furthermore, set

We have

~ = ~ k* v-I ~ k: r ~ -vP +~ v'

p 1'=1 1'=1 P

~ = ~ f* pV-l + ~ f: n ~ -II L....J v'

P 1'=1 v=I P

_f_ = ~ l v-I ~ lv n+r ~ -vP + ~ v'

P 1'=1 1'=1 P

(Lv=o for v=1,2, ... ),

_k_ = ~ 1. v-I ~ 1.1' n+r ~ -vP + ~ v'

P 1'=1 1'=1 P

(1.-1' = 0 for v = 1,2, ... ).

k* = [_k_] (mod p) = kr+v for v::; n, -v - prpv-l

k~v = 0 for v ~ n + 1,

(mod p) = { o[ k ] pr-v+l-l (mod p) = L(r-v+l)

for

for

273

v ~ r + 1,

1::;v::;r,

274 CHAPTER 11

i.e., n r k* n r k !..- = ~ k* v-I + ~ --.!!.. = ~ k_ v-I + ~ . -(r-v+l) .

r L..J -vp L..J v L..J (r+v)p L..J v P 1'=1 1'=1 P 1'=1 1'=1 P

Similarly,

£ r n £* r n £ _ = ~ £* v-I + ~ --.!!.. = ~ £ v-I + ~ -(n-Hl)

n L..J -vp ~ V ~ -(n+v)P ~ v ' P 1'=1 1'=1 P 11=1 1'=1 P

k n+r k- n+r k __ = ~ ~ = ~ -(n+r-II+I) n+r L....J v ~ v '

P 1'=1 P 11=1 P

n+r - n+r £ _£_ = ~ £11 = ~ -(n+r-lI+l). Pn+r ~plI ~ p"

1'=1 11=1

Using the definition (1.5.33) of the function X(x, y), we obtain

(11.2.3)

( k £) 27l'i ~(k* £* k*£*) X --;:,--;;- =exp-~ _II 1'+ II-V

P P P 1'=1

(11.2.4)

and

(11.2.5)

The sum on the right side of (11.2.5) differs from the one in (11.2.4) only in the order of summation. This can be seen by changing variables wi th v = n + r - Vi + 1. Similarly, changes of variables v = Vi - r in the second sum in (11.2.3) and v = r + 1- Vi in the third sum transform the right side of (11.2.3) into the right side of (11.2.4). This proves 11.2.3 .•

Using the notation (11.2.2) and applying 11.2.3, we see that when PI = P2 ... = P ~ 2, the transforms (11.2.12) and (11.1.13) can be written in the form:

£ = 0,1, ... ,pn +r - 1,


pn+r -1 Xk = lr L YtXt ( :+r)' k = 0,1, ... ,p,,+r - 1.

P £=0 P

Since nand r are arbitrary natural numbers, replacing n + r by n and transferring the factor in front of the second sum to the first one leads us to the formulae

(11.2.6) £=O,I, ... ,pn-l,

pn_1 Xk = L YtXi (:),

l=O p (11.2.7) k = 0,1, ... ,pn - 1.

We conclude that if W = (Xk (p~ )), £, k = 0,1, ... ,pn - 1 is the DDMT matrix,

then the IDMT matrix W* is the complex conjugate of W, i.e., W* = W. In particular, computing the DDMT is essentially the same as computing the IDMT and we shall confine our attention to one of these, namely the DDMT (11.2.6).

As we remarked above, the matrix W = (Xk (p~)), £, k = 0,1, ... ,pn -1 of the

transform (11.2.6) can be written as a product of n matrices

w(n)w(n-1) ... W(l)

of order pn X pn in which each row and each column contains exactly p entries different from zero, and each of these entries is a power of the quantity q = exp( - 27ri / p). Although in the general case W(j) =I- W(/I) for j =I- v, this representation of vTf as a product of sparse matrices is not unique. In fact, each different representation generates its own algorithm for computing the DMT. Following the work of Zukov [1], we shall show that the matrix lV can be represented as W = CBn, where C is a permutation matrix and B is a sparse block-type matrix. (Recall that a permutation matrix is a matrix whose rows have only one non- zero entry and that every non-zero entry is 1. Multiplication of a column vector by such a matrix results in a permutation of the coordinates of the vector.)

11.2.4. For n 2: 2 there exists a sparse matrix B of order p" x pn such that the

matrix W = (Xk (pin))' £, k = 0,1, ... ,pn - 1 of the discrete transform (11.2.6) can be written in the form W = CBn, where C is a permuta.tion ma.trix of order pn x pn.

PROOF. For £, k = 0,1, ... , pn - 1, denote the elements of the unknown matrix B by bi,k. Write e and k in the form

n n

I! '" I!* /1-1 I!* + '" I!* /1-2 I!* + {, = L...J {,/IP = {,1 P L...J {,/IP = {,1 P'Ye, /1=1 /1=2

276

n n-l

k = L k:p"-l = L k:p"-l + k~pn-l = fh + k~pn-l. 11=1 11=1

Here, k~ = k_II' £~ = £_11' and If = 13k = 0 for n = 1. Set

(11.2.8)

where

Since

( 11.2.9)

we have

-2rri q=exp--,

p {

I for v = 11, 1i",11 = 0 for v =111.

for n;::: 2,

for n = 1,

{ 0,

1i-y/,j3k = 1, if e~ =I k;_1 for some v = 2,3, ... ,n,

if e~ = k;_1 for all v = 2,3, ... ,n.

Therefore, for the case n 2:: 2,

'f n n* ",n k* II-I I t. = <:1 + L..JII=2 "II-IP ,

for all other pairs (e, k).

CHAPTER II

We shall prove by induction that the elements b~~) of the matrices Bm can be written in the form '

(11.2.10) {

n ",m ("k. 11 · lie· k. q L..J. =1 • "-m+. b(m) = }=m+1 j' j-m

e ,k "'" e" k· qL..J"=l II "

for m::; n - 1,

for m = n.

The proof of (11.2.10) begins with the case m = 2. Notice that B2 = B·B implies

p"-I

b~~l = L be,IIbll,k, £,k = 0,1, ... ,pR - 1. 11=1

Thus in the case n;::: 3 it follows from (11.2.8) and (11.2.9) that


n

V = L Vjpj-1. j=l

In the case n = 2 we have

p-1 p-1

b~~l = L L (bi,v;+pv; 6v; ,kr )qV; k;. v;=o v~=O

Since e and k are fixed, the product of Kronecker delta symbols above is different from zero only when vj = kj_1 for all j = 2,3, ... ,n. Thus we see that

(2) {L~:-~O (bfV*+"~ k~ pi-I) qv;k~ for n 2: 3, b - 1 , 1 W)=2 )-1

e,k - p-1 v* k* Lv;=o be,v;+k~pq 1 2 for n = 2.

Applying (11.2.8) and (11.2.9) again, we obtain

p-1

b(2) _ "" (b ) v; k~ i,k - ~ (*+p"~ rpi-2,v*+"~ k~ pi- 1 q 1 WJ=2 J 1 W;=2 }-l v;=o

p-1 _ "" (c ) frk~_I+v;k~ - ~ u"n i~pi-2 v*+ "n k* pj-l q . Wj =2 J 1 1 L.Jj =2 ) - 1

v;=o

Unless v; = e~, the symbol 6e;, IJr = O. Thus

() { rr~- 6l~ k~ qe~k~_IH;k~ b 2 _ )-3), )-2

e,k- l*k*+e*k* q 1 1 2 2

i.e., (11.2.10) holds for m = 2.

for n 2: 3,

for n = 2,

Suppose now that (11.2.10) holds for all m ::; n - 1. Thus

pn_1 p-1 p-1 n

b~mk+l) = "" b( vb(mk) = "" ... "" bill IT 611~ k~ qL:, "I k~_m+i . .(., L...t, v, ~ ~ I ) 1 )-rn

v=o v~=O "i=O j=m+l

Since this product of Kronecker delta symbols is different from zero only when vj = kj_m for all j = m + 1, ... , n, it is evident that

p-1 p-1 b(m+1) "" "" (b ) "m v~k* +. i,k == L..-t ... ~ t;'+pa,,B q6i=1 ' n-rn l

v~=O "i=O p-1 p-1

= L ... L 6a,{iqlrk~-m+L:,"lk~-m+i, v~=O v~=O

278 CHAPTER 11

where n m n-1

a = L£jpi-2, (3 = L vipi-1 + L kj_mpi-1 j=2 j=1 j=m+1

and the empty sum (i.e., when m + 1 = n) is interpreted to be zero. The factors involving the quantities vi will be different from zero only when vi = £j+1 for all j = 1,2, ... ,m. Consequently,

for m < n -1,

for m=n-1

for m < n -1,

for m = n -1,

which establishes (11.2.10). Recall that

( £ ) -27Ti ~ ( £ ) Wl,'" = X'" ---; = exp --~ k_1I ---; , p p 11=1 P II

and use the relationship

(pen) II - [pfnPIl ] (mod p) = L(n-lI+l)

to write

(11.2.11) "n k l "n ",or W(,k = qL..JV=1 -v -(n-v+l) = qL..JV=1 v n-v+'

for e, k = 0,1, ... ,pn - 1. Compaling these expressions with the identities

( n) "n ",or b - qL..J -1 v v i,k - 11- , £,k=0,1, ... ,pn-1

we see that the elements of the matrix Bn can be obtained from the elements of the matrix TV by a 'P-adic permutation of the entries of ea.ch row, i.e., we can write

where C = (Ct,k), £, k = 0,1, ... ,pn - 1, and

n

C( k = II {jl~ kO , • , J' n-J+l

j=1


In fact,

pn_l

Wl,k = L cl,vbS~l v=o p-l p-l

'" '" "n" k" ~ ... ~ Cl,vqLj=l Vj 'j

V~=O V;=O p-l p-l n

= '" ... '" II bl~ v" . q~7=1 v; kj . ~ ~ .' n-.+l ,,~=o v; =0 i=O

Since br v", ::j:. ° only when vn* -i+l = f.*,., it follows that I' n-I+l

"n "k" "n l"k" Wl,k = qLj=l V n_j+1 n-j+' = qL;=l ; n-;+l,

and the proof of (11.2.11) is complete .•

279

Writing the matrix IV of the transform (11.2.6) in the form W = C Bn has several advantages over the factorization W = w(n)w(n-I) ... W(I) in which W(j) ::j:. W(v):

a) The product of the matrix W with a vector X, i.e., the computation WX = C B n X, can be accomplished by n iterations of a single instruction calculation, namely, successive multiplications by the matrix B of the vectors Z v = B" X, V = 1,2, ... ,n - 1, followed by a P-adic permutation. This significantly simplifies the algorithm and programming necessary to perform this product.

b) Since each of the p rows of the matrix B:

kp ::; f. ::; (k + l)p - 1, k = 0,1, ... ,pn-l -1,

can be obtained by stretching out the matrix

-27l'i q=exp--,

p

ml == m (mod p),

( m) m, q * = q ,

o ::; ml ::; p - 1

by judiciously placing zero columns in it, then the computation can be executed in parallel by single instruction processors, the number of which is pj where j = 1,2, ... ,pn-l.

280 CHAPTER 11

c) Multiplication of the elements of B} by corresponding components of the vector X (or the vectors Zv) can be done in the complex basis (l,i) using complex multiplication by the quantity qV = exp( -2rril//p), 1/ = 0,1, ... ,p - 1, followed by addition. Each complex multiplication can be accomplished by four real multiplications and two additions. However, it is more appropriate to carry out these computations in the pseudo-complex basis (1, q, . .. ,q1'-2). In this case, we use the fact that 1 +q+ ... q1'-} = 0 or q1'-} = -(1 +q+ ... q1'-2) to verify the relationships

(a} + a2q + ... + a1' _2q1'-2)q = -a1' -2 + (ao - a1'-2)q + ... + (a1' -3 - a1'_2)q1'-2,

(a} + a2q + ... + a1'_2q1'-2)q2 = (a,,_2 - a1' -3) - a1' -3q + ... + (a1' -4 - a1' _3)q,,-2,

(a} + a2q + ... + ap _ 2qP-2)qP-}

= -a} (1 + q + ... q1'-2) + a2 + a3q + ... + a1' _2q1'-3

= (a2 - ad + (a3 - adq + ... + (a1' -2 - adqP-3 - a}q1'-2.

Observe that calculating coordinates in the pseudo-complex basis (1, q, . .. q,,-2) requires only the addition operation. Moreover, notice that if p is a composite number, say p = PIP2 = 2·3, then components of the basis which have different signs are considered different from one another (q3 = exp(-2rri. 3/6) = -1).

We examine this computation in the special case when p = 3, i.e., in the pseudocomplex basis (1, q), where q = exp( -2rri/3). Let

B} = (~ ! q\) 1 q2 q

represent the corresponding block which operates on the components of X or Zv, and represent the components of Zv in the pseudo-basis (l,q) by z} = a} ·1 + f3}q, Z2 = a2 ·1 + f32q, and Z3 = a3 . 1 + f33q. Using the relationships

zq = (a· 1 + f3q)q = aq + f3q2 = aq - 13(1 + q) = -13 + (a - (3)q,

zq2 = (a· 1 + f3q)q2 = aq2 + f3 q3 = 13 - a(l + q) = (13 - a) - aq,

we obtain

APPLICATIONS TO DIGiTAL INFORMATION PROCESSING 281

Consequently, all the pseudo-components of the vector BIZ in the basis (l,q) can be computed by using 16 additions followed by 16 multiplications and 12 additions in the basis (1, i).

In addition to eliminating some multiplications, we notice that using the pseudobasis (1, q, ... , qP-2) to compute the DMT allows us to simultaneously compute the discrete transform of the data in n steps from only pn(p - 1) real accounts. To do this, we need to consider each component of the vector X = (XO, Xl, .. . , Xpn_d T as a (p - 1 )-dimensional pseudo-vector

for v = 0,1, ... ,pn -1, in which the numbers xV), j = 0,1, ... ,p-2 are determined in a natural way from the original signal in pn(p-1) readings. In this way the time to compute the DMT can be reduced still further by a factor of p - 1.

§11.3. Applications of discrete multiplicative transforms to information compression.

Thanks to the growing availability of high speed computers, the discrete Fourier transform (DFT) and the discrete Walsh transform (DWT) have been used in a wide variety of applications for both theoretical and practical problems. The latest achievements in radio electronics, micro electronics, and scientific computing (see, for example [9]) have fostered applications of general discrete multiplicative transforms (11.1.12), (11.1.13) (DMT), in particular, the DFT and the DWT. This is especially true for digital information processing, including compression of information, and coding theory. Thus research into the properties of the DMT is both practical and timely.

Mathematically, information compression by means of discrete orthogonal transforms can be described as follows. Let T be a non-singular N x N matrix. A given vector X of dimension N is "transformed" to a vector Y = T X which is altered further by some method to produce a "smaller" vector Y. An approximation to the input vector X is reconstructed by X = T- I Y which results in a reconstruction error 8 = IIX - X II. We define the compression coefficient of the information to be T = IYI/IYI, where IVI is some measure of the "size" of the vector Y. We shall call that method which minimizes 8 for a given T the optimal method of compression of information. The final objective of information compression is data reduction.

In one method of information compression, which we shall call zone coding [9J, the vector Y is obtained from the vector Y by replacing some of the coordinates of the vector Y by zero. "Compression" of the vector Y is measured by the number of coordinates not replaced by zero (evidently, IVI = N).

For the DFT, the "transformed" vector Y takes on the form

1 N-I {27rikn} Yn = N L Xk exp -~ ,

k=O

n = 0,1, ... , N - 1.

282 CHAPTER II

For the class of vectors

Efimov [3] obtained an estimate

(11.3.1 ) IYnl :::; 1)./ sin 7:, n = 1,2, ... ,N - 1,

which was improved by him in [5] for integers n = N/2 and n = N/4. Based on this estimate, optimal methods of zone coding by means of the DFT involve replacing the central coordinates of the vector yT = (Yo, YI, ... ,YN -t) by zero. Earlier [4], he also estimated the coordinat.e restoration error 6 = Ix k - oX k I for this method of zone coding.

Similar results for the DMT are contained in the work of Kanygin P]' where an analogue of (11.3.1) was obtained. More detailed investigations of the DMT in the uniform and integral norms and for various restrictions on the class of input vectors were conducted by Bespalov in [3], some of whose results will be given below.

We shall restrict ourselves to the case Pk = P for k = 1,2, ... and consider zone coding by means ofthe discrete transforms (11.2.6) and (11.2.7). The DMT (11.2.6) takes a vector X T = (Xo, xl, ... ,Xpn_I) to a vector yT = (Yo, YI,'" ,Ypn -1) by the formula

i.e.,

(11.3.2) e = 0,1, ... ,pn - 1.

The inverse tr.msform

(11.3.3) k=O,I, ... ,pn-1,

takes the vector Y to the vector X, i.e., X = H!-Iy. Let Y be a vector obtained from Y by replacing k of its coordinates by zero.

Then the compression coefficient T equals pn / (pn - k), and the reconstruction error is 6 = IIX - XII, where JY = W-IY.

For a given compression coefficient, the reconstruction error of a vector X depends on the method of zone coding and on the norm chosen to measure the error. The strategy is to replace by zero those coordinates of the vector Y which are sma.ll in absolute value, because these coordinates make smaller contributions to the reconstructed input vector. At this point it is important to recall (see §11.2) that


there is a program for computing the DMT which uses only the internal structure of the transformation W. Namely, the matrix lV is divided into single-type blocks, which lend themselves readily to parallel computing. Therefore, it is appropriate to replace by zero coordinates of the vector Y which correspond to an entire block ( or several blocks) of the matrix W. From the calculations below it will be evident that this strategy brings about the highest reconstruction accuracy. We shall also consider the case when the last coordinates Yt are replaced by zero, which is very convenient for calculation on a computer.

In connection with the preceding remarks we introduce the following notation: we shall call the vector

pa pn_p .s+l

T ~ ~ Ys =(O, ... ,O,Yp·,Yp·+l, ... ,Yp·+l _I,O, ... ,O)

the s-th packet of the vector Y (in the sequel we usually omit the superscript T ); for j = O,I, ... ,p - 1, we shall call the vectors

1j~ = (0, ... ,0, Yjp', Yjp·+I,· .. , Y(j+l)p' _1,0, ... ,0)

the j-th subpacket of the s-th packet of the vector Y. We shall let V(j) = W- 1 Yj,n-l represent the vector of dimension pn reconstructed from the j-th subpacket of the last packet of the vector Y, and let

( p-l )

X(j) = W- l Y - LYv ,n-l V=J

represent the vector reconstructed from the vector (Yo, YI, ... , Yjpn-l_l, 0, ... ,0) (during the reconstruction which does not replace the first j pTI -1 coordinates of Y by zero).

Also we will use the notation

(11.3.4)

p-l (v) _ ~ ~ -

zJ1 - L...J x J1P+rXvpn-l p r=O

where J-l = 0,1, ... ,pn-l and 0 :-s: v :-s: p - 1.

11.3.1. For k = 0,1, ... ,pn - 1, the coordinates vii) of the vector V(j) satisfy

284 CHAPTER II

where p,p ::; k ::; (p, + l)p - 1 and z~j) are defined by (11.3.4).

PROOF. Substitute the formula (11.3.2) for Ye into the expression

to verify

. 1 U+I)pn-l_1 (pn_1 _ (e)) ( k ) v(J) = - '"' '"' x v - X - . k n ~ ~ rAr n f n

p f=jpn-l r=O p P

Using 11.2.3 and changing the order of summation, we obtain

But

{

n-l P ,

0,

k T 1 if 0 < -8- <--- pn pn pn-I'

1 k r if -- < - 8 - < 1. pn-I - pn pn

Since k E [p,p, (p, + l)p - 1], the inequality ~ 8 ~ < ~l holds only for those pn pn pn

T E [p,p, (p, + 1)p - 1 J. Consequently,

We notice for Il = I:~;;;:: ILvpv-1 that Il-n = 0 and

X n-1 (_Il_) = exp 27ri 1 . (_Il_) = exp 27rill-n = 1. P pn-l p pn-l n P

Therefore,

From the definition of V(j) and X(j) it follows that

X(j) = V(O) + V O) + ... + V(j-l).

Consequently, the following result is a corollary of 11.3.1.

11.3.2. The coordinates of the vector X(j) satisfy

where IlP $ k $ (Il + l)p - 1 and kl = k - IIp.

285

•

Notation (11.3.4) and this last relationship will be used for the case s = n - 1. We shall introduce further notation to be used in the cases s $ n - l.

Write an arbitrary k, 0 $ k $ pn - 1, in the form

(11.3.5) k = k(ll, r, 1]) = Ilpn-. + rpn-.-l + 1],

where 0 $ Il < p', 0 $ r $ p - 1, and 0 $ 1] $ pn-s-l - 1. As we saw in (11.2.2), the vector X can be considered as values of a step function J(t) which takes the value Xk on the k-th interval bk(n) of rank n, and moreover, that the tk's can be distributed on the interval [0,1] across even subintervals. The coordinates of the vector X can be divided into collections of pn-.-l elements whose coordinates lie in the packets J.+ 1 (k) of rank s + 1. We shall denote the arithmetic means of these collections of coordinates by

(11.3.6) 1 p"-'-'_1

a(ll, r) = pn-.-l L Xk(Jl,r,1/)' 1/=0

calling them the mean values (of the vector X) over packets of rank s + 1. Thus

1 p-l

a(ll) = - L a(ll, r) P r=O

286 CHAPTER II

is the mean value over the fl-th packet Js(fl) of s-th rank. Similar to (11.3.4), we shall also use the notation

(11.3.7)

Since

1'-1 () LXjp.-l rs = 0, r=O p

it is easy to see by Abel's transformation that the right side of (11.3.7) can be written in the form

i.e.,

(11.3.8)

The following result, which we call the localization principle, is a simple conscqucnce offormula (11.3.8).

11.3.3. For all 1 ~ j ~ p - 1 the values of the quantity A( fl, j, s) depend only on tile '~umps" of the mean values over packets of rank s + 1, equal to D.a(fl,7°) = a(ll, r) - a(fl' r + 1) for indices 0 ~ r ~ p- 2, and depend neitl]er on the distribution of the individual coordinates x k nor on the mean values a(,l) over packets of rank s.

PROOF. Indeed, if for some fl one adds € to all quantities a(ll, r), where 0 ~ r ~ p - 1, then a(fl) will be changed by € but the quantities D.a(ll, r) do not change for alII ~ r ~ p- 2 and thus neither does A(fl,j,S) .•

Let 3,6. be the class of vectors a = (ao, al,' .. ,ap-l) with real coordinates which satisfy the condition

11.3.4. For any p ~ 2 and 1 ~ v ~ p - 1 we have

m'lx IR(a, v)1 ~ p/ sin 1W. aE=-l p

PROOF. By Abel's transformation,

p-I k-I p-I

R(a, v) = L(ak-I - ak) Lw"S + ap_1 Lw"' k=1 .=0 s=O

p-I

= _1_ "(ak-I _ ak)(1 _ W" k). I-w" L

k=1

Estimating this expression, we find for any a E 3 1 that

p-I

IR(a,v)l< 1 "II-w"kl< . 2p = p . I - II-will ~ - 2sm(7rv)/p sin(7rv)/p

287

Now we shall estimate the spectrum and reconstruction error which these methods of zone coding generate. We will use the following notation: if a = (ao, al , ... , aN -I) is an N dimensional vector then

1 ~ q < 00,

and Ilall(oo) = max lakl.

O~k~N-I

11.3.5. For packets y., s = 1,2, ... ,n - 1, and subpackets Yj,s, j = 1,2, ... , p - 1, of the spectrum of a pn dimensional vector XT = (xo, XI, .. " xpn_d E 3Ll, the following estimates hold:

(11.3.9) 2 ~ q ~ 00,

(11.3.10)

PROOF. It suffices to establish estimate (11.3.9) for the case q = 2 because for 2 < q ~ 00 the Cq norm of a vector is less than or equal to its £2 norm. Let o ~ v < p' and s ~ n - 1. Consider the quantity

pn_1 1" _ (jp. + v) YjP'+1I = ---;:;- L XkXk n '

p k=O P

288

which by 11.2.3 can be written in the form

1 pn_l ( k ) Yjp'+v = pn L XkXjp'+v pn .

k=O

Using notation (11.3.5), we have

CHAPTER 11

Since jp' + v < p.+l, IJ/pn < pn-.-l /pn = l/p·+l, we have Xjp'+v(17/pn) = 1, and since v < p', r/p·+l < l/p', we have Xv(r/p·+l) = 1. Moreover, for fl < p' we have

and thus ( / .) -27ri.( / .) Xjp' Ii P = exp --) Jl P .+1 = 1.

p

Hence, it follows from the notation (11.3.6) and (11.3.7) that

p'-1 () p-l 1 _ fl _ l' n-.-l Yjp'+v = -; L Xjp'+v -; LXjp' (~) p a(jJ., r)

p 1'=0 P r=O p

p'-1

= '~1 LXv (~) pA(fl,j,s + 1) P 1'=0 P

p'-1

= ~ L A(II,j,S + l)Xv (I:) p 1'=0 P

I.e.,

p'-1

Yjp'+v = ~ L A(fl,j,s + l)Xv (fl.). p 1'=0 P

(11.3.11)

Using properties of the function Xv (x), we see that

IYjp'+vl 2 = Yjp'+vYjp'+v p'-1 p'-l

= !. L A(/l,j,s + l)Xv (fl.) . L A(R,j,s + l)Xv (R.) p ,,=0 p (=0 P

1 p'-l p'-l (k) = 2s L IA(/t,j,s + 1W + LXv -; B(k,j,s + 1),

p 1'=0 k=l P

where the quantities B( k, j, s + 1) can be expressed by the A(ll, j, S + 1) 's and their complex conjugates. Using this representation and the identity

p'-l

LXv(k.)=o, v=o p

1 :::; k :::; p' - 1,

we have for 2 :::; q :::; 00 that

[ . ]1/2 P -1

< ~ p;s ~'§:: IA(/-l,j,s + 1)12p' v-o 0$.I'$.p'-1

= max IA(/-l,j,s + 1)1. XE::::~

O$.I'$.P' -1

Since (11.3.6) implies

it follows from the restriction X E 3 A that

(11.3.13) Ill. a(/t, r)1

pn-s-l_ 1

:::; pn!'-l L IX/lpn-'+rpn-.-l+'1 - X/lpn-·+(r+1)pn-.-l+1/1 '1=0

< __ 1_ pn-~_l n-.-1ll. = n-.-1ll.. - pn-.+1 L.." p P

1/=0

Suppose Xjp.-l(V/P·) = w jv , where w = exp( -27ri/p). Notice that

290 CHAPTER 11

for 0 ::; r ::; P - 2. Thus it follows from 11.3.4 and (11.3.13) that

max XES.,.

O::;I'::;p'-1

p-2 r

IA(II,j,S + 1)1 = m1!.x ~ I L~a(ll,r) Lw jll I XE:::.,. p

O::;I'::;p'-1 r=O ,,=0

p-2 r

= m~x ! I L~a(p"r) Lw j " I 6a(l',r)E='pn_._I.,. p r=O ,,=0

_max ~IR(a,j)l::; ~pn-s-l~. p. aE:::pn-,_I.,. P P sm(7rJ/p)

Putting this estimate into (11.3.12), we obtain (11.3.9):

To obtain estimate (11.3.10), consider the vector X' whose coordinates J:i: arc obtained from the coordinates Xk of the vector X in the following way:

Xk = xk(/t,r,'l) = a(ll, r) - a(ll)·

Clearly, we have packets of 1'n-s-l identical coordinates arranged in order. Let Y* = 1'-nlVX·. Using (11.3.11) and the localization principle 11.3.3, we conclude that Y' = Ys '

Use Parscval's identity IIflll; = IIfll12 and take into account the form of the vector Ys and the fact that its preimage is the vector X'. We have (IIXII;. =

2

-n ,",pn_1 I 12) l' L..Jk=o :Z:k

(11.3.14)

pn_l

IlYslit = IIX*II~; = 1'-n L IXkl2 k=O

p'-lp-l

= S~1 L L la(/l, r) - a(IlW· l' 1'=0 r=O

Fix ll, r, nand s and temporarily set


for v = 0, ... ,p - 1 and 17 = 0,1, ... ,pn-s-l. Applying (11.3.6) we find

la(p,r) - a(p)1

1

= n!s-l pni-I (x(r, 17) - ~ ~ xCv, 17))

p 1]=0 p v=O

1 =---x pn-s-l

Since X E 3.6. we have

la(p,r) - a(p)1

n-8-1 1 ( 1) :::; __ 1_ p '"' - rpn-s-l,6. + ~ ~ vpn-s-I,6.

n-s-l L-t ~ p 1]=0 P v=O

n-$-l 1

= P ~ - (1' + P(P2; 1)),6. = (1' + p; 1) pn-s-I,6..

Putting this estimate into (11.3.14), we conclude that

IlYsllt:::; S~l Pi ~ (1' + p; 1)2 p2(n-s-I),6.2 p /1=0 r=O

= ~p2(n-s-1),6.2 ~ (1'2 _ rep _ 1) + (p ~ 1?) p r=O

= 2(n-s-I),6.2~ (P(P-1)(2P-1) +p(p-1? +P(P-1?) p p 6 2 4

= p2(n-s-l),6.2 (p - 1)(13p - 11) = p2(n-s-I),6.2 (Cp _ 1)2 + p2 - 1) 12 12 '

i.e., (11.3.10) is true .•

291

Inequalities (11.3.9) and (11.3.10) show that the smallest contribution to the inverse transform (11.3.3) comes from the last packet (8 = n - 1) of the spectrum, and the smallest contributing subpacket is the one whose index j is nearest to [p/2]. We shall now estimate the maximal contribution this j-th subpacket makes to reconstructing the input vector by using formula (11.7.3), i.e., we shall estimate the quantity V(j) = W- 1 Yj,n-I.

292 CHAPTER 11

11.3.6. For a111 :5 j :5 P - 1 and 1 :5 q :5 00, the following inequality holds

(11.3.15) max IIV(i)llt. < . 6.. . XES", • - sm(7rJ/p)

PROOF. Let 0:5 JL:5 pn - 1 _1 and JLp:5 k :5 (JI+1)p-1. Using the representation of the vector V(j) given in 11.3.1 and the norm

we have

Combining representation (11.3.4) with estimate 11.3.4, we obtain

p-l ( ) mflx IIV(i)lIt. = max ml!.X ~ I LX/,p+rXjpn-, .!....

XE::.", • O$/'$pn-I_l XE::.", p r=O pn

< ~ p6. _ 6. - psin(7rj/p) - sin(7rj/p)" I

We shall now estimate the error induced by replacing the vector X with the vector X(i), which has been reconstructed from the vector (yO, YI, ... Yjpn-I_l, 0, ... ,0).

11.3.7. The estimate

(11.3.16) p-j

ml!.X IIX - X(i) lit. :5 6.". 1 XE::.", • ~ sm(7rv/p)

1'=1

holds for any 1 ::; j ::; p - 1 and 1 ::; q ::; 00.

PROOF. By the definitions, we have

X - X(i) = W- 1 Y -W- 1 (Y -~ Yv,n-l) 1'=;

p-l p-l

- "W-1y' - "V(v) - L...t v,n-l - L...t . v=j v=j

Applying 11.3.6 for 1 :s: q :s: 00, we have

p-l

m~x IIX - X(j) lie· :s: ~ m~x IW(v) lie· XE.::", q 6 XE.::", q

v=)

p-l 1 p-j 1

<~~.( /)=~~.( /)"1 - ~ sm 7rV p ~ sm 7rV p

293

Thus we have obtained estimates, in the I!q norm, of the coordinates of the vector Y and the error which comes from reconstructing the input data X. Since many applications involve quadratic estimates, we shall examine these results in the £2 norm.

The choice of the optimal method of zone coding has great practical significance. For a given compression coefficient T it is easy to determine the number of coordinates of the vector Y which should be replaced by zero. Depending on which coordinates of the vector Yare replaced by zero (i.e., depending on the method of zone coding), we obtain various values of the reconstruction error D. As was mentioned earlier, the method of zone coding which guarantees the minimal error is called optimal. We shall compare the methods of zone coding generated by the DFT and the DMT.

We begin by examining compression of information by means of the DFT. The spectrum in this case is defined by the formula

1 N-l -27rike Ye = N L Xk exp N .

k=O

For 1 :s: £ :s: N - 1, we have by (11.3.1) that

~ max Iyel < --XE2", - sin(7rC/p)

Hence it follows from Parseval's identity IIYlliz = IIXlli; that the optimal method of zone coding by the DFT is that in which the central coordinates of the vector Y are replaced by zeros.

To estimate the £; norm of the reconstruction error generated by this method, let N be an even integer (the difference between the even and odd cases is insignificant). Replace the 28 + 1 central coordinates of the vector Y by zero, i.e., the coordinates

Y4- s ' Y4- s+1'···' Y4'···' Y4+ s '

N 8 < 2.

Then the reconstruction error D is the £2 norm of the vector formed by these coordinates. Hence it follows that the maximal error generated by the DFT on the class

294

of vectors 3.c. is exactly

( (N/2)+s 1 ) 1/2

DF = fl 1 + L -;;-2--

l=(N/2)-s sin (,rr€/N)

CHAPTER 11

S· h f' / 2 • d h'd .. . (7r 7rs) 71"S mce t e unctIOn y = 1 cos X IS monotone an tel entItIes sm 2" ± N = cos N

are well-known, we obtain

Thus we see that

J 2N 71"S fl 1 + - tan - < DF < fl 7r N - -2N ( 71"( S + 1) 71" )

1 + -;- tan N - tan N .

In particular, an approximation to the maximal error of the optimal method of ZOlle coding by means of the DFT is given by

(11.3.17)

In contrast to zone coding by means of the DFT, when compressing informatioll by means of the DMT it is appropriate to replace the coordinat.e's of the last packets of the vector Y by zero. In this case the basic factor pn-s-J in estimates (11.3.9), (11.3.10) reduces to 1. Based on the results of Theorem 11.3.7, optimal methods of zone coding by means of the DMT can be obtained by three different methods. Choosing one in a specific problem often depends on the compression coefficient T

and the fixed number p. THE FIRST METHOD. We replace every element in the last packet by zero, i.e.,

pn _ pn-I coordinates of the vector Y. Then T = pn /pn-I = p. The maximal reconstruction error in the DMT case as measured by the £2 norm can be estimated as follows:

p-j 1 D}\[ < fl L -.---

- 11=1 sm(7rv/p)

{ p~ In(p - j + 1)

~ pfl (p p + 1) 2 In( 2 + 1) + In 2:J

for r. <J' <p-l 2 - - ,

1:::: j < ~. for


For the optimal method of zone coding in the DFT case with initial information of size N = pn and compression coefficient r = p, it is necessary to replace the

n-l

28 + 1 = pn - pn-l central coordinates of the vector X by zero, i.e., 8 ~ ~ (p - 1).

Thus, it follows from (11.3.17) that

/ pn-l bF ~ tJ.y 2-2-(p -1) ~ PVpn-2 tJ..

THE SECOND METHOD. We replace every element in the central subpacket of the last packet by zero, i.e., pn-l coordinates of the vector Y. Suppose that p is even. Then

and by (11.3.15), bM :s: ~. In the DFT case with the same Nand r we have

pn-l 8'" --- 2 '

CF bF = tJ.y2T-2- = pJpn-3~.

THE THIRD METHOD. We replace some subsequence lying in the central subpacket of the last packet of Y by zero. This method is used for large p, p 2': 6, because a sharp increase in the estimate (11.3.9) is observed only for j approaching 1 or p - 1.

Unlike the DFT case, notice that for all three methods the quantity bM does not depend on the size N of the given input vector, and for large N is significantly smaller than b F. Thus for compression of information by means of orthogonal transformations, the method of zone coding which uses the discrete multiplicative transform is more accurate for restoration of the input vector than that which uses the discrete Fourier transform.

§11.4. Practicalities of processing two-dimensional numerical problems with discrete multiplicative transforms.

\Ve saw in the previous section that estimates of the spectrum can be employed for zone coding, i.e., for compression of information. However, for the problem of pattern recognition, the spectrum of the original signal is of primary importance. Moreover for multidimensional signals, pattern recognition uses the multidimensional spectrum. Since multi-dimensional signals can be transformed into one-dimensional signals at the expense of extending the size of the system, it is natural to consider the problem of establishing a correspondence between the spectrum of multidimensional numerical signal and its one-dimensional counterpart. In this context, it is important to notice that the transformation from a multi-dimensional signal to its one- dimensional counterpart is not unique. For example, a two- dimensional signal {xm,d, m, k = 0,1, ... ,N - 1, can be converted to a one-dimensional signal by means of row scanning while preserving the order in each row:

Xm,k = zmN+k, m, k = 0,1, ... ,N - 1.

296 CHAPTER 11

Since the difference Z(m+1)N - ZmN+N-l corresponds to the difference of the value at the left end of the m + 1-st row and the right end of the m-th row, the one dimensional signal obtained in this way can have large jumps for k = N - 1 . Thus the inverse Fourier transform of the one-dimensional signal (ZmN+k) can exhibit Gibbs phenomenon at points of large jumps. Consequently, when converting signals from two-dimensions it is necessary to choose a scanning method which preserves "continuity" when passing from point to point, i.e., choose a method in which distances between data points which are near in the two-dimensional sense remain relatively so in the converted one-dimensional signal. However in this case it is practically impossible to establish some kind of dependence between the twodimensional spectrum of the initial signal and the one-dimensional spectrum of the converted signal. Moreover, one must keep in mind that computing the DFT of a one-dimensional digital signal of size N 2 takes more calculations than computing the two-dimensional DFT of an N x N array, including approximately N(N - 1) values of the trigonometric functions at points of multiples 7l" / N2. Therefore, from the point of view of efficiency, the two- dimensional DFT of a system is preferred over the one-dimensional DFT of that same system.

We shall examine the situation in the discrete multiplicative transform case. Existence of dependence between the multiplicative spectrum of a two-dimensional signal and the multiplicative spectrum of its one-dimensional counterpart (obtained through row by row scanning) was discovered by Zukov [2J. We shall define the spectrum {Ym,j'}' m = O,l, ... ,pn -1, I' = O,l, ... ,pT -1 of a two-dimensional signal {ak,v}, k = 0,1, ... ,pn -1, v = 0,1, ... ,pT - 1 by

(11.4.1)

11.4.1. Let {Ym,j'}' m = 0,1, ... ,pn - 1, I' = 0,1, ... ,pT - 1 be the spectrum of a. two-dimensional signal {ak,v}, k = 0,1, ... ,pR -1, v = 0,1, ... ,pT -1, and let {Zt}, £ = 0,1, ... ,pn+T - 1 be the spectrum of the corresponding one-dimensional signal {x'l}' 7J = 0,1, ... ,pn+T -1 obtained by row scanning, i.e., x'l = Xkpr+v = ak,v for 7J = kpT + v, 0 ~ k ~ p" - 1, and 0 ~ v ~ pT - 1. Then

for £ = I'pn + m, 0 ~ I' ~ pT - 1, 0 ~ m ~ pn - 1, i.e., the two-dimensional spectrum {Z,.,m}, obtained from the one-dimensional spectrum {Zt} by placing its elements one by one in pn rows (pT elements to each row) is the transpose of the initial spectrum {Ym,,.}.

PROOF. By formula (11.3.2) we have

(11.4.2) £.=O,l, ... ,pn+r_1.

Using property 11.2.3 and the fact that the function n(x) is multiplicative, we obtain

Setting £. = Itpn + m, 0 s:; It s:; pr - 1, 0 s:; m s:; pn - 1, we find that

Zc = Z/tpn+m = z/t,m pn_l pr_l

= pn~r ~ ~ ak,vX,.pn+m (:n) X/tpn+m (p:+r) pn_lpr_l

- pn1+r ~ ~ ak,vX/tpn (:n) Xm (p:) X/tpn (p:+r) Xm (p:+r) .

Since 0 s:; II s:; pr - 1 implies IIjpn+r < 1jpn, we can write

where 0 s:; IIj s:; P - 1 for all j, and thus see that

- -- -1 ( II) Xm pn+r -

for 0 s:; m s:; pn - 1. Using 11.2.3 we obtain the identity

X,l.pn (p:+r) = Xv (;!:nr) = Xv (; ) = X" (;r) .

298 CHAPTER II

Moreover, since the p-adic coordinates of the numbers flpn and k/pn satisfy

and

we have

Therefore,

(11.4.3)

(mod p) = fl-(n-q) = { 0 fl-(q-n)

= { ~-(n-q+l) for q:::: n + 1,

for q '5: n,

for

for

( k ) 21l"i ~ n ( k ) --;- = exp - L)flP )-q --;- = 1. P P q=l P q

q '5: n,

q:::: n + 1,

In particular, comparing (11.4.1) with (11.4.3) we conclude that z/.,m = Ym,!" • Notice that although the transforms (11.4.1) and (11.4.2) can be computed in

roughly the same number of arithmetic calculations, nevertheless in practice it takes considerably more time to compute the system {Ym,/.} by formula (11.4.1) than by formula (11.4.2). This is due to the fact that computation of the transform (11.4.1) additional reorganization of all the calculations when one moves from calculating the inner sum to calculating the outer sum.

§11.5. A description of classes of discrete transforms which allow fast algorithms.

In the last decade interest in the study of discrete orthogonal transforms has grown significantly. This interest has been fueled by the appearance of a whole series of algorithms, similar to the fast Fourier transform (FFT) and to the algorithm we looked at in §11.2 for computing the DMT, which allowed these transforms to be computed quickly and efficiently (in real time) on high speed computers, even for problems which involved a large number of calculations. Investigations of the various properties of orthogonal transforms have been influenced by their applications to image processing and speech signals, by identification of tests for pattern recognition, by analyzing and designing communication systems, by generalized Wiener filtering, and by a number of other applied questions.

Because of this, there is keen interest in the problem of determining conditions on an n x n matrix so that its product with a column vector uses roughly n log n


calculations instead of the n 2 operations used in the general case. Solutions to this problem involve creating an algorithm of type FFT for the given matrix.

In this section we shall find necessary and sufficient conditions for which a square matrix A of order n x n, n = pq, can be represented as a product of two sparse matrices, each having a similar sparse structure.

We shall say that a square matrix A of order n x n, n = pq, is (p, q)-factorable if it can be written as a product A = BC where the elements of the matrices

B = (bi,j) = (bidi 2P,j,+hq),

C = (Ck,m) = (Ck I+k2q,ml+m2P)'

satisfy the conditions

O~il,h~p-l, O:Si2 ,h:Sq-l,

O~k2,ml~p-1, O~kl,m2~q-1

(11.5.1) { bi,+i2P,it+hq = 0

Ckl +k2q,ml +m2P = 0

for it =f. i 2 ,

for ml =f. k2 •

The structure of the matrices Band C are shown in Figures 2 and 3, where all elements except those denoted by stars are zero.

c=

B=

Figure 2 Figure 3

11.5.1. Elements of a (p, q)-factorable matrix A satisfy the relationship

(11.5.2)

for k = i2 + jIq.

PROOF. Indeed, by (11.5.1) we find

n-I

aj = L bi,vCv,j

v=o q-I p-I

= L L bil+i2P,V,+V2qCVI+V2Q,jl+hp VI=O V2=0

= bil+i2P,i2+itqCi2+itq,it+j2P = bikCkj .•

300 CHAPTER 11

11.5.2. Let Aij, 0 ~ i ~ q - 1, 0 ~ j ~ p - 1, be submatrices of order p x q of a matrix A = (am,k) organized in the following way:

(11.5.3) o ~ r ~ p - 1, 0 ~ s ~ q - 1.

Then the matrix A is (p, q )-factorable if and only if

(11.5.4) Rank Aij ~ 1

for all 0 ~ i ~ q - 1, 0 $ j $ p - 1.

PROOF OF NECESSITY. Let A be (p,q)-factorable. By 11.5.1 we have

If for fixed i and j we denote

r = 0,1, ... ,p- 1,

s = 0,1, ... ,q - 1,

then ar+ip,j+.p = a r f3 •. In particular, it follows that Rank Aij $ 1. PROOF OF SUFFICIENCY. Suppose that (11.5.4) holds for every 0 $ v $ q - 1,

and 0 $ f1 $ p - 1. Then choose numbers a r+vp and f3/.+. p so that the elements of the matrix Av,1' from (11.5.3) are written in the form

(11.5.5)

Set

(11.5.6) bi,i = bi,+i,p,j,+hq

= { ~il+i2P for j1 =I i2, j2=0,1, ... ,p-1,

for )1 = Z2,

and

(11.5.7) Ck,m = Ck, +k,q,m, +m,p

= { ~m'+m'p for m1 =I k2, k1 = 0,1, . .. ,q - 1,

for m1 = "~2'


It is obvious by construction that the matrices B = (bi,j) and C = (Ck,m) are sparse as defined in (11.5.1). It remains to verify that their product coincides with the matrix A. But by (11.5.6), (11.5.7), and (11.5.5) we find that

n-l q-l p-l

Lbi,IICII,m = L L bi,+i2p,V,+1I2qCII,+v2q,m,+m2P v=O 111=0112=0

= bil +i2P,i2+m, q C i 2+m 1 q, m, +m2P

= OIi,+i2P(3m, +m2P = ai, +i2P,m, +m2p· •

It is fairly difficult to verify condition (11.5.4) for matrices oflarge order. Because of this the condition of (p, q)- factorability of a matrix will be described in other terms.

We shall associate two bases of the linear space en with each non-singular matrix A : the canonical basis {ed, k = 0,1, ... , n -1, where the k-th component of ek is 1 but all other components of ek are zero, and the basis {ad, k = 0,1, ... ,n - 1, where ak is the k-th column of the matrix A. We shall denote the linear hulls (in en) generated by certain collections of these vectors in the following way:

7'=

f{ q

f;-~(n--~-j~p ~-_--_.VL-__ -~ I I I 0 I 0" 0 10! 0 I p I b I: I '-_____ .1..JI ____ J

Figure 4

i = 0,1, ... ,q - 1,

j = 0,1, ... ,p - 1.

~--~--~ ,g 0 I I· I P I 1-'------1 1 0 I 10! 0 I ,: I L.._!!... ___ .J

11.5.3. If A is a non-singular (p, q)-factorable matrix, then

(11.5.8) dim (Li nLj) 2: 1

for i = 0,1, ... , q -1, and j = 0,1, ... ,p - 1.

PROOF. For each i = 0,1, ... ,q -1 and j = 0,1, ... ,p -1 it suffices to show that there exists at least one non-zero vector which belongs simultaneously to the spaces

302 CHAPTER 11

Li and Lj. Consider the vector bi+jq, i.e., the vector which occupies the i + jq-th column of the matrix B. Since A is (p,q) factorable, we have bV\+v2P,i+jq = 0 for V2 =j:. i. Consequently,

p-l

bi+ jq = L bV\+V2P,i+jqe v \+ip ELi.

v\=o

On the other hand, since the matrix A is non-singular we can write B = AC- 1•

We shall show that the matrix C- 1 = (2(}',;3) is sparse with

(11.5.9)

where 0 S; 0:1,/32 S; P -1, and 0 S; 0:2,/31 S; q -1. Indeed, set T = (ti,j) where

(11.5.10)

for ZI = J2 and Z2 = JI,

OS; i l ,j2 S; p-1, 0 S; i 2 ,h S; q -1,

in the remaining cases,

are elements of a permutation matrix whose structure is shown in Figure 4. Set C' = CT and notice by (11.5.10) that

n-I

c~,m = c~\+k2q,m\+m2q = LCk,vtv,m v=o

p-I q-I

L L Ck\ +k2q,v\ +V2PtV\ +V2P,m\ +m2q = Ck\ +k2q,m\ +m2Q' v\=o V2=0

By (11.5.1) we obtain

i.e., the only elements different from zero have the form

o:=O,l, ... ,p-1.

Thus the matrix C' is block diagonal and its structure is shown in Figure 5. Since C' can be obtained from C by a permutation of columns it must also be non

singular. It follows that each block of C is non-singular, since det C is the product of the determinants of these diagonal q X q blocks. Hence the inverse of C' is also block diagonal (see Figure 6). Since C' = CT we have (C')-I = (CT)-l = T-1C- 1, i.e.,

T(C')-l = TT-1C-1 = EC- 1 = C- 1

c'=

r---T---' 10 ' 0 1 I 1 I I I 1 L ___ .J... ___ .J

,*"*"7.":""*' 1** ... *1 1 ••••••. 1

l!-*..:.:: .. .*J

Figure 5 Figure 6

[*T.-:-*j I~.~·::~I k*...::..:ttl

303

where E is the identity matrix. Consequently, C-I = T(C')-I. Since (C')-I has a similar block structure, the product T( C')-l, obtained from (C')-l by a permutation of columns, is also sparse with

n-l

cer ,/3 = Cer ,+er 2P,/3,+/32q = L ter,+er2p,vC~,/31+/32q, v=o

where <,j are the elements of the matrix (C')-l. Using the definition (11.5.10) of the matrix T, we obtain

q-I p-I

cer ,/3 = L L t er , +er 2P,", + V2q c:, +V2q,/3, +f32q = C~, +n2Q,f3,+f3,q' V1=0 V2=0

Since the elements of this block diagonal matrix vanish if the second indices are different, i.e., if 0'1 =f. (32, we see that (11.5.9) holds as promised.

It remains to show that along with bi+jq E L j we also have bi+jq E Lj. But using (11.5.9) we find

n-1

bi+ jq = L Cv,i+jqav

V=O p-1 q-1

= L L cv,+v2P,i+jqaV1+V,p

v,=o V2=0

q-1

== L Cj+v,p,i+jqaj+v,p E Lj .• v,=o

11.5.4. Let A be an n X n non-singular matrix. Suppose that the subspaces L j ,

i = 0,1, ... ,q -1, and Lj, j = 0,1, .... ,p-l, satisfy the conditions

(11.5.11) dim (Li nLj) = 1

304 CHAPTER 11

for all i,j. If li,j E Li nLj are non-zero vectors, then the system {!i,j}, i = 0,1, ... , q - 1, j = 0, 1, ... ,p - 1, is a basis for the space en.

PROOF. Since the number of vectors li,j, i = 0,1, ... , q - 1, j = 0,1, ... ,p - 1, is exactly pq = n, it suffices to prove that they are linearly independent. Let

q-l p-l

L L AU/i,j = 0. i=O j=O

Since ~~:~ Ai,j!i,j E Li then ~~:~ Ai,j!;,j = 0. But for each fixed i the vector !i,j belongs to Lj. Therefore, the previous relationship implies that Ai,j = ° for all i,j .•

11.5.5. Let A be an n x 11 non-singular matrix. Then A is (p, q)-factorable if ana only if (11.5.11) holds for i = 0,1, ... ,q - 1, j = 0,1, ... ,p-1.

PROOF OF NECESSITY. Let A be a non-singular, (p, q)- factorable matrix. Then by relationship (11.5.8) we have

q-l q-l p-l q-I p-l

dim en = L dim Li = L L dim (Li nLj) ~ L L 1 = pq = n. i=O ;=0 j=O ;=0 j=O

Consequently, we must have equality in (11.5.8), i.e., dim (L; nLj) = 1. PROOF OF SUFFICIENCY. Let A be a non-singular matrix which satisfies con

dition (11.5.11). Then by 11.5.4 the system of vectors {Ji,j}, i = 0,1, ... ,q - 1, j = 0,1, ... ,p - 1, is a basis for the space en.

vVe introduce the notation

B = {eo, el , ... ,en -1 },

B' = {aO,al,'" ,an-d, B" = {/0,0,!1,1,'" ,!g-l,p-d·

Then SB~B' = SB~B"SB"~B" where S.~. represents the matrix which takes one basis into another, for example SB~B' represents the matrix which takes the basis B to the basis B'. By definition the columns of this matrix are the coordinates of the basis B' as vectors in the basis B. But B is the canonical basis. Thus the corresponding coordinates of the vectors from B' coincide with their components. But by construction, the vectors of the basis B' are just the columns of the matrix A. Consequently, the coordinates of any vector in B' with respect to the basis B is just a corresponding column of A. In particular, SB~B' = A.

Set B = SB~B" and C = SB"~B" Then A = BC and it remains to show that the matrices Band C arc sparse matrices as defined by (11.5.1).


We notice that the vectors li,j, j = 0,1, ... ,p - 1, belong to the space Li and consequently have the form

p-I

li,j = L !v+ip,i+jp . ev+ip· 1'=0

Thus if we assume bk = bi+ jq = h,j we find that the components

0:::; mI, k2 :::; p -1, 0:::; m2, kI :::; q - 1,

of the vector bk satisfy the relationship bm,k = ° for m2 i- kl . Consequently, the matrix B = S B-B", whose columns are the vectors {bko k = 0, 1, ... , n -I}, satisfies (11.5.1).

Similarly, the vectors li,j, i = 0,1, ... ,q - 1, belong to the space Lj. Thus the vectors bk, k = jq, jq + 1, ... ,jq + q -1 form a basis ofthis subspace. From this we obtain the following decomposition:

q-I q-I

aj+ip = L cv+jq,j+ip . bv+jq = L cv+jq,j+ip . Iv,j'

1'=0 1'=0

Introducing the notation

Cj+ip = (0, ... ,0, Ciq,j+ip, Ciq+I ,j+ip, ... ,Cjq+q-I ,j+ip, 0, ... ,0),

we find that cv,/t = CV1+V2q,1'1+/1.2P = ° if PI i- 1I2' Therefore, the matrix C = SB"-+B'

whose columns are the vectors Ck, k = O,I, ... ,n -1, also satisfies (11.5.1). In particular, the matrix A, as a product of Band C, is (p, q)-factorable .•

For applications, it is frequently sufficient to use matrices whose elements are obtained by a permutation of rows and columns of some (p, q)-factorable matrix. Conforming to the terminology in digital methods of information processing, we shall call the set of matrices A of the form A = {A' : A' = T1ATd, where Tl and T2 are permutation matrices, the discrete transformation A = ACA) generated by the matrix A.

It is evident that a discrete transformation A is generated by any matrix it contains, i.e., if A = A(A) and A' = TIAT2 for some permutation matrices Tl and T2 ,

then A = ACA'). It is clear by definition that any discrete transformation A is a union of matrices

which differ from one another by permutations of rows and columns. A discrete transformation A is called (p, q)-factorable if at least one of its gener

ators is (p, q )-factorable.

306 CHAPTER 11

Suppose that A is the discrete transformation generated by a matrix A'. We shall use this matrix and certain kinds of partitions to generate special subspaces and triangular matrices in the following way.

Suppose n = pq. Let r = {.6.; : i = 0,1, ... , q - I} be a partition of the set of numbers {O, 1, ... , n - I} such that each .6.; contains exactly p elements chosen from the collection of indices {O, 1, ... ,n - I}. Clearly, .6. il n .6. i , = 0. Similarly, let r' = {.6.j : j = 0,1, ... ,p -I} be a partition of the set of numbers {O, 1, ... , n - I} where each.6.j contains exactly q elements. Again, .6.jl n.6.j, = 0. Let ti =L(.6.;)

be the linear hull of the collection of vectors e v , v E .6.;, and let t~ = L(.6.j) be the linear hull of the collection vectors which form the I-l-th columns of the matrix A' (which generates the discrete transformation A), where It E .6.j. And, let A(i,j)

be the triangular p x q matrix made up of the elements of the matrix A' which lie in the intersections of the v-th rows, v E .6. i , and the It-th columns, II E .6.j, i = 0, 1, ... , q - 1, j = 0, 1, ... ,p - 1.

11.5.6. A discrete transformation A = A( A') is (p, q )-factorable if and only if there exist partitions r and r' of the set {O, 1, ... ,n - I} such that

(11.5.12) Rank A(i,j) $ 1, i=0,1, ... ,q-1, j=O,l, ... ,p-1.

PROOF OF SUFFICIENCY. Let A be a (p, q )-factorable discrete transformation. Thus there is a (p, q )-factorable matrix A which generates the transformation A (we may suppose that the matrix A is different from the matrix A'). Hence by definition there exist permutation matrices T) and T2 such that A' = TJ AT2 • The matrices T) and T2 induce permutations on the set of indices {O, 1, ... ,n - I}. For each i = 0,1, ... , q - 1, let .6. i represent the subset which results from applying the permutation TJ to the set {ip, ip + 1, ... ,ip + (p - I)}. Similarly, let .6.j represent

the image of {j, j + p, ... ,j + (q - l)p} under the permutation T2 . Then A (;,j) = Aij

and we conclude by Theorem 11.5.2 that

Rank A(i,j) = Rank A;j $ 1.

PROOF OF NECESSITY. Suppose that the discrete transformation A is generated by a matrix A' for which there exist partitions r = {.6.; : i = 0,1, ... ,q - I} and r' = {.6.j : j = 0,1, ... ,p -I} of the set {O, 1, ... , n -I} such that the submatrices

A(i,j) satisfy (11.15.12). Let T) and T2 represent the matrices induced by the permutations on the set {O, 1, 00 • ,n - I} which take, respectively, the subsets .6.; and.6.j to the sets {ip,ip+ 1,oo.,ip+ (p-l)} and {j,j +P,oo.,j + (q -l)p}.

The matrix A = Tl A'T2 also generates the discrete transformation A and is (p, q)factorable by 11.5.3. Indeed, Aij = A(i,j) and consequently, RankAij $ 1. In

particular, the discrete transformation A is (p, q)-factorable. •


A discrete transformation will be called non-:Jingular if it is generated by a nonsingular matrix. Evidently, all the generators of a non-singular discrete transformation are non- singular since the determinant of a permutation matrix equals ±1.

11-5.7. A non-singular discrete transformation A A(A') is (p, q)-factorable if and only if there exist partitions 7 and 7' of the set {O, 1, ... ,n - I} such that

(11.5.13) i = 0,1, ... , q - 1, j = 0,1, ... ,p - 1.

PROOF. The proof is similar to that of 11.5.6 but is based instead on 11-5.5. Necessity of condition (11.5.13) is obtained from construction of the permutation matrices TI and Tz as in 11-5.6 and necessity of (11.5.13) from construction of the matrices TI and Tz. We notice that if the matrices TI and Tz are already chosen then one can suppose that TI = (T1)-1 and Tz = (T2 )-I .•

The difference between factorability of discrete transformations and matrices can be illustrated by the following example. Let TV be the discrete 'Walsh transformation generated by the matrix W = (Wn,k), n, k = 0, 1, ... , 2r - 1, where if

then Wn,k = exp ( 7ri I:::~ nvkr- v- l ). We shall show that TV is (2 r - l , 2)-factorable

but that the matrix W is not (2 r - l , 2)- factorable. Let

k = 0,1, ... , 2r - 1

denote the column vectors of the matrix W. Below, we shall construct two partitions 7 = {~i : i = 0,1} and 7' = {~j : j = 0,1, ... ,2r -I} of the set of numbers

- -, {O, 1, ... , n - I}. Using these partitions, we define subspaces Li = L(~i) and L j = L( ~D which are linear hulls of vectors of the canonical basis and vector columns of the matrix l¥ whose indices belong, respectively, to the sets ~i and ~j. We must verify that

(11.5.14) i = 0,1, j = 0,1, ... , 2r - 1.

Let the partition 7 be determined in the following way: ~o = {O, 1, ... , 2r - 1 -I}, and ~l = {2 r - 1 -1, ... ,2r -I}. We shall construct the partition 7' so that (11.5.14) holds. Each ~j consists of two elements j(1) and j(Z). For any fixed j we will have

308 CHAPTER 11

-I

Consequently, each vector z of the space Lj has the form z = Alaj(1) + A2ap), where

Al and A2 are real numbers. If z E L; then z = lOCo + IIeI + ... + 12r-Lle2r-LI. - -I

In particular, if z ELi nLj is a non-zero vector then

(11.5.15)

and not all of these coefficients are identically zero. The right side of the vector equation (11.5.15) has the form

10

o

Thus (11.5.15) will be satisfied if and only if we can find coefficients Al and A2, not both zero, such that the last 2r - I coordinates of t.he vector AlajO) +A2aj(2) are identically zero. Sinceaj(\) = (wO,j(\), ... ,W2r_l,j(\)) andaj(2) = (WO,j(2), ..• ,W2r_l,j(2»),

we obtain the following system:

(11.5.16) II = 2r-I, ... , 2r - 1.

If II E {2r-I, ... , 2r - 1} then in the decomposition II = 110 + 2111 + ... + 2r- I llr _ 1 we have IIr-I = 1. Consequently,

( r-I )

WI/,I' = exp tri L 118Ilr-.-I

8=0

for Il = j(I) , j(2). If we set

then (11.5.16) can be written in the form

( r-2 )

= exp tri L 1I.llr-.-I + Ilo

.=0


for v. = 0,1, s = 0,1, ... ,r-2. Since j(I) and j(2) are distinct, the numbers (-1 )j~l) and (_1)j~2) have different signs. Therefore, (11.5.16) will be satisfied when A2 = AI.

We have shown that if we take the partition 7' = {6.j : j = 0, 1, ... ,2 r - I - 1} where 6.j = {2j, 2j + 1} for each j, then (11.5.14) holds for i = 0, i.e.,

- -, A similar construction shows that dim (LI nLj) = 1. Hence by 11.5.7 the discrete Walsh transformation is (2 r - I , 2)-factorable. However, we notice by construction -, that the subspaces L j do not coincide with the subspaces Lj or, equivalently, that the partition 7' does not satisfy the conditions of proposition 11.5.7. Therefore, we conclude that the matrix l'V is not (2 r - I , 2)-factorable.

Chapter 12

OTHER APPLICATIONS OF MULTIPLICATIVE FUNCTIONS AND TRANSFORMATIONS

§12.1. Construction of digital filters based on multiplicative transforms. Filtering is one of the fundamental techniques of digital signal processing. By

a digital filter we shall mean a transformation which takes a sequence of numbers {x(n)}, called the input, in another sequence {yen)}, which is called the output, or the filter response. In general, the relationship between the input sequence and the filter response can be written in the form

During the last decade, rapid expansion of digital methods of signal processing has been a huge stimulus for the development of both a theoretical basis and practical methods to apply digital filtering in various situations. Investigations in this direction are well documented, both in a large number of journal articles and in a series of monographs. Hence instead of dwelling explicitly on the execution of these ideas we shall consider only digital filtering based on spectral analysis of the input signal. Widespread use of such digital filtering began at the end of the sixties with development of the so-called fast algorithms for computing the discrete Fourier transfOlID (DFT). The idea behind such filtering is that the DFT can be used to compute the spectrum Y of an input signal X, i.e., the quantity Y = FX. Then by using a diagonal matrix D one can obtain from Y a filtered signal X = F- 1 DFX. The matrix D compresses, strengthens, weakens, or otherwise affects various spectral components of X.

We shall be especially interested in spectral filtering by means of the discrete multiplicative transform (DMT), and confine ourselves to the case when the DMT is generated by the sequence Pk = P ~ 2 for k = 1,2, ... and some fixed p. Let lV denote the direct DMT of size pn and let (W)-1 denote its inverse. Then the filtered signal generated by spectral filtering using the discrete multiplicative transform can be described by the formula

(12.1.1)

where the transformations Wand (W)-1 are defined by (11.2.6) and (11.2.7). Let W denote the matrix of the transform (11.2.6) written by 11.2.4 in the form W = C Bn , where the elements bl,k of the matrix B are defined by (11.2.8), and the elements

310

OTHER APPLICATIONS 311

CR,k of the matrix C are defined by (11.2.12). Then (12.1.1) can be written in the form

x = (CB n)-1 DCBn X.

But (CB n)-1 = (Bn)-IC- l = (B- l tC- l and C- IDC = D*, so

(12.1.2)

1 - T We shall show that B-1 = -(B) . Denote the elements of the matrix B by bl k and P ,

use identity (11.2.8) for the elements b't,k of the matrix (B)T to write (for n ~ 2)

(12.1.3)

h - - 27ri k - ~n k IJ-l d e _ ~n n IJ-l were q - exp -, - 61J=1 -IJP an - 61J=1 >:--IJP . P

Let Cl,k represent

the elements of the matrix product (BfB, i.e.,

p"-1

Cl,k = L b;,mbm,k. m=O

Using (12.1.3) and (11.2.8) we obtain

Since for each index m the quantities 8m _(V+l),c_V are different from zero only when m-(v+l) = £-IJ for all v = 1,2, ... ,n - 1, it follows that

(12.1.4)

Using _ 27ri -27ri(p-1) p-l q = exp - = exp = q ,

P P

312

we can write

= {P, 0,

CHAPTER 12

if k-n + (p - l)Ln == 0 (mod p),

if k-n + (p - l)e_n =I- 0 (mod p).

But Ln + (p - l)Ln == 0 (mod p) is equivalent to Ln - Ln == 0 (mod p), i.e., k-n = Ln. Hence it follows from (12.1.4) that

{ p,

Ce k = , 0, if k_n=e_ n , v=1,2, ... ,n, i.e.,ife=k,

in all other cases.

Thus we see that (B)T B = pE, where E is the identity matrix. Consequently,

B- 1 = ~(Bf and the filtering algorithm (12.1.2) becomes p

(12.1.5)

We now consider practical methods for computing the filter (12.1.5). As we noticed in remarks a), b), and c) in §11.2, the computation Bn X can be done using repeated mtlltiplications by the matrix B and can be conducted in the basis (1, q, . .. , qP-2) on single instruction parallel processors using only the addition operation.

We shall find the form of the matrix D* = C- l DC. Since C is a symmetric orthogonal permutation matrix, we have C- l = C. Let Cl,k, e,~: = 0,1, ... ,pn - 1, represent the elements of the matrix C. We have by (11.2.12) that

n

Cl,k = II OLj ,L(n-HI) j==1

for e = 2:;'=1 L j pJ-l, k = 2:;'=1 L jpi- l . The elements d;,l of the diagonal matrix D can be written in the form di,l = O;,IA; where A; are the diagonal elements of the matrix D. Consequently, the elements di,k of the matrix DC have the form

pn_l pn_l n

J. k - ~ d· -c· k - ~ o· ,A· II O. k I,' - ~ 1,(. (.,' - ~ I,e: I (.-VI -(n-v+l)

1=0 1=0 v=l n n

= Ai II OL.,L(n_.+l)' . ~. v-I l = ~Z-vp .

v=1 v=I

OTHER APPLICA nONS 313

Let di,j represent the elements of the matrix D* = CDC. Then

The first product is different from zero only for those k E [O,pn - 11 which satisfy k-(n-/L+l) = i_ ll , i.e., k-v = i-(n-v+l), v = 1,2, ... ,n. Consequently,

In particular, we see that dr,j is different from zero only when j = i, in which case d* \ £ .* "n . n-v

i,i = Ai* or z = L,.,v=l Z-vP .

We have shown that by multiplying the i-th element of the column vector Bn X by di i = Ai*, for i = 0,1, ... ,pn -1, we obtain the product D* Bn X, and furthermore, th'at the components of this column vector are written in the basis (1, q, . .. ,qP-2). It remains to multiply the corresponding column vector by the matrix (B)T a total of n times. We note that each p columns of the matrix (B)T can be realized by stretching the rows of a matrix Bl (see remark b) from §11.2). Thus using the identities

(-)v 27ri -27ri(p - v) p-v q =exp-=exp =q ,

p p

for v = 1,2, ... , n, we conclude that multiplication by (B)T by a column vector can be carried out in the basis (1, q, . .. ,qP-Z). This means that to carry out the computations necessary for the digital filtering (12.1.5), the multiplication operation is used only for taking the product of the elements of the diagonal matrix D* and the elements of the column vector Bn X, the division operation only for dividing the final components by pn, and all remaining calculations can be performed using only addition and subtraction.

§12.2. Multiplicative holographic transformations for image processing. At the present, a number of problems concerning storage, transmission and repro

duction of information are solved by so- called digital holography (see [41 and [14], for example), i.e., analysis and synthesis of wave fields, using the discrete Fourier transform implemented on digital computers. This technique has been used to solve problems of visual information to create optical devices for processing visual signals and for pattern recognition. The use of digital computers for analysis and synthesis of wave fields gives an alternative of analog methods including the methods of physical holography. This alternative brings with it several advantages intrinsic to

314 CHAPTER 12

the digital techniques of signal processing: high accuracy and absolute reducibility of processing, simple algorithms of processing, accessibility of results. Moreover, algorithmic flexibility of digital techniques allow the extension of these techniques to other discrete transforms, in addition to the Fourier transform, which can be realized in optical systems and can be used in digital holography. In other words, there is now the capability to build analog digital holograms on the basis of other linear transforms which preserve all the advantages of Fourier holograms.

The discreet multiplicative transforms (11.1.12) and (11.1.13), which were introduced in §11.1, including the discrete transforms (11.2.6) and (11.2.7), and the special Walsh transform case (when P = 2) are of significant interest from this point of view. In this context, we mention the results of Lisovec and Pospelov p], [2].

Every two-dimensional discTete multiplicative transform is part of a class of spatially separable transformations (see [27], §2.2). In fact, the two-dimensional multiplicative transform generated by a sequence {Pk = p} bl reduces to a oncdimensional transform (see §llA). Thus construction of holograms based on thcse transforms can be reduced to the one- dimensional case. Moreover, the two-dimensional extension does not present additional difficulties. Let PI, P2, ... , Pa be a sequence of whole numbers, PI! ~ 2, v = 1,2, ... ,so Let N = rna = PI ... Pa and No = rna_I = PI ... Pa-I, i.e., N = PaNO. Using definitions (11.1.12) or (11.1.13), we correspond to each vector X = (xo, Xl , ••• , .T N -1) T in the real Euclidean spac(~ EN, a vector Y = (Yo, YI, ... , YN-I )T, whose components have the form

(12.2.1) 1 N-I ( e)

ydX) = Yk = ,IN t; xeXk N ' k = O,I, ... ,N - 1.

The inverse transform is written in the form

(12.2.2) 1 N-I ( C)

.Te = ,IN L YkXk N ' k=O

C=O,I, ... ,N-1.

Each component of the vector Y is a complex nnmber of the form

(12.2.3) Yk = ,fii; exp{27Tifh},

1 8k = 27T arg Yk, k = 0,1, ... , N-1.

Analogous to the Fourier transform case, the vectors M = (Mo,MI, ... MN-lf and 8 = (80 ,81 , ... , 8N_dT are called the spectral capacity and the phase spectT1lm.

It is well-known that the spectral capacity M by itself does not give enough information to reconstruct the input vector X. Nevertheless, there exists a sufficiently large class of vectors X, whose components can be uniquely determined using only the spectral capacity AI and the inverse multiplicative transform. For a description


of one of these classes in the vector space £N = {X : X = (Xo, Xl, ... , XN_t)T}, we shall construct a linear manifold of dimension No = NiPs in the following way.

Use the components of the vectors X E £N to form vectors

which constitute an No-dimensional subspace of £N. Consider the vector Z(O) whose components zkO) are defined by the relationship

(12.2.4) (0) _ { Xv zk -

° if k = vps, v = 0,1, ... ,No - 1,

if k = VPs + n, 1::;: n ::;: Ps - 1,

and the vector E = (0, "fN, 0, ... ,O)T which has only one non-zero component, the second one. Then the set of vectors Z = Z(O) + E is a linear manifold in £N. We shall prove the following result:

12.2.1. Let X(Z) = (Xo, Xl, ... ,XN_I)T be a vector in EN obta.ined from tbe vector Z = Z(O) + E by means of tbe formula

(12.2.5) n = O,l, ... ,N -1,

wbere Mk(Z) is tbe spectral capacity of tbe vector Z defined by relationsbips (12.2.2) and (12.2.3). Tben for any R = 0,1, ... , No - 1 tbe coordinates Xl of tbe vector X(O) are generated by tbe coordinates xn in tbe following way:

(12.2.6) { eitber Xn = Xnp.+l, n = 0,1, ... , No - 1,

or XO=XN-l> Xn+I=X(n+l)p.-I, n=0,1, ... ,No-2.

PROOF. We apply the discrete transform (12.2.2) to the vector Z = Z(O) + E. Using (12.2.4) and the definition of the vector E we have

(12.2.7) N-I

Yk(Z) = Yk = ~ ~ znXk (;)

~ ~ I C~.-' zox. G)) _ vP. n 1 No-Ip.-l (+ )

= "fN ~ ~ zvp.+nXk N

316 CHAPTER 12

Consequently, the spectral capacity of the vector Z satisfies

(12.2.8) Mk(Z) = 1vh = IYk(ZW

~ ~ II x.x, (;.) +,IN x, u )1' = ~ (~~I XvXk (;J + -IN Xk (~ ) ) (~~I XvXk (;J + -IN Xk (~ ) )

~ ~ {I;' x"", (;,)1' + N +2R, (,IN x, U);' x.x< (;.)) }

~ ~ IN~' x"x, (;.)1' + 1+ ~R+' U) :~' x.x< (;,) )

= M(I) + 1 + M(2) k k •

To compute the components of the vector X(Z) by the formula (12.2.5) write

(12.2.9)

where

and

N-I () (I) 1 (I) e u( = -IN t; Mk Xk N '

N-I () (2) 1 e ue = -IN L 1Xk N '

k=O

(3) __ 1_ ~ M(2) , (~) u( - r,;;r L k Xk N

vN k=O

for e = 0,1, ... ,N - 1. Obviously,

(12.2.10)

OTHER APPLlCA nONS 317

Furthermore, for 0 :::; C :::; N - 1 we have

Since XL,No (;J = [XNo (;J r-' = (exp ~i (;0) J k, = 1 (by the defi-

nition and fundamental properties of the function Xm(x) and the identity

it follows that

But

(1) 1 No-1 11 No-1 _ (1/) 12 ( C) p,-l ( C ) (J"t = VN ~ VN ~ XI/XI" No X" N k~O XL,No N .

p,-l (C) p,-l ( ( C ))k_, L XL,No N = L XNo No k_,=O k_.=O

p,-l ( 2rri ( C) )k' L exp- -k_,=O Ps No s

for (CIN)s == 0 (mod Ps),

for (CIN)s i' 0 (mod Ps).

318 CHAPTER 12

Consequently, we obtain

No-l INO-l () 12 ( f ) IN ~ ~ XvX~ ;0 X~ N

(12.2.11) ~(l) _ Vi - if f == 0 (mod P3)' i.e., for f = mps ,

m = 0,1, ... , No - 1,

o if f ':/: 0 (mod Ps).

It remains to evaluate the third expression U~3). First we write

2 N-l ( ( 1 ) No-l ()) ( e) U~3) = N ~ Re Xk N ~ XnXk ;0 Xk N

1 p,-l (k_.+l)No-l No-l ( n ) ( f 1 )

= N L L L XnXk No Xk N 8 N k_.=O k=k_, No n=O

2. p,-l (L,+l)No-l No-1 _ (..!!:...) (!:... 2.) + N L L L xnXk No Xk N 8 N

k_,=O k=k_, No n=O

- T(l) + T(2) - t t·

Since XL,No(n/No) = 1 for n = 0,1, ... ,No -1 we have

1 p,-l No-1 No-1 () ( f 1 ) Til) = N k~O k, ?; XnXk' ;0 Xk-. No+k' N 8 N

1 No-1 No-1 ( n ) ( f 1) p,-l ( f 1 ) = N ~ Xn k"?; Xk' No Xk' N 8 N k~O XL,No N 8 N .

Since

Us == [uN] (mod Ps)

for 0 ::; u < 1, and

== e (mod Ps) 8 1

= { Ps-1 f~s -1

f~s = 0,

for 1::; f~s ::; Ps - 1,

for

OTHER APPLICATIONS

where c~. == C (mod Ps), 0 ~ C~. ~ P. - 1, we also have

Pfl XL,No (~ 8 ~) = Pf (XNo (~ 8 ~)) L, k_.=O k_.=O

Consequently,

P.-l( 27ri(C l))k_. = 2: exp- -8-

L,=O P. N N

~ C' if £~s = 1, i.e., for C = ,Ps + 1,

,= 0,1, ... , No - 1,

for all other £.

{

Ps No-l No-l ( nIPs + 1) 2: Xn 2: Xk' no EB N r?) = ON n=O k'=O

for C = IPS + 1,

for all other C.

Furthermore, using properties of the Dirichlet kernel we have

:~: Xk' (;0 EB IP:V+ 1) = DNo (;0 EB (~ + ~ )) if ;0 EB (~ + ~ ) < ~o' if ;0 EB (~ + ~ ) ~ ~o

for n = " for all remaining n.

Substituting this expression into (12.2.12) we obtain

for £=IPs+1, 1=0,1, ... ,No-1, (12.2.14)

for all other e.

319

320 CHAPTER 12

But

and

if e~8 + 1 == 0 (mod P.), i.e., for e~8 = Ps - 1

e = IPS + Ps - 1, I = 0,1, ... , No - 1,

for all other f.,

No-l ( ) ( ) IPS + Ps - lIn I + 1 lIn :L Xk' EB - 8 - = DNo -- - - EB - 8-k'=O N N No No N N No

for n = 1+1,

for all remaining 7l.

Consequently,

and

(12.2.15) { X..,+l

(2)

7"t = ~o

for f. = h + l)ps - 1, ,= 0,1, ... , No - 2,

for e = No -1,

for all other e.

We have an expression for 0";1) from (12.2.11), for 0";3) from (12.2.12), (12.2.14) and

(12.2.15), and for 0";2) from (12.2.10). Substituting these expressions into (12.2.D) we obtain

No-III No-l () 12 ( f. ) .IN ~ .IN ~ xIIX/. ;0 X/. N +VN if e = 0,

(12.2.16) No-III No-l () 12 ( e) .IN ~ .IN ~ X"x/. ;0 Xp N

if e = mps, m = 0,1, ... , No - 1,

x.., if f. = IPS + 1, 1= 0,1, ... , No - 1,

X..,+l if e=(,+1)ps-1, 1=0,1, ... ,No-2,

Xo if f. = N-1.

Therefore, either n = 0,1, ... , No - 1,


or Xo = :IN-I, Xn+I = X(n+I)p,-I, n = 0,1, ... , No - 2. •

The result obtained in 12.2.1 allows us to give a definition of the discrete multiplicative holographic transform which is analogous to the Fourier holographic transform (see [4], §1.4). Let A = (ao,all ... ,aNo-dT be an information vector, i.e., a vector whose components represent coded information subject to compression. We shall replace it by a corresponding vector Z(O) = Z(O)(A) E eN in the following way:

ziO) = { aOn if k = np., n = 0,1, ... ,No - 1, for all remaining k E [1, N -1].

This mapping brings about a unique embedding of each information vector A in the Euclidean space eN. We shall find the spectral capacity M of the sum of the vectors Z(O) and E. The vector M will be called the hologram of A and the linear transformation A -7 M will be called the multiplicative holographic transformation.

We shall establish a connection between the multiplicative holographic transformation and the discrete multiplicative transform of the information vector A. Notice that the sum which appears in (12.2.7) (written for the vector A), namely

No-I

L a'/Xk(v/No), k = 0,1, ... , N - 1, ,,=0

is periodic in J( of period No, i.e., for each k- s = 1,2, ... , P. - 1 and each k = 0,1, ... ,No - 1 we have

(see the representation for /1~1) which appears below (12.2.10». Thus set

No-I

(12.2.17) bk = L a"Xk(v/No), k = 0,1, ... ,No - 1. ,,=0

322 CHAPTER 12

for k = 0, 1, ... , No - 1 and ° :::; L .. :::; P .. - 1. Formulae (12.2.18) and (12.2.17) can be used to write down algorithms to obtain

digital holograms. However these algorithms can be simplified considerably in the case when Pv = P for 1/ = 1,2, ... ,s. As follows from 11.2.4, calculation ofthe quan-

tities bk from formula (12.2.16), i.e., multiplication of the matrix TV = (Xk (;0) ), k,1/ = 0,1, ... ,No - 1, and the column vector A, can be obta.ined by multiplying the (s - 1) power (No = pS-I) of the sparse ma.trix B by A, followed by a p-adic permutation using the matrix C. Moreover, if these calculations are carried out in the basis (1, q, . .. , qP-2), q = exp( -27ri/p), then the arithmetic operation of multiplication can be excluded and the final resulting vector can be written using this basis. For example, for p = 3 we have

k = 0,1, ... ,No - 1.

Using the relationship

and the identity

( 1 ) ( 1 ) - 27ri XHL.No N = XL,No N = exp -3-k- s, k:::; No -1,

for k- s = 0,1,2, we have

Re (bkXk+k_.NO (~ ))

R {( 27rk_.. .. 27rk_s ) (0) (I) 27r . (I) . 27r)} = e cos-3- - lSln-3- Ok +Ok cos 3 - W k SIn 3 (0) 27rk_.. (I) 27rk_s 27r . 27rk_.. (I) . 27r

= Ok cos -3- + ok cos -3- cos 3 - SIn -3-0k SIn 3 (0) 27rL.. (I) 27r

= ° cos -- + ° cos -(1 + k ) k 3 k 3 - .. (0) 1 (1)

Ok - 20k for k- s = 0,

= for k- s = 1,

for k- s = 2.

OTHER APPLICA TlONS


2 +--x

VN

323

for k- s = 0,

for k- s = 1,

for k- s = 2

for k = 0,1, ... , No - 1. Substituting these values for My, v = 0,1, ... , N - 1, into (12.2.5), we can compute the values of xn . Moreover, these calculations again can be carried out in the same way using the basis (1, q) and the N x N matrix (Xk(n/N)).

§12.3. Solutions to some optimization problems.

In the previous sections we have examined the questions of applying multiplicative functions to digital processing, i.e., to problems in which both the input and output were discrete. Now we shall consider applications of multiplicative functions to problems where the input data is discrete but the solutions are continuous processes. Such problems can arise in the physical world. For example, let a linear radiator of electromagnetic waves (e.g., an antenna) consist of N elementary dipole radiators, arranged along some straight line. We wish to find, among all practical feasible (i.e., discretely placed) distributions of currents along such a radiator, a distribution whose corresponding radiation pattern is the nearest, in some sense, to a specified radiation pattern.

Another example of such a physical problem is the problem of diffusion of matter. Under circumstances frequently arising in the manufacture of electronic instruments, the concentration of matter before the execution of a diffusion process can only be assigned discretely. The problem we refer to consists in determining, among all possible (i.e., discrete) initial distributions, that distribution which at the end of the diffusion process, i.e., after time T, yields a concentration which is closest, in some sense, to a desired concentration of matter in the sample.

In order to give a precise mathematical formulation of these physical problems and their solutions, we shall introduce several new concepts.

The Chrestenson-Levy system Xn(x), defined by (1.5.10) for some integer p::::: 2,

324

will be used to generate a system of functions

for (12.3.1)

1 1 --<u<-2 - - 2'

CHAPTER 12

for u E (-00, - t) U (!, (0) ,

for n = 0,1, ... , which like the Chrestenson-Levy system on the interval [0,1]' is orthonormal on the interval [-1/2,1/2] and complete in the space V( -1/2,1/2). We also form the system of inverse Fourier transforms of this system (12.3.1), i.e., the system

(12.3.2) 1 100 1 11/2 Rn(w) = -- fn(u)e iwu du = -- fn(u)e iwu du, ",j2; - 00 ",j2; - 1 / 2

and establish a number of its properties.

12.3.1. The system Rn(W), n = 0,1, ... , defined by (12.3.1) and (12.3.2) is orthonormal in L2(R) iind complete with respect to the L2(R) norm in the class W 12/ 2 of all functions in VCR) of exponential type with exponent::; 1/2.

PROOF. We shall use the following corollary of Planchercl's Theorem (see [17], p. 442, for example): if 1>1,1>2 E L2(R) and «1>1, «1>2 represent their respective Fourier transforms, then

( 12.3.3)

Applying (12.3.3) to the functions fn(u), fm(u), which are orthonormal on the interval [-1/2,1/2]' and to their Fourier transforms Rn(w), Rm(w), we obtain

for m = n,

for m =f:. n.

This proves that the system {Rn(w)}~=o is orthonormal in L2 (R). Suppose now that R(w) is a function which belongs to the space W;/2' By a the

orem of Paley-Wiener (see [22], p. 26) there exists a function fey) E V( -1/2,1/2) such that

(12.3.4) 1 11/2

R(w)= ftC. f(y)eiwYdy. v27l' -1/2


Moreover, since the system {In(Y)} is complete in the space U(-1/2,1/2), the Fourier series of the function fey) in this system must converge to fey) in the L2( -1/2,1/2) norm, i.e.,

N-1 lim Ilf(y) - L cnfn(y)IIL2(-1/2,1/2) = 0,

N-+OCJ n=O

where

11/2 Cn = f(y)fn(Y) dy,

-1/2 (12.3.5) n = 0,1, ....

But by Plancherel's Theorem,

N-1 N-1 (12.3.6) IIR(w) - L cnR n(w)IIL2(R) = IIf(y) - L cn fn(y)IIL2(-1/2,1/2).

,,=0 n=O

Since R(w) E Wl/2 was arbitrary, it follows from (12.3.5) and (12.3.6) that the system {Rn(w)} is complete .•

Notice by (12.3.3) that

(12.3.7) 11/2 100 Cn = f(y)fn(Y) dy = R(w)Rm(w) dw,

-1/2 -OCJ

n = 0,1, ... ,

i.e., the Fourier coefficients of any function R(w) E l¥l/2 with respect to the system {Rn(w)} coincide with the Fourier coefficients of a function fey) E L2( -1/2,1/2) with respect to the system {In(Y)}, where the functions R(w) and fey) are related by (12.3.4). Consequently, it is possible to use this system as a device for obtaining approximations in various problems connected with the set of functions quadratically integrable on the whole real line which have a finite Fourier transform.

12.3.2. The Fourier series of any function R(w) E Wl/2 witll respect to the system {Rn (w)} converges to R( w) uniformly on the entire real axis and for any w E R,

N-1 1 N-1 1 R(w) - ~ cnRn(w) I::; y'2;"R (w) - ~ cnRn(w)IIV(R).

PROOF. By (12.3.2), (12.3.7), and (12.3.4) we can write

N-1 1 {1/2 . N-1 1/2 . } R(w) - L cnRn(w) = I?= 1 f(y)e'WY dy - L Cn 1 f,,(y)e'WY dy

n=O V £7r -1/2 n=O -1/2

1 1/2 [ N-1 l' = I?= 1 fey) - L cnfn(Y) e'WY dy.

V £7r -1/2 n=O

326 CHAPTER 12

Hence by the Cauchy-Schwarz inequality we conclude that

N-l 1 11/2 [ N-l [2 11/2 R(w) - :E cnRn(w) = tn= fey) - :E cnfn(Y) dy lei "'Yl2 dy

n=O V 211" -1/2 n=O -1/2

1 N-l

= ..n:;lIf(Y) - ~ cnfn(y)IIL2(-1/2,1/2)

1 N-l

= ..n:;IIR(w) - ~ cnR n(w)IIL2(R)' •

We shall now find a closed form for the function Rn(w).

12.3.3. Let n = E!=1 n_vpv-l, k = k(n), n_k oF O. Then

_ ( . w ) k sm-- . 2 1 w 2 v-I 1I"m

R.(w) ~ ~~ ,in 2 .II . (w p. ) exp (---.=") P v=1 sm - + -n-v 2pv p

PROOF. Recall that the function

2 . k 1I"Z"",

Xn(Y) = exp - ~ n-vYv p v=1

is constant on the intervals ht(k) = (llpk, (l + l)lpk), e = 0,1, ... ,pk - 1. Thus

(12.3.8)


pk_1 {iW (e + I _ Ipk) exp iw .. - exp -iv: } 2 '"' - 2 2 2pk 2pk

= tn= ~ Xn(b((k))exp k 2' v~w ~o P I

W 2 sin -. pk -1 . k

_ 2pk '"' (f (k-)) Iw(2e + 1 - P ) - rn-- ~ Xn ve exp . V 27rW (=0 2pk

= f3.~ sin ~S·.(w), V -:; W 2pk k

where

( 12.3.9)

pi_1 . '"' _ iw(2e + 1 _ pk)

Skew) == L.J Xn(be(k))exp k . (=0 2p

To calculate Si,; (w) consider the sum

(12.3.10) O"n,II(W) =

pk-"+t 1 •

L - (f (k"':.)) iw(2e + 1 - pk-II+1) Xn II V( exp . , 2pk

where

k

(12.3.11) Xn,Abe(k)) = II (Xpm-l(y)t- m , v=1,2, ... ,k. m=JI

It is clear that

(12.3.12) O"n,1(W) = Skew).

We shall establish the recursive relation

(12.3.13)

for v == 1,2, ... , k - 1. Changing the index of summation in (12.3.10) by

e == rpk-v + et , r = O,l, ... ,p -1, et = 0,1, ... ,p"'-v -1,

328 CHAPTER 12

we obtain

(12.3.14)

But definition (12.3.11) implies

Therefore,

(12.3.15)

The function Xn,v+l(Y) can be written in the form

Since

T e+8 then Y = - + --- for some 0 < 8 < 1 and we have

pI! pk

(mod p) == [(!...- €~8) m] /I + k P

P P (mod p) == [e; 8 pm] (mod p)

for any m 2: 7/ + 1. This means that for m 2: v + 1 the coordinates Ym of the munber y E bl+ rpk -"

coincide with the coordinates Xm of a number x E be. Consequently,

(12.3.16)


for £ = 0,1, ... ,pic-v -1. Furthermore, since y E fil+rpk-v and e = 0,1, ... ,pk-v_1

Yv == [ypV] (mod p) == [(;v + £; 0) pI'] (mod p)

[c + OJ == r + -_- (mod p) = r, pk-v

then

(12.3.17)

Substituting (12.3.16) and (12.3.17) into (12.3.15) we have

Combining this last identity with (12.3.14), we obtain

k- '" 1 -p - iw(2e + 1 _ pk-v+l)

O"n,v(W) = L exp - X 2pk

(=0

p-l .

x L(XpV-l(firpk-v)t-vxn,V+1(fit)exp z;: r=O

p-l .

= L(XpV-l(fi k_v)t- v exp lwr X r=O rp pI'

This completes the proof of (12.3.13).

330 CHAPTER 12

Write (12.3.13) in the form

O"n,v(W) = O"n,v+l(whv,

and apply (12.3.12). We obtain

v=1,2, ... ,k-1,

k-2 k-I

Skew) = O"n,I(W) = O"n,2(whl = ... = O"n,k-I (W) IT IV = O"n,k(W) IT IV' v=1 v=1

Substituting the value for 0" n,k(w) from (12.3.10) and the value of IV we find

k-Ip-I

IT ~ n_ iw(2r+1-p) = L(Xp.-I(crpo-.))· exp 2 v •

v=lr=O p

But it is easy to see that

~ (iwr 27rin_vr) LexP - + = r=O pV p

1 - exp (~ + 27rin_ v )

( iW 27rin_v ) 1-exp -+---

pV P zw

exp--2pv-l

. W sm--

2pv-1

( iw 7rin_ v ) exp -+--

2pv p sin -+--. (w 7rn_v )

2pv p

Consequently, it follows from (12.3.17) and the previous identity that

S () ITk ~ iw iw(l - p) 27rin_vr k W = L exp ---;; exp 2 v exp

v=1 r=O p p p

k . (1 ) (. .) . sin _w IT zw - p zw zw -7rzn_v 2pv-1 = exp exp -- - - exp

2pv 2pv-1 2pv p. (w 7rn_ v ) v=1 sin - +--

2pv p

k . sin-W-IT -7rzn_v 2pv-1

= v=1 exp p sin (~ + 7rn_ v ) . 2pv p


Substituting this expression for Skew) into (12.3.8), we conclude that

_ . W I! k . SIn--2 1 . W II -7Tzn_'11 2p"-l

Rn(w) = --sm-- exp ( ) .• 7TW 2pk 11-_1 p. W 7Tn_ 1I

sm -+--2p" p

If we set p = 2 in the identity for Rn(w) found in 12.3.3 and use the fact that

sin - = e1rm-. sin - + 7Tn_ W . (W ) 2" 2" II

1rin_ • (W 7Tn_ lI ) (W 7Tn_ lI ) = 2e • sm 2"+1 + -2- cos 2"+1 + -2- , n_ 1I = 0 or 1,

then we see that the function Rn (w) which corresponds to the Walsh system is given by

2k+l 1. W Ilk 7Tin_1I (W 7Tn_ 1I ) Rn(w) = y"27r ~ sm 2k+l 11=1 exp -2- cos 2"+1 + -2- .

In particular, for the case p = 2 the first six functions RnCw) are given by:

D~() 2 1 . W ~t.Q W = --- sIn-,

y"27r W 2

4i 1 . 2 W Rl(W) = ---SIn -,

y"27r W 4

8i 1 2 W W R2( W) = --- sin - cos -,

y'2;w 8 4

-8 1 . 2 W • W R3(W) = --- sm - sm-,

y'2;w 8 4

16i 1 . 2 W W W R4(W) = ---SIn -cos-cos-,

y"27r W 16 8 4

R () -16 1 . 2 W W. W 5 W = -- - SIn - cos - SIn -,

y"27r W 16 8 4

R () -16 1 . 2 W . W W 6 W = ---SIn -SIn-cos-.

y"27r W 16 8 4

We shall give a mathematical formulation of optimization problem concerning discrete radiators of electromagnetic waves, i.e., linear radiators which are the aggregate of elementary dipole radiators, located along the straight line segment -R/2 ~ x ~ R/2. Suppose that to an elementary radiator of length b.x situated in

332 CHAPTER 12

a neighborhood of the points x E [-t'/2,t'/2]' we supply current which changes in time t according to the sinusoidal law

f(x)eiwt[),.x, w = const,

where f(x) is a complex function interpreted in the following way: If(x)1 represents amplitude and arg f(x) represents the phase of the current supplied at the point x. In this case, amplitude of the current emitted at an angle 'I/J depends on the angle 'I/J in the plane passing across the X axis of the electromagnetic field and is related in a unique way to the radiation pattern of the radiators, written in the form (see [29], p. 13)

(12.3.18) 1 jtl2 D('I/J) = - f(x)ei(wxsin 1/-'l/e dx. A -t12

Here 'I/J is the angle between the perpendicular to the X axis and an arbitrary din'ction (-7r /2 ::; 'I/J ::; 7r /2), c = const, A is a normalizing factor which for convenience we shall write in the form A = 2eJ7r/2.

If we let It = (wesin'I/J)/c in (12.3.18) and make the change of variables y = x/e then we obtain

(12.3.19) D( 1/') = D (arcsin :~) = R( u)

R jtl2 = - f(Ry)e iuy dy A -t12

1 lt/2 . = -- f*(y)e IUY dy. J2i -t12

The function R(u) is defined for u E [-wR/c,we/c]. The function r(y) = f(ey) is piecewise constant for y E [-1/2,1/2] since the radiators we arc considering arc discrete. Moreover, r(y) E V( -1/2,1/2) and thus the integral (12.3.19) representing R(u) can be analytically continued to the whole complex plane C. Hence it follows from the Paley-Wiener Theorem that a solution r(y) of the equation (12.3.19), identically zero outside the interval [-1/2,1/2], exists if and only if the function R( u) belongs to the class W;/2' Consequently, we consider the cases R( u) E TV;/2 and R( u) E V (R) separately.

12.3.4. Let R( u) E Wi12' There exist piecewise constant solutions Sn(Y) to equation (12.3.19) which converge to the solution r(y), as n --+ 00, in the L2 [-1/2, 1/2] norm.

PROOF. To approximate solutions to the equation (12.3.19) we apply the method of partial diagrams (see [29], p. 28). This involves expanding the left side of (12.3.19) in a Fourier series with respect tot the system {Rn(u)}:

(12.3.20) 00

R(u) rv LCnRn(U), n=O

cn = 1 R(x)Rn(x) dx.

Notice by (12.3.7) that the series in the system {fn(Y)} with the same coefficients Cn is the Fourier series of the function f* (y) with respect to the system {f n (y )} . Since the system {fn(Y)} = {Xn(Y + 1/2)} is complete, it follows that the series

00

Cn = L R(x)Rn(x) dx (12.3.21) f*(y) '" L cnfn(Y), n=O

converges to f*(y) in the L2 [-1/2, 1/2] norm .• The main advantage of considering questions using the system {Rn(w)} defined in

(12.3.2) is that the functions fn( u) = Xn( U + 1/2) are piecewise constant and therefore the approximate solutions SN(Y) = '£:';01 cnfn(Y) to the equation (12.3.19), being a linear combination of these functions, are also piecewise constant. Hence an approximate solution to the problem of distribution of current which is found using the system {Xn(Y + 1/2)} can be reproduced in practice, i.e., a solution of equation (12.3.19) by the method of partial diagrams, using the system {Rn (w )}, allows for the discrete structure of the radiators.

Notice also that if a solution to equation (12.3.19) is written as a partial sum of some series involving the functions Xn( U + 1/2), for some p, consisting of N terms, pk-l :5 N < pk, then for an exact reproduction of this distribution of currents the radiator must consist of pk elementary radiators of identical length, because this partial sum has pk intervals of constancy.

Suppose now that the given radiation pattern R*(u) E L2(R), i.e., it does not necessarily belong to the class Wi/2' Let S::, represent the set of piecewise constant functions which are identically zero outside the segment [-1/2,1/2] and have m identical intervals of constancy

( 1 k-1 1 k) h = -2 + ----;:;-' -2 + m ' k = 1,2, ... ,m.

12.3.5. Suppose a linear radiator of electromagnetic waves consists of pk elementary radiators of identical length. Among all practical feasible, i.e., those whose distributions of current belong to S~k' there is a distribution j(y) which gives the radiation pattern

- 1 11./2 - . R(w) = /iL f(y)e'wy dy v27r -1./2

is nearest, in the least squares sense, to a given radiation pattern

1 11./2 R*(w) = - f*(y)e iwy dy .

.j2i -1./2

Moreover, the distribution j(y) is defined by the formula

(12.3.22)

334 CHAPTER 12

where

(12.3.23) Cn = L R(x)Rn(x) dx, n=0,1, ... ,pk_1,

and Rn( u) is defined in 12.3.3.

PROOF. We need to find a function iCy) E S~k such that

IIR*(w) - R(w)llv(R) = inf IIR*(w) - R(w)IIV(R), fES O

pk

1 1(/2 . where R(w) = tr>= f(y)e'wy dy. Thus this problem about best approximation

V 27r -(/2

(in the U(R) norm) reduces to specifying a radiation pattcrn with the help of a systcm of pk elementary radiators.

Let f*(y) be the inverse Fourier transformation of the given function R*(w). Since any function from the set SO k can be represented exactly as a linear combination

p

of the first pk Chrestenson-Levy functions Xn(Y + 1/2), n = 0,1, ... ,pk - 1, then using the extremal property of partial sums of Fourier series we have

p'-I

11f*(y) - 2::: CnXn (y + ~) IIV(R) = in~ 11f*(y) - f(y)IIV(R), n=O fESpk

Cn = 11/2 f*(Y)Xn (y + ~) dy -1/2

(recall that the functions Xn(Y + 1/2) have been extended to be zero on the whole real line). Since the Fourier transform is an isometry in L2(R), it follows that

I.e.,

pk_1 p'_1

1If*(y) - ?; CnXn (Y + ~) IIv(R) = IIR*(y) - ?; cnRn(u)IIV(R)

= inf 11f*(y) - f(y)llv(R) fES;.

= inf IIR*(u) - R(u)IIL2(R), fES;.


where by Plancherel's Theorem the coefficients Cn are determined by the given functions by means of the formula

Cn = L R*(u)Rn(u) du, n=O,l, ... ,pk_l.

Therefore, to define the best distribution (in the indicated sense) of currents j(y) in radiators it is sufficient to find the pk Fourier coefficients (12.3.23) and put them in the sum (12.3.22) .•

We shall presently consider the problem of optimizing an initial condition in a diffusion process. First we introduce some more notation.

Using the system {fn(u)}, which is given by (12.3.1), we define functions

(12.3.24) k = 0, ±1, ±2, ... , n = 0,1, ... ,

which evidently coincide with the functions Xn(u + k + 1/2) on the intervals u E [k - 1/2, k + 1/2), and are identically zero outside these intervals.

We shall represent the Fourier transforms of these functions by

(12.3.25) 1 lk+1/2 .

Rn,k(W) = rrc. «I>n,k(u)e'WU du V 27l' k-1/2

1 lk+1/2 . = rrc. fn(u + k)e'WU duo v27l' k-1/2

Making the change of variables u + k = v we obtain

e-iwk 1+1/ 2 ( 1). Rn,k(W) = rrc. Xn V + - e'wv dv.

v27l' -1/2 2

Therefore, we have by (12.3.2) that

(12.3.26)

12.3.6. The system {Rn,k(W)} is orthonormal and complete in the space L2 (R).

PROOF. Orthonormality of the system {Rn,k(W)} follows from (12.3.3) and the fact that the functions «I>n,k(y) are orthonormal in L2 (R).

To prove the system {Rn,k( w)} is complete let R( w) E L2 (R) and c > ° be arbitrary. We shall show that there exists a finite linear combination of the functions Rn,k(W) which is near the function R(w) in the L2(R) norm in the following sense:

M Nk

(12.3.27) IIR(w) - L L Cn,kRn,k(W)IIL2(R) ~ c. k=-Mn=O

336 CHAPTER 12

Let ley) be the inverse Fourier transform of the function R(w) E L2(R). By Plancherel's Theorem, l(y) E L2 (R), i.e., the integral of the square of this function is convergent. Thus there exists a number M depending on c > 0 such that

(12.3.28) + Il(y)1 2 dy::; ~. (1-M - 1/ 2 100 ) -00 M+l/2 v 2

Write the function l(y) in the following form:

00 ley) = L f(yh,

k=-oo

{ ley) where f(Y)k = 0

for y E h, for y ¢ Jk •

Evidently f (y h E V ( .h ) for all k, and thus using the fact that the system { n ,k (y )} is complete in t.he space L2 (Jk), we can choose coefficients Cn.k such that

(12.3.29) Nt

IIf(yh - ?; Cn,kn,k(y)lIv(Jt) ::; ';4; + 2'

But the functions n,dy) vanish off the interval Jk. Consequently,

M Nt

lIi(y) - L L Cn,kn.k(W)lIi'(R) k=-1\1 n=O

M Nk

= L IIf(Y)k - L Cn,kn,k(W)lIi'(JkJ k=-M n=O

(1-M - 1/ 2 100 ) + + Il(y)1 2 dy. -00 M+l/2

Thus it follows from (12.3.28) and (12.3.29) that

M Nt 2 2

' ' ''''''' 2 c c 2 IIf(y) - L... L... Cn,kn,k(W)IIV(R) ::; (2M + 1) 4M + 1 + '2 = c . k=-M n=O

Since the Fourier transform is an isometry in L2(R), we conclude that

M Nk

c 2 IIl(y) - L L Cn,kn,k(W)lIi2(R) k=-M n=O

M Nk

= IIF-1[l(y)]- L L Cn,kF - 1 [n,k(y)]lIi2 (R) k=-M n=O

M Nk

= IIR(w) - L L Cn,kRn,k(W)lIi2 (R) k=-M n=O


where F-1 represents the inverse Fourier transform. In particular, inequality (12.3.27) holds and it follows that the system {Rn,k(W)} is complete on VCR) .•

Let H -y represent the set of functions tP( x) E L2 (R) whose Fourier transforms have the form

(12.3.30) A 2 tP(u) = f(u)e--YU

for some feu) E L2 (R). Let Sm represent the set of piecewise constant functions in L2(R) which have m identical intervals of constancy on each of the segments Jk for k = 0, ±1, ±2, ... , and let S~ represent the set of piecewise constant functions in Sm which vanish outside Jk.

The following theorem gives a solution to the problem mentioned above concerning optimization of an initial condition of a diffusion process.

12.3.7. There is a function tPo(x) in the set Sp~ which is the best approximation in Spm with respect to the L2(R) norm of the initial condition

(12.3.31) u(x, 0) = tPo(x) E L2(R),

such that the solution u(x, t) of the differential equation

(12.3.32) 8il(x, t) 2 8 2u(x, t)

8t = a 8x2 '

satisfying the initial condition (12.3.31) coincides with a given function tPT(X) E Ha2T at time t = T, i.e.,

u(x,O) = tPo(x) and u(x,T) = tPT(X).

Moreover, the function tP~(x) can be written in the form

p~-l

¢~(x) = L Cn,kq,n,k(X), x E Jk , k = 0, ±1, ... , n=O

where

Cn,k = fa Rn,k(W)F[tPT)(w)ea2w2T dw,

q,n,k(X) and Rn,k(W) are defined by (12.3.24) and (12.3.25), and

1 l . F[tPT)(W) = r.c tPT(u)e'WU du v27r R

is the Fourier transform of the function tPT( x).

The solution in 12.3.7 is understood in the distributional sense when the function ¢o(x) satisfies no other condition except tPo(x) E L2(R) ([19], p. 202).

338 CHAPTER 12

PROOF. We shall show that the initial condition ¢>o(x) E L2(R) for which u(x, T) = ¢>r(x) E Ha2r exists and is unique. For this we choose a solution u(x, T) of equation (12.3.32) which satisfies the initial condition ¢>o( x) by using the Fourier transform ([17], p. 408):

2r 2 F[u(x, T)](w) = F[¢>r](w) = e-a W F[¢>o](w).

Since ¢>r( x) E H a2r, it follows that

(12.3.33)

i.e., the function ¢>o(x) as an element of L2(R) can be uniquely determined by using the inverse Fourier transform.

Wri te the functions ¢>o (x) and ¢>o (x) in the form

00

¢>o(x) = L ¢>o(xh, k=-oo

{ Jo(x) where ¢>o( x h = 0

and

00

where ¢>o(X)k = { ~~(X) ¢>~(x) = L ¢>~(x)k, k=-oo

Since 00

k=-CX)

for x E Jk,

for x ~ h,

for x E h, for x ~ h.

the posed problem about finding the best approximation reduces to finding functions ¢>(i(X)k which satisfy the property

We notice that any function from the set S;m is by definition a linear combination of the functions <Pn,k( u) which were defined in (12.3.24). Taking into account the extremal property of the partial sums of a Fourier series, we can write

pm_l

inf //¢>o(xh - ¢>(X)//L2(Jk ) = inf //¢>o(xh - L Cn ,kES;m if>ES;m n=O

But this means that pm_l

¢>~(X)k = L Cn,k<Pn,k(X), n=O


where

for k = 0, ±1, ... , and n = 0,1, ... ,pm -1. Hence to prove 12.3.7 it suffices to find the coefficients Cn,k. Using the definition of the function 4>~(X)k and applying the generalized Parseval identity (12.3.3), we see that

(12.3.34) Cn,k = f 4>O(Xh~n,k(X) dx JJk = fa 4>O(Xh~n,k(X) dx

= fa .r[4>ol(u).r[~n,kl(u)du, where .r is the Fourier transform operator. But by (12.3.26),

(12.3.35)

Hence we obtain the expression of Cn,k introduced above by identities (12.3.33), (12.3.34), and (12.3.35) .•

APPENDICES

ApPENDIX 1

ABELIAN GROUPS

A1.1. A group is a set G together with a binary algebraic operation which satisfies the associative property and for which there exists an inverse operation.

Here we shall consider only groups whose binary operation is commutative. Such groups are called commutative or abelian. The algebraic operation of a commutative group is usually called addition and is denoted by the sign +. On occasion we shall use other notation.

Thus a commutative (or abe/ian) group is a non- empty set G which satisfies the following axioms: a) There is an addition operation defined on G which assigns to each pair a, b in G a third element

a + b in G which will be called the sum of the elements a and b. b) The operation "+" is associative, i.e., for any three elements a, b, c in G the identity (a+b) +c =

a + (b + c) holds. c) The operation "+" is commutative, i.e., for any pair of elements a, b in G the identity a+b = b+a

holds. d) There is a zero element in G, i.e., an element 0 such that a + 0 = a for all a E G. e) For each a E G there is an additive inverse, i.e., an element (-a) such that a + (-a) = O. It is not difficult to verify that the zero element of a group is unique. One can also verify that for

each a E G the inverse element (-a) is uniquely determined (see [18)). If A and B are subsets of a group G then we shall represent the set of sums a + b where a E A

and b E B by A + B. If b is a singleton, i.e., B = {b}, then the set A + {b} will be written as A + b. We shall also denote the set of elements (-a) where a E A by -A.

One of the simplest examples of a commutative group is the set Z of integers where the usual addition of real numbers serves as the group operation.

Examples of finite groups are given by Z(n), where for each positive integer n Zen) is the set of integers 0,1, ... ,n - 1 and the group operation (which we shall denote by EB) is defined to be addition modulo n, i.e.,

aEBb={a+b a+b-n

For other examples of groups, see §1.2 and §1.5.

if a+ b < n, if a +b ~ n.

A 1.2. Subgroups. Equivalence classes. Factor groups. A subset H of a group G is called a subgrollp if it is itself a group with respect to the group operation on G

Let H be a fixed subgroup of a group G. We shall call an element a E G equivalent to an element bEG with respect to the subgroup H if a - b E H. This is an equivalence relation, i.e., it satisfies the following properties:

a) the reflexive property: each element a EGis equivalent to itself (since a - a = 0 E H)j b) the symmetric property: if a is equivalent to b then b is equivalent to a (since if a - b E H then

b - a = -(a - b) E H)j c) the transitive property: if a is equivalent to b, and b is equivalent to c, then a is equivalent to

c (since a - bE Hand b - c E H imply that a - c = (a - b) + (b - c) E H). An equivalence class in G is a set of points which are all equivalent to a single element in G. It

is easy to verify that given two equivalence classes (with respect to the same subgroup H) either these classes are disjoint or they are identical to each other. Consequently, the group G can be decomposed into pairwise disjoint subsets, the eqllivalence classes of the group G with respect to the subgroup H.

341

342 APPENDICES

It is clear that a given equivalence class is completely determined by anyone of its elements a. Indeed, all remaining elements of this equivalence class have the form a + h, where h ranges over the subgroup H. For this reason such an equivalence class is denoted by a + H (see A1.I). Sometimes the equivalence class a + H is said to be generated by the element a. It is clear that an equivalence class is generated by each of its elements.

The set of equivalence classes of a group G with respect to a subgroup H is denoted by G/H. One can define an addition operation on G / H as follows. The sum of two equivalence classes a + H and b + H is the class (a + b) + H. It is easy to see that this sum does not depend on the choice of the representatives a and b of the class a + Hand b + H and that this addition operation satisfies the axioms for a commutative group (see A1.I). Thus the set G/H is a commutative group which we shall call the factor group of the group G with respect to the subgroup H.

A1.3. Isomorphisms and automorphisms. n:anslations. A map q, from one group G to another G I is called an isomorphism if it is one to one and preserves the group operations, i.e., for any elements a, bin G,

q,(a + b) = q,(a) + q,(b),

where ¢(a) + ¢(b) is a sum in the group G I . If G = G I such a mapping is called an automorphism of the group G.

Let a be fixed in some group G. It is easy to verify t.hat the map x -+ x + a is an automorphism on G. It is called trill/slation of the group G by the element a.

For a fixed subgroup Jl the translation x -+ x + a of t.he group G corresponds to a translation of the factor grollp G/ II by the equivalence class a + H. Indeed, for the translation x -+ x + a, all elementoS of a given equivalence class b + ]{ are carried to clements of t.he class a + b + ]{. These element.s can be written in the form (b + II) + (a + II), i.e., the equivalence class a + b + II can be viewed as the image of the class b+ H by the translation of the factor group G/ H by the equivalence class a + H.

ApPENDIX 2 METRIC SPACES. METRIC GROUPS

A2.1. A set X is called a metric space if there is a function p which takes each pair x and y of X to a non-negative real number p(x, y) which satisfies the following conditions:

a) p(x, y) = 0 if and only if x = y; b) p(x,y) = p(y,x) (the symmetric property); c) p(x, y) :::; p(x, z) + p(z, y) (the triangle inequality). The function p(x, y) is called the distance between the clements x and y, or the metric of the

space X. The set of points

U,(xo) = {x EX: p(x,xo) < C},

where Xo is a fixed point in the space X and ( > 0, is called the open ball of radius ( and center Xo or an (-neighborhood of the point Xo.

A point x E X is called a clllster point of a set m C X if every neighborhood U,(x) contains infinitely many points of M.

A set !If C X is called closed if it contains all of its cluster points. Denote the union of a set 111 C X and the set of cluster points of 111 by M. Then M is a closed

set and is called the closure of the set 111. The set M is called dense in X if M = X. A point x is said to be interior to the set 11,[ if there is a neighborhood U,(x) which is contained

inM.

APPENDICES

A set is called open if all of its points are interior to it. The following result is well-known (see [17)):

A2.1.1. A set M is open if and only if its complement X \ M is closed.

343

The following theorem describes the situation for open subsets of the real line with respect to the usual metric p(z, y) = Iz - YI (see [21]):

A2.1.2. Every open subset of the rea1line can be written as a finite or countably infinite union of pairwise disjoint open intervals (i.e., sets of the form (a,oo), (-00, a), (-00,00), or (a,b»).

Combining this result with A2.1.1, we obtain the following corollary:

A2.1.3. Each closed subset E of the real line can be written as the complement of a finite or countably infinite union of pairwise disjoint open intervals.

These intervals are called the contiguous intervals of the set E. The collection of sets which can be obtained from a countable (Le., finite or count ably infinite)

combination of unions and intersections of open or closed sets is called the collection of Borel sets. A subset M of the space X is called compact if given any open covering {Aa} (i.e., given any col

lection of open sets {Aa} which satisfy M C Ua Aa) there is a finite sub collection Ak) ,AI:., ... ,AkN which also covers M.

If the set X is compact then it is called a compact space. A space X is called locally compact if each of its points z has a neighborhood U,(z) whose closure

is compact. A sequence {Zn} of elements of a space X is said to converge to an element Z E X if p(z, zn) --t 0

as n --> 00. A sequence {Zn} of elements of a space X is called Cauchy if given f> 0 there is a number nO(f)

such that p(zm' Zn) < f for all n, m ~ no(c). A metric space is said to be complete if each of its Cauchy sequences converge.

A2.2. A group G is called a metric group if it is a compact group and a metric space, and the group operations are continuous with respect to th.e metric in the following sense:

a) If z = Z + y then given any neighborhood U(z) of z there exists neighborhoods U(z) and U(y) of the points Z and y such that

U(z) + U(y) C U(z).

b) Given any neighborhood U(-z) of the element -z there exists a neighborhood U(z) of the point Z such that -U(z) C U(-z).

The concept of a metric group is a part.icular case of the more general idea of a topological group (see [23)).

ApPENDIX 3 MEASURE SPACES

A3.1. A class A of subsets of a set X is called a O'-algebra (or countably additive) if 1) 0 E A, where 0 represents the empty set; 2) if A E A then X \ A E ..4; 3) if {An} is any countable collection of sets with An E A then Un An E A. A set function Jl(A) defined on all sets in some 0'- algebra A is called a measure if a) 0 S Jl(A) S +00 for all A E ..4; b) Jl(0) = 0;

344 APPENDICES

c) the identity 00 00

Jl( U An) = L p(An) n=1 n=1

holds for all sequences {A n }::"=! which satisfy An E A and An nAm = 0 for n "I m. A measllre space is a set X together with a IT-algebra A and a measure Jl on A.

A3.2. Measure on the real line. The measure most frequently used on the real line is Lebesgue measure. It can be defined in the following way.

First define the ollter measllre of any subset A of the real line by

(A3.2.1 ) mes'(A) = inf{L IInl: UIn:> A}, n n

where each In is an interval and IInl represents its length. It is clear that mes'I = III for any interval I.

A3.2.1. A set A is called Lebesgue measllrable if given any ( > 0 there is a closed set F and an open set Q such that. Fe A c Q and mes'(Q\ F) < (.

A3.2.2. The collecUon of Lebesgue mea.~urahle sets is a IT-algehra which contains the Borel sets and the restriction of meso to this collect.ion is a mea.sure a.~ defined in A3.1.

This measure is called Lebesgue measure on t.he real line and will be denoted by mes, i.e., the Lebesgue measure of a measurable set A is defined to be mes A = mesO A. (See [21]).

A3.2.3. If a set A is measurable then

mes A = inf mes Q = sup mes F,

where the infimum is taken over all open sets Q lI'hich satisfy A C Q and the supremum is taken over all closed sets F which satisfy F CA. (Sec [21)).

A3.3. Lebesgue-Stieltjes measlll·e. Each function 1/'(x) which is c071tillIlOIlS from the right and 71071- decreasi7lg on (-00,00) gives rise to a generalization of Lebesgue measure on the real line in the following way.

Define a function m on int.ervals (a, b) by

m(a,b) = 1I'(b-) -1/,(a).

This function generalizes the concept oflength since when 1/;(x) = x we have m(a, b) = b-a = I(a, b)l. Substitut.ing III In for IInl in formula (A3.2.l), we define an out.er measure mes;j, and as in A3.2.1,

we form t.he class of measurable sets with respect t.o this out.er measure and obtain a measure mes"" which is called the Lebesglle- Stielfjes measure generat.ed by t.he fundion 1/;.

We notice by construction that

(A3.3.1) {

mes",(a, h) = m(a, b) = 1/,(b-) -1/'(a),

mes", [a, h) = 1/;( h) - 1/'( a- ), mes",(a, b) = 1/'(&) -1/;(a),

mes.p[a, b) = 1/;(b-) -1/'(a-).

In particular, the measure of a point satisfies mes",{a} = 1/,(a) - 1/;(a-). In case t.he function is continuous from t.he left, instead of cont.inuolls from the right, we can define

the function 111 by

APPENDICES 345

With slight changes, the program outlined above can be carried out for this case as well. A corresponding analogue of (A3.3.1) holds in this case. For example,

(A3.3.2) mes", [a, b) == 1/J(b) -1/J(a).

A3.4. Lebesgue measure on the plane can be defined as it was on the line with only one change: III

formula (A3.2.1) use two-dimensional intervals in place of the In's, i.e., open rectangles whose sides are parallel to the axes, and let IInl represent the area of the rectangle In.

A3.5. Haar proved the following theorem concerning construction of measure on groups (see [13]).

A3.5.1. On any locally compact group (see A2.2) there exists a measure J1, defined on the u- algebra of Borel subsets of this group, such that

a) J1(A) > 0 for any non-empty open set A; b) J1( a + A) == J1( A) for any Borel set A (this property is referred to as the translation invariance

of the measure J1); c) J1( -A) = J1(A).

Such a measure is called Haar measure on the group G. Lebesgue measure on the real line R provides a simple example of Ham measure. Here the group

structure is provided by the usual addition of real numbers.

ApPENDIX 4 MEASURADLE FUNCTIONS. THE LEDESGUE INTEGRAL

A4.1. Let X be a measure space and J1 be a measure on some u-algebra A of subsets of X. In this case we shall refer to the elements of the class A as A-measurable sets.

A function f defined on X is called A-measurable (or II-measurable) if for every real number C the set {x : f(x) > C} is measurable. If in place of t.he collection A we use t.he collection of Lebesgue measurahle sets (see A3.2), t.hen we obtain the definition of a Lebesgue measurable function.

A function f defined on a metric space X is called lower semi-continuous on X if for every real number C the set {x: f(x) > C} is open.

Two functions f and g defined on the same measurable set E are called equivalent if

J1{x: f(x) f. !lex)} == o. In connection with this recall the following terminology. If some property holds on the set E, except possibly on some set of measure zero, then we say

that this property holds almost everywhere (abbreviated as a.e.). Thus we see that two functions are equivalent if they coincide almost everywhere.

A4.1.1. Egoroff's Theorem. Let E be a set of finite J1 measure and let Un(x)} be a sequence of measurable functions which converges alIna>t everywhere on E to a function f(x). Then for any t> 0 there exists a measurable set E, C E such that J1(E,) > J1(E) - t and such that the sequence Un(x)} converges uniformly to f(x) on the set E,. (See [17)).

In the ca.~e of Lebesgue measurable functions, the set E, can be chosen to be closed.

A4.1.2. Lusin's Theorem. Let f be a function defined on an interval [a,b]. Then f is Lebesgue measurable if and only if for each t > 0 there exists a closed set F, C [a, b] and a function <J> continuous on [a,b] such that mesF, > (b - a) - t and f(x) == <J>(x) [or x E F,. (See [21]).

A sequence of measurable functions {fn(x)} defined on some measurable set E is said to converge in measure to some measurable fuuction f(x) if limn _ oo J1{x : Ifn(x) - f(x)1 > t} = 0 for every t> O.

We notice that every sequence which converges almost. everywhere also converges in measure (see [21]).

346 APPENDICES

A4.2. The Lebesgue integral. Let X be a measure space and p. be a measure on some .,.-algebra A of subsets of X. Let XA(X) denote the characteristic function of the set A, i.e.,

A function of the form

(A4.2.l)

for x E A,

for x rt. A.

EaiXA,(X), i=O

where {Ad is a sequence of pairwise disjoint sets, is called a simple function. It is easy to verify that a simple function is measurable if and only if all the sets Ai which appear in its definition are measurable.

A simple measurable function f of the form (A4.2.1) is called Lebeo.gue integrable (or summable) on a set A = Ui Ai if the series Li ail/(A;) colltlcrges absolutely. In this case we define the Lebesgue illtegral of the function f to be the sum of this series, i.e.,

1 f dlt == E ail/(A;). A i

An arbitrary function I is called Lebesgue jlltegmbie (or summable) on a set A of finite measure if there exists a sequence {fn} of simple funct.ions snmmable on A which converge uniformly to f. In this case we define

1 f dJl == lim 1 In dp.. A n-oo A

One can show (see (17]) that the value of the integral so defined does not depend on the choice of the sequence Un}.

Among t.he collection of iI/finite measures we restrict our attention to .,.-finite measures, i.e., those measures whose space X can be written as a COUll/able union of sets of finite p.-measure:

(A4.2.2) n

A monotone increasing sequence {Xn} of measurable subsets of the space X which satisfies condition (A4.2.2) will be called an exhausti-lle sequellce.

A function f defined on a space X wit.h .,.-finite measure II is called sum mabie on X if it is summable on each measurable subset A C X of finite measure and for each exhaustive sequence {Xn} the limit

lim r fdp n-oo}x ...

exists and does not depend on the choice of this sequence. This limit is called the integral of the function I on X and is denoted by fx f dp.

Thus we have defined an integral for each space with a .,.- finite measure. In the case of a concrete measure, this definition gives rise to a concrete form for the integral. The classical Lebesgue Integral on the real line or in the plane is obtained if the const.ruction ca.rried out above is applied to the measures introduced in A3.2 and A3.3. In this context., the Lebesgue int.egral corresponding to the measure "mes" is usually written in the form fA I dx rat.her tha.n fA I d meso

APPENDICES 347

A4.3. Functions of bounded variation. The Lebesgue- Stieltjes integral. A function 1 defined on an interval [a, b] is called a function of bounded variation if there exists a constant C such that for any partition a = Xo < x! < ... < Zn = b of the interval [a, b] the inequality

n

L If(xi) - f(xi-dl < C i=l

holds. The quantity n

V;[J] == sup L If(zi) - f(Xi-1)1, ;=1

where the supremum is taken over all partitions of the interval [a, b], is called the total variation of the function f on the interval [a, b].

We shall list a number of properties concerning functions of bounded variation and their total variation (see [17]).

A4.3.1. Ifa < b < c then V;[J] + VnJ] = Vi[!].

A4.3.2. For any numbers 0< and fJ we have

A4.3.3. Each monotone function 1 is a function of bounded variation and V;[J] = I/(b) - f(a)l.

A4.3.4. If f is a function of bounded variation then the functions v(z) == Va"'[J] and v(z) - fez) are monotone non-decreasing. Moreover,

fez) = v(z) - (v(x) - fez»~,

i.e., each function of bounded variation can be written as a difference of t wo monotone non-decreasing functions!.

A4.3.5. Every function of bounded variation has a finite derivative almost everywhere.

A4.3.6. If f is a function of bounded variation then

dv,,"'[J] = If'(x)1 dz

holds almost everywhere. (See [25], p. 121.)

A4.3.7. Let'l/>(z) be a function of bounded variation on an interval [a, b]. Then

(A4.3.1)

for almost every z E [a, b].

dVa"'['I/>(t) - 'I/>'{z)· t] = 0 d:c

PROOF. Let A be the set on which the derivative 'I/>'(z) does not exist. By A4.3.5, mesA = O. Fix a real number c and apply A4.3.6 to see that

(A4.3.2) dV."'['I/>(t) - ctj = "I,'(z) _ cl dz

1 H the function f is continuous at a point Xo from one side or the other, then the same i. true of the functions vex) and vex) - lex).

348 APPENDICES

almost everywhere on [a,b]. Let Ec denote the subset of [a,b] on which (A4.3.2) fails to hold. Then we have mes Ec = 0 for each c.

Let {Tn} be the set of rational numbers. Set

Clearly, mes E = O.We shall prove that (A4.3.1) holds everywhere outside of E. Indeed, suppose :e ¢ E. Then vJ'(:e) exists. Let ( > 0 and choose a rational number Tn such that

(A4.3.3)

Then by properties A4.3.2, A4.3.3, and inequality (A1.3.3) we have

(A4.3.4)

kV:+h[vJ(t) - vJ'(:e)· t]::; kV:+h[if,(t) - Tnt] + kV:+h[(Tn - vJ'(:e))t)

1 ::; hV:+h[.p(t) - Tnt] + I'·n - vJ'(:e)1

1 ::; hV:+h[1/,(t) - '·n t] + (.

Since :e ~ E we have :e ¢ Ern. Thus applying (A4.3.2) for the case C = Tn we see that if h is sufficiently small then

I~V:+h[1/,(t) - ,·"I]-IVi(:e) - Tnll < f.

Thus it follows from (A4.3.3) that

1 - Vr+h[.I,(t) - ,. t] < 2( h x 'f n' - .

Putting this estimate into the right side of inequality (A4.3.4), we conclude that

for h sufficiently small. In particular (A4.3.l) holds .•

A4.3.8. Let vJ be a function of bounded variation, continuous (for example) from the right, and written (by A4.3.4) as a difference of two monotone non-decreasing functions which are also continuous from the right:

Form the Lebesgue-Stieltjes measures (see A3.3): mes,i't and mes"". Then the Lebesgue-Slieltjes integral with respect to the fUllction 1/, is defined by the formula

1b f(:e)dvJ(:e) == 1b f(:e)dmes"" -lb f(:e)dmcs.p,.

A funct.ion f is considered integrable with respect 10 1/' if both integrals on the right side of this formula exist and are finite. It is easy to verify that the value of this integral does not depend on the choice of the representation of 1/J as a difference of two monot.one functions.

APPENDICES 349

A4.3.9. The Lebesgue-Stieltjes integral satisfies the following inequality (see [16], p. 206):

Il l(x)dlP(x)I ~ ll/(X)ldVaX[IP].

A4.4. We shall list several well-known properties of the Lebesgue integral (see [17), [21]).

A4.4.1. The Lebesgue Dominated Convergence Theorem. If a sequence of measurable functions {In} converges in measure on a measurable set A (in particular, if it converges a.e.) to a function I and if there exists a function </>, summable on A, such that

I/n(x)1 ~ </>(x)

for all n and all x E A, then the limit function I is also summable on A and

lim [ In dp. = [ I dp.. n_oo JA JA

A4.4.2. Levy's Theorem. Suppose that A is a mea.5urable set and

h(x) ~ hex) ~ ... ~ In(x) ~ ...

is a sequence of summable functions whose integrals are uniformly bounded on A, i.e.,

LIn d/l ~ J(.

Then the limit I(x) = limn~oo In(x) is almost everywhere finite on A and

lim [ In dll = [ I dp.. n_co iA iA

A4.4.3. A corollary of A4.4.2. If

IPn(x) ~ 0, and ~ L IPn dp. < 00

for almost every x E A, then the series L:;:"=l1j'n(x) converges almost everywhere on A and

A4.4.4. Fubini's Theorem. Let p. represent two- dimensional Lebesgue measure (see A3.4) and let P.x, Jly represent one-dimensional Lebesgue measure on, respectively, the X -axis and the Y-axis. Let I(x, y) be a function integrable with respect to the measure Jl on the square [a, b] x [a, b]. Then

1 l(x,y)dJl = 16 (16 l(x,y)d/lY ) d/l x = 16 (16

I(X,y)d/lx) dlly. [a,6Ix[a,6l a a a a

(A4.4.1 )

In this situation, the inner integrals exist for almost all values of the variables and are themselves mtegrable with respect to the remaining variable.

On tIle otller hand, if even one of the integrals

exist, then I(x,y) is integrable on the square and (A4.4.1) holds.

A4.4.5. For every function I summable on [a,b] the function F(x) = J: I(t)dt is a function of bounded variation and rex) = I(x) almost everywhere on [a,b].

A proof of the following theorem can be found in [21], p. 461:

350 APPENDICES

A4.4.6. Let f be summable on [a,b]. Then given c > 0 there is a lower semi-continuous function u(z) on [a, b] (see A4.1) which satisfies the following properties:

a) u(z) > -00 everywhere on [a,b]; b) u(z) ~ f(z) everywhere on [a,b]; c) u(z) is summable on [a, b] and

b 16 1 u(z)dz < c+ a I(z)dz.

ApPENDIX 5 NORMED LINEAR SPACES. HILBERT SPACES.

AS.1. A set X is called linear (or a vector space) if it satisfies the following conditions: a) X is an abelian group (see Al.l)j b) there is a scalar product which takes a number a and an z E X to an element az E X such

that bI) a(j3z) = (aj3)z for all numbers a, j3 and all z E X; b2) a(z + V) = az + av, (a + f3)z = az + j3x for all numbers a, j3 and all z,v E Xj b3) I . z = z for all z EX. If the scalar product is defined for all complex scalars then the space is called a complex linear

space and if it is defined for all real scalars then the space is called a real linear space. A normed linear space is a linear space X toget.her with a funct.ion IIzll, called the norm of X,

which takes the set X into the real line and satisfies t.he following conditions (the norm axioms): I. IIxll ~ 0 and IIxll = 0 if and only if x = O. II. lIaxll = lalllxll· III. IIx + vII ~ IIxll + IIvll· Every normed linear space is a metric space where the distance between two elements x, V E X is

defined to be p(x, V) = liz - vII. Hence all the concepts which were introduced in A2.1 for metric spaces also make sense for normed linear spaces, in particular, the idea of completeness. A complete normed linear space is called a Banach space.

AS.2. An important example of a Banach space is provided by the space of continuous bounded functions defined on some metric space X with the usual function addition and scalar multiplication of functions for the linear space structure and

IIfll = sup If(z)1 .,ex

for the norm. This space is denoted by C(X). Other examples include the various illtcgrobility spaces ollunctions with the integral norms. Let

X be some space with a measure p. The collection of all functions sum mabIe on X form a linear space under usual function addition and scalar multiplication of functions. For any 1 ~ p < 00 we consider the set of functions f measurable on X such that the power 1111' is summable. It is easy to see that the quant.ity

11/111' == {[ 1111' dP} 1/1'

satisfies axioms II and III of a norm. IIowever, axiom I can fail since from the definition of the Lebesgue integral it easily follows that changing a function on a set of measure zero does not change the value of its integral. Hence in order for axiom I to hold it is necessary that functions which are equivalent to each other on X (see A4.1) are considered equal as elements of these integrability

APPENDICES 351

spaces. In particular, we consider any function which is almost everywhere zero to be the zero element of these spaces. We shall let LP(X) represent the space determined by the norm Ilflip .

In the case p = 00, the norm is defined somewhat differently. Namely, let

Ilflloo = inf{a : If(x)J ::; a almost everywhere on X}

and define Loo(X) to be the space of equivalence classes of funct.ions for which the quantity IIflloo is finite. This quantity is a norm for this space.

We shall use these spaces only in the case when X is an interval of real numbers, either [a, b] or the positive real axis [0,(0). Thus we shall denote these spaces by LP[a,bJ or LP[O,ooJ.

Here are some of the fundamental properties these spaces enjoy.

A5.2.1. The space of continuous functions is dense ill the spaces LP[a, bJ for any 1 ::; p < 00. (See [2])

A5.2.2. Holder's inequality. Iff E LP[a,b], p > 1 and 9 E Lq[a,b], where! +! = 1 then the p q

product fg is summable on [a, b] and

(see [21], p. 214).

In the case when p = q = 2 this inequality is called the Cauchy- Schwarz integral inequality to distinguish it from the well-known Cauchy-Schwarz inequality for sums:

A5.2.3. For the spaces LP[a,b], p 2': 1, axiom III of normed linear spaces (see A5.1) becomes

(b) IIp ( b ) IIp ( b ) IIp

I. lJ(x)+g(x)JPdx ::; I. Jf(x)J1'dx + I. Jg(x)JPdx

and is called the Minkowski inequality. (See [2].) There is a similar inequality for iterated integrals which is called the generalized Minkowski inequality (see [30], Volume I, p. 19). It is

( bid IP ) IIp d ( b ) IIp

I. 1 f(x,y)dy dx ::;1 I.lf(x,Y)JPdx dy,

where the function f(x, y) is summable on the rectangle [a, bJ x [c, d] and p 2': l.

A5.2.4. If a sequence {fn(X)} converges ill the L[a,bJ norm to a function f(x) then there is a subsequence {fn.(x)} which converges to f(x) almost everywhere. See [17].)

A5.3. A linear functional on a linear space X is a real or complex-valued function F(x) which satisfies F(x + y) = F(x) + F(y) for all x, y E X and F(ax) = aF(x) for all x E X and all numbers a.

For a normed linear space X we consider only those linear functionals F(x) for which the following quantity is finite:

(A5.3.1) JJFJJ = sup IF(x)l· 11"'119

This quantity is ca.lled the norm of the linear functional F. Such fUllctionals are ca.lled continuous (or bounded) since the quantity in (A5.3.1) is finit.e if and only if the function F is continuous on X (see [17]).

352

A5.3.I. IfC> 0 is a constant and F is a linear functional which satisfies

then the functional F is continuous and 11F1I:5 C. See [17], p. 176.)

A5.3.2. For each fixed function f E IJ'[a,b] the map

F(g) == l f(x)g(x)dx

is a linear functional on the space Lf[a, b] where! + ! = 1. Moreover, IIFII = IIflip . p q

APPENDICES

A5.3.3. The Banach-Steinhaus Them·em. Suppose that {F,,} is a sequence oflinear functionals on some Banach space X which is bounded at each point x EX. Then the sequence of norms of these functionals {llFnlll is also bounded.

A5.4. A Hilbert space is a complex linear space H together with a map (x,y), called an inner product, which takes each pair of elements x, y E H to a complex number and satisfies the following conditions:

a) (x, y) = (y, x) (in particular, (x, x) is always a real number); b) (Xl + X2, y) = (Xl, y) + (X2' V); c) (ax, y) = a (x, y) for all complex numhers a; d) (x,x) ~ 0 and (x,x) = 0 if and only if x = O. It is easy to see that the number IIxll == ~ satisfies the norm IIxioms (see A5.1). It is also

required that H be complete with respect to this norm, i.e., that II is in particular a Banach splice. Some authors also require that a Ililhert space be infinite dimensiollll/. If the given linear space is real then in a similar way we can define a real lIilbert space. In this

case the scalar product is assumed to be real for all elements. The space L2(a, b) is an important example of a (real) IIilbert space and its inner product is given

by

(J,g) == l f(x)g(x)dx.

The Lebesgue integral of a complex-valued function q, = u + iv is defined by the identity

Using this integral we can define a complex Hilbert splice L2(a,b) by using the inner product

(J,g) == l f(x)g(x)dx.

A system 1'1, e2,"" en, ... of elements from a Hilbert space H is called orthonormal if

if i = j, if i # j.

An orthonormal system {e;} is called complete if there is no non-zero element x E H which satisfies (x,e;) = 0 for all i = 1,2, ....

An orthonormal system {ei} is called closed if the set of aU finite linear combinations of the elements of this system is dense in H.

APPENDICES 353

S.4.1. Let {ej} be an ortllOnormal system. Tllen {ej} is complete if and only if it is closed. (See [15].)

For any element x E H the numbers Ci = (x, ei) are called the Fourier coefficients of the element x with respect to the system {ej} and the series L~l Cjej is called the Fourier series of the element x in this system.

AS.4.2. Let {ei} be a complete orthonormal system in a Hilbert space H. Then the Fourier series of any element x E H converges to it in the norm of H and its Fourier coefficients satisfy Parseval's identity:

00

(A5.4.1) LC; = Ilxll2. i=l

(See [17]).

AS.4.3. The Riesz-Fischer Theorem. Let {ej} be any orthonormal system in a Hilbert space H and let Cl, C2, •.• be any sequence ofnumbers such tllat the series L~l c; converges. Then there is an element x E H such that the sequence {cd is the Fourier coefficients of x, i.e., Cj = (x, ei). Moreover, if {ei} is complete, then Parseval's identity (A5.4.l) holds. (See [17].)

The Hausdorff-Young-F. Riesz Theorem. Let {,pn(t)} be a uniformly bounded, orthonormal system in tIle space L2(a, b), i.e., an orthonormal system which satisfies l,p,.(t) 1 ~ M for alIt E [a,b],

II = 1,2, ... , and some fixed constant M. If! + ! = 1 and 1 n} satisfy the condition

2. If a sequence of numbers {cn} satisfies the condition

then tllere exists a function I(t) E Lq(a, b) such that {cn } are its Fourier coefficients and

(See [2], Vol. I, p. 218 .)

AS.6. Mercer's Theorem. If {,pn(t)} is a uniformly bounded, ortllOnormal system in the space L2[a, b] then the Fourier coefficients of any summable function with respect to this system converge to zero. (See [2], Vol. I, p. 66 .)

We notice that Theorem 2.7.3 is a special case of this general theorem.

COMMENTARY

In this commentary we include a very brief hist.orical discussion of t.he themes we have touched upon, some references to scholarly articles from which proofs were adapted which we used in this book, and supplemental information for some of the theorems. For further information about these topics we refer the reader to the articles of BalaSov and Rubinstein [I], Wade [4], and the survey found in the monograph [1].

CHAPTER 1

The Walsh functions were first defined by J.L. Walsh Pl. The enumeration we used here was given by Paley [I]. For other enumerations see nala.~ov and Rubinstein [I] and Schipp [3]. The idea of viewing the Walsh system as a special case of systems of characters on zero-dimensional groups was first int.roduced by Vilenkin [I], and somewhat lat.er by Fine [I]. For more about the topological structure of the group G, on which the Walsh funct.ions are defined and continuous, see (1] and [23]. The concept of the modified interval first appellred in the work of Sneider [3]. The connection between the Walsh system and the IIaar system (sec §1.3) was first. not.iced by Kaczmarz (1); see also [Ui]. Estimate (1.4.16) for the Walsh-Dirichlet kernel WIlS obtained by Sneider [1] and Fine [I). The mult.iplicative systems introduced in § 1.5 are a special case of a more general collection of systems which Vilenkin [I) considered. For the case Pj = P =F 2, where {pj} is the sequence (1.5.1), these generalized Rademacher and Walsh functions can be found in the work of Levy [I) and Chrestenson [I). The systems defined in (1.5.6) and (1.5.10) were considered by Price [I] and Efimov [I).

The Walsh system with continuous index set was introduced by Fine (2). The groups G(P) introduced in §1.5 are examples of locally compact zero-dimensional groups whose characters were studied by Vilenkin (3) in the general case; see also (1). Extension of the system of Price to a continuous index set., i.e., the functions (1.5.35), were considered by Efimov and Karakulin [I). under condition (1.5.35), and in the general case by Pospelov F]. lIe also considered the functions (1.5.27) on the group G(P). See also Efimov and Pospelov [I). Relationship (1.5.38) and Proposition 1.5.6 are due to Bespalov.

CHAPTER 2

Propositions 2.1.1, 2.1.2 and formula (2.1.9) hold in general for all Fourier t.ransforms on character groups (see [I], [24)). The Lebesgue constants for the Walsh system were in fact estimated by Paley [I). They were studied further by Fine [I] and Sneider [3). The analogue of the Dini- Lipschitz Theorem for the Walsh system (see 2.3.6) was proved by Fine [I]. The modulus of continuity (2.3.5) was introduced in the general case of a zero-dimensional group by Vilenkin [1). and the modulus (2.5.1) for the space LP by Morgenthaler [1). Theorem 2.4.1 was established by Onneweer [I). The analogue of Dini's test 2.5.7 was proved by Fine [I). Theorem 2.5.12 can be found in the work of Kaczmarz PI, and convergence oft.he sums S2' (x, f) at points of continuity of the function f was not.iced already in the work of Walsh [I). For furt.her information concerning completeness and closure of the Walsh system see the more detailed account in [15]. Elementary estimates of WalshFourier coefficients were obtained by Vilenkin [I] and Fine [I). Concerning 2.7.6 through 2.7.8 see Yoneda [1], also Bljumin and Kotljar [1), McLaughlin [I], Vilenkin and Rubinstein [I). Theorem 2.7.9 was proved by Fine (1). We notice that Bockarev [1) established the finiteness of condition (2.7.11) in Theorem 2.7.10. Concerning Theorem 2.7.10 see also Vilenkin and Rubinstein [1). In §2.8 we have given only a very elementary introduction to Fourier series with respect to multiplicative systems. For more details see [1).

354

COMMENTARY 355

CHAPTER 3

The term by term integral tf;(x) of a Walsh series was used by Fine [1] to study properties of general Walsh series. This series plays the same role for Walsh series that the Riemann functions plays for the study of general trigonometric series (see [2], Vol. I, p. 192).

Propositions 3.1.2 and 3.1.3 can be obtained for Walsh series from corresponding results from the theory of Haar series (see Skvorcov [1]). The idea of using upper and lower derivatives through nets and continuity through nets also is borrowed from the theory of Haar series (see Skvorcov [1][3]). The generalized Riemann-Stieltjes integral which was introduced in §3.1 is a very special case of the Kolmogorov integral (see Kolmogorov [1]).

The study of uniqueness for Walsh series is heavily influenced by the well developed theory of uniqueness of trigonometric series (see [2], Chapter XIV). A simple form of uniqueness for Walsh series, namely 3.2.6 in the case that the excluded set E is empty and the limit function f(x) = 0 everywhere, was proved by Vilenkin [1]. In the case when f(x) E L[O, 1) and E is countable, Theorem 3.2.6 was established by Fine [1]. In the case when E is a countable set, it was obtained by Arutunjan and Talaljan [1] as a corollary of an analogous theorem for Haar series, and (by another method) by Crittenden and Shapiro [1]. The proof given here and the more general one for 3.2.5 are due to Skvorcov [4]. For other results concerning uniqueness see Arutunjan P]' Skvorcov [4], [1], [8] and Wade [1], [3], and [5].

The formal product introduced in §3.3 was first mentioned by Sneider [2]. He also proved Theorem 3.3.2. Concerning 3.3.3 see Skvorcov [5]. The first example of a null series in the Walsh system and the first example of an M- set of measure zero were constructed by Sneider [2] and Coury [1]. The first example of a perfect M-set of measure zero is contained in the work of Skvorcov [5]. The example which was constructed for the proof of 3.4.2 is a simplified version of an example found in the work of Skvorcov [6]. An uncountable U-set for the Walsh system was constructed by SneIder [2]. For further results concerning U-sets and lit-sets for the Walsh system see Skvorcov [9], [11], Wade [2], [4], and the survey in the book [1].

CHAPTER 4

More details about linear methods of summation can be found in [8]. The kernel Kn(x) for the (C, 1) method was studies by Fine [1]. In particular, he discovered formula

(4.2.10). Formula (4.2.6) and Theorem 4.2.2 were proved by Yano [1], [2]. The proof of the theorems in §4.3 can be found in the article of Skvorcov [10] which deals with more general questions. Theorem 4.3.2 about uniform (C, 1) summability of the Walsh- Fourier series of a continuous function was proved by Fine [1] and earlier for character systems on zero-dimensional groups by Vilenkin [1]. Theorem 4.3.3, for 0 < a < 1, was proved by Yano [2], and for a = 1, by Jastrebova [1]. For analogous results in the multiplicative case see Efimov [1], Bljumin P], [2]. Theorem 4.4.5 was proved by Fine [3]. Theorem 4.4.4 was formulated without proof in the work of Fine [4] and the complete proof, as well as that of 4.4.3, appears here for the first time.

CHAPTER 5

The results about sublinear operators which appear in §5.1, in particular Theorem 5.1.2, appear in a more general form in the monograph [30], Vol. II, Chapter 12. The dyadic maximal operator (5.2.1) is a special case of maximal operators with respect to differential bases (see [7]). The modified Dirichlet kernel (5.3.1) is in some sense analogous to the conjugate Dirichlet kernel for the trigonometric case. It was introduced and used by Billard [1]. The part of Theorem 5.3.2 which deals with strong type inequalities for the operators Sn was proved by Paley [1]. That weak type inequalities were proved by Watari Pl. For analogous results for multiplicative systems see Young [1].

356 COMMENTARY

CHAPTER 6

The Fourier transform for generalized Walsh functions (6.1.1) (which in this case also coincides with (6.1.2)) was introduced by Fine [2], and for locally compact, zero-dimensional abelian groups, by Vilenkin [3].

For symmetric P, i.e., the case when P-i = Pi, j = 1,2, ... , Propositions 6.1.3, 6.1.4, 6.1.5, 6.2.2, 6.2.3, 6.2.4, and 6.2.9 were formulated in the work of Efimov and Zolotareva [1]. Propositions 6.1.7,6.3.1,6.3.2 were proved by Bespalov [1]. We notice that using the Plancherel identity 6.2.4, Proposit.ion 6.1.7 can easily be established using t.he interpolation theorem of Riesz- Thorin (see [13], pp. 227-229). Propositions 6.2.5 and 6.2.6 were proved in the work of Efimov and Zolotareva [2], Propositions 6.2.10, 6.2.11, 6.2.13 were proved by Zolotareva [1], and Propositions 6.2.12, 6.2.14, 6.2.15 were proved by Bespalov [3].

CHAPTER 7

Theorem 7.1.1 about uniform convergence on each interval (6,1),0 < [, < 1, of Walsh series with coefficient.s which decrease monotonically to zero evident.ly was first proved in the work of Sneider [1]. The theorems about integrability, in some sense, of the sum of a Walsh series (including the A-integral and the various classes LP(O, 1),0 < P < 1) were proved in the work of Rubinstein [1]. lIe also const.ructed the first example of a Walsh series with coefficients which decrease monotonically to zero whose sum is not Lebesgue integrable on (0,1). An analogue of Theorem 7.1.3 for the Kaczmarz enumeration was proved by Balasov [1] and Theorem 7.1.4 is due to Moricz [1].

The main result of §7.2 (Theorem 7.2.4) is due to Yano [1]. The main result of §7.3 (Theorem 7.3.6) in the general case of character systems is due to Timan

(see Timan and Rubinstein P]). For the Walsh syst.em this was established by Moricz [1].

CHAPTER 8

Theorem 8.1.2 W35 proved by Rademacher P]' 8.1.3 by Kolmogorov, 8.1.4 and 8.1.5 by Khintchin (see Kolmogorov and Khint.chin P]). We notice that lIaagerup [1] found exact. values for the constant.s Ap and Bp in Khintchin's inequality

for 0 < p < 00.

The results in §§8.2 and 8.3 were proved by Morgenthaler [1], and the special case of Theorem 8.2.5 when Ilk = 2k (i.e., for the Rademacher system) was proved by Kaczmarz and Steinhaus [1].

For more concerning general results about lacunary series see t.he survey articles of Gaposkin [1] and Wade [4].

CHAPTER 9

Theorem 9.1.2 is a vValsh analogue of a theorem of Kolmogorov, which shows that there exist everywhere divergent trigonometric Fourier series (see [2], Vol. I, pp. 455-464). The example introduced in §9.1 is a simplified version of a construction of Moon [1] which he used for a more general result. The first examples of everywhere divergent Walsh- Fourier series were constructed by Schipp [1], [2], see also Heladze [1]. These examples were preceded by an example of an almost everywhere divergent Walsh-Fourier series (see Stein [1]). For mult.iplicative systems, examples of everywhere divergent Fourier series were constructed by IIeladze [2], in the case when the sequence

COMMENTARY 357

{Pi} from (1.5.1) is bounded, and by Simon [I], in the general case. For other generalizations of Kolmogorov's theorem see Bockarev [I] and U1'janov [3].

Theorem 9.2.1 is due to Billard [1], who transferred the methods of Carleson [1] from the trigonometric case to the Walsh case. The proof presented in §9.2 has been adapted from Hunt [1]. Concerning more general results see Sji:ilin [1]. For multiplicative systems under the condition that Pi = P for all i, where {Pi} is the sequence from (1.5.1), Theorem 9.2.1 appeared in Hunt and Taibleson P]' and for the case when the sequence {Pi} is bounded, in Gosselin [1].

CHAPTER 10

The results in §10.1 are due to Golubov [3]. However, inequality

n = 1,2, ... ,

which is less exact than the left side of inequalit.y (10.1.10), can be obtained as a corollary of a theorem of Nagy [I]. Necessity of the condition in Theorem 10.1.4 can also be obtained as a corollary of a result of Nagy [1]. We notice that the left most inequality in (10.1.10) is exact in the following sense, that

cannot hold for all continuous functions f and all natural numbers n if C < 1. Theorems 10.2.1 through 10.2.6 were proved by U1'janov [1], and Theorem 10.2.7 by Haar [1].

Theorems 10.2.8-10.2.10 were established by Golubov [3]. Propositions 10.3.1-10.3.3 were proved by U1'janov P]' and 10.3.4-10.3.11 by Golubov [3]. All the results in §lOA, except 10.4.7, 10.4.8, and 10.4.15, are due to U1'janov [1]. Propositions

10.4.7 and 10.4.8 are well-known (see, for example, the book of Zygmund [30], Vol. I, pp. 29-31), and 10.4.15 was established by Golubov [3]. For character systems see also Timan and Rubinstein [1].

Proposition 10.5.1, in the case of approximation by Walsh functions, was established by Watari [2], and in the general case by Efimov [1]. Proposition 10.5.2 was formulated in the work of Efimov [5].

CHAPTER 11

The proofs of 11.1.1 and 11.1.2 and also the idea behind the proof of 11.1.5 can be found in the work of Efimov and Zolotareva [2]. Propositions 11.1.7, 11.1.8, and 11.2.3 appear here for the first time. Theorems 11.2.1 and 11.2.2 were formulated in the work of Efimov and Karakulin [1]. The proof of 11.2.4 was given by :lukov P]' although the representation of the matrix W as a product of matrices of the same type and t.he passage for calculations to the new basis first appeared in the work of Efimov and Kanygin [1]. Est.imates of the spectral coefficients in §11.3 were given by Bespalov [3], [4], but various cases of these estimates are contained in the work of Kanygin [1]. The results of § 11.4 are due to :lukov [2], the theorems in § 11.5 were formulated in the work of Pospelov [2] (see also Efimov and Pospelov [ID, but the proofs of these theorems, courtesy of A.S. Pospelov, appear here for the first time.

CHAPTER 12

The foundations of digital filtering are given in an article by Kaiser [1], in the book [6], and the monograph [12]. Construction of digital filt.ers based on the Walsh functions and general multiplicative transforms were considered in the work of Robinson and Grander [1], Harmuth [1] (see also the

358 COMMENTARY

monographs [9] and [10], the work of Good [1], Tatachar and Prabhakar [1], and Thbol'cev [1], [2]). The algorithm for digital filtering presented here is the information algorithm of Thbol'cev.

Commentary to §12.2 can be found in the text of this section. The results from §12.3 are due to Lesin [1] (see also Efimov and Lesin [1]).

REFERENCES

Note: For articles and books in Russian, the titles have been translated into English.

I. MONOGRAPHS AND TEXTBOOKS

1. G.N. Agaev, N.Ja. Vilenkin, G.M. Dzafarli, and A.I. Rubinstein, "Multiplicative systems and harmonic analysis on zero- dimensional groups," Baku "ELM", 1981.

2. N.K. Bary, "A Treatise on Trigonometric Series," Pergamon Press, London, 1964. 3. M.M. Dirbaiijan, "Integral Transforms and Representation of Functions in the Complex Plane," Moscow

"Nauka", 1966. 4. B. F. Fedorov and R.I. El'man, "Digital Holography," Moscow "Nauka", 1976. 5. B. V. Gnenenko, "A Course on the Theory of Probability," Moscow "Nauka", 1969. 6. B. Gol'd and C Reider, "Digital Signal Processing," Moscow "Sov. Radio", 1973. 7. M. Gusman, "Differentiability of Integrals in If!'," Moscow "Mir", 1978. 8. G.H. Hardy, "Divergent Series," Oxford, 1949. 9. H.F. Harmuth, "Transmission of Information by Orthogonal Functions," Springer-Verlag, Berlin, 1972.

10. H.F. Harmuth, "Sequency Theory," Academic Press, New York, N.Y., 1977. 11. R.V. Hemming, "Numerical Methods for Technicians and Engineers," Moscow "Nauka", 1968. 12. R.V. Hemming, "Digital Filters," Moscow "Sov. Radio", 1980. 13. E. Hewitt and K. Ross, "Abstract Harmonic Analysis I," Springer-Verlag, Heidelberg, 1963. 14. L.P. Jaroslavskiiand H.S. Merzljakov, "Methods of Digital Holography," Moscow "Nauka", 1977. 15. S. Kaczmarz and G. Steinhaus, "Theorie der Orthogonalreihen," Monogr. Mat. Vol. 6, Warsaw, 1935. 16. E. Kamke, "The Lebesgue-Stieltjes Integral," Moscow "Fizmatgiz", 1959. 17. A.N. Kolmogorov and S.V. Fomin, "Elementary Theory of Functions and Functional Analysis," Moscow

"Nauka", 1981. 18. A .G. Kuros, "A Course in Higher Algebra," Moscow "Nauka", 1975. 19. O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural'ceva, "Linear and Quasi-linear Equations of Para-

bolic Type," Moscow "Nauka", 1967. 20. M. Loeve, "Probability Theory," Van Nostrand, Princeton, 1963. 21. I.P. Natanson, "Theory of Functions of a Real Variable," 3rd Edition, Moscow "Nauka", 1974. 22. R.E.A.C. Paley and N. Wiener, "Fourier Transforms in the Complex Domain," Amer. Math. Soc.,

Colloquium Publications, Vol. XIX, 1934. 23. L.S. Pontryagin, "Topological Groups," Princeton Univ. Press, Princeton, N.J., 1946. 24. W. Rudin, "Fourier Analysis on Groups," Interscience Pub., Wiley and Sons, 1967. 25. S. Saks, "Theory of the Integral," Dover Publications, New York, N.Y., 1964. 26. M.H. Taibleson, "Fourier Analysis on Local Fields," Princeton Univ. Press, Princeton, N.J., 1975. 27. A.M. Trahtman and V.A. Trahtman, "Foundations of the Theory of Discrete Signals on Finite Intervals,"

Moscow "SOy. Radio", 1975. 28. V. S. Vladimirov, "Equations of Mathematical Physics," 3rd Edition, Moscow "Nauka", 1981. 29. E.G. Zelkin, "Construction of Radiation Systems by a Given Diagram of Directions," Moscow "GEl",

1963. 30. A. Zygmund, "Trigonometric Series," Cambridge University Press, New York, N.Y., Vols. I, II, 1959.

II. JOURNAL ARTICLES

F.G. Arutunjan. 1. Recovery of coefficients of Haar and Walsh series which converge to functions which are Denjoy integrable, Izv. Akad. Nauk SSSR 30 (1966), 325- -344.

F.G. Arutunjan and A.A. Talaljan. 1. On uniqueness of Haar and Walsh series, Izv. Akad. Nauk SSSR 28 (1964), 1391-1408.

L.A. Balasov. 1. Series with respect to the Walsh system with monotone coefficients, Sibirsk. Mat. Z. 12 (1971),25-39.

359

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L.A. Balaiiov and A.I. Rubin§tem. 1. Series with respect to the Walsh system and their generalizations, J. Soviet Math. 1 (1973), 727-163.

M.S. Bespalov. 1. Multiplicative Fourier transforms in U', in "ltogi Nauki. Mat. Anal.," Bee. Inst. Naue. Tehn. Inf., No. 100-82, Moscow, 1981.

2. On Parseval's identity and approximation of functions by Fourier series in multiplicative systems, in "Applications of Functional Analysis to Approximation Theory," Kalinin: KGU, 1982, pp. 28-42.

3. Compression of information using discrete multiplicative Fourier transforms, Sbor. Trud. MIET, Moscow (1982), 91-97.

4. Multiplicative Fourier transforms, in "Theory of Functions and Approximation," Proc. Saratov. Zim. Sk., Jan. 24 - Feb. 5, Sarat. Inst., 1982, pp. 39-42.

P. Dillard. 1. Sur la convergence presque partout des series se Fourier- Walsh des fonctions de espace L2(0, I), Studia Math. 28 (1967), 363-388.

S.L. Dljumin. 1. Linear methods of summation of Fourier series with respect to multiplicative systems, Sibirsk. Mat. Z. 9 (1968), 449-455.

2. Certain properties of a class of multiplicative systems and problems of approximation of functions by polynomials with respect to those systems, Izv. Vuzov Mat., No.4 (1968), 13-22.

S.L. Dljumin and D.O. Kotljar. 1. Hilbert-Schmidt optrators and absolute convergence of Fourier series, Izv. Akad. Nauk SSSR, Ser. Mat. 34 (1970), 209-217.

S.V. Dockarev. 1. "A method of averaging in the theory of orthogonal series and some problems in the theory of bases," Trudy Mat. Inst. Steklov., No. 146, 1978.

2. The Walsh Fourier coefficients, Izv. Akad. SSSR, Ser. Mat. 34 (1970), 203-208.

L. Carleson. 1. On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157.

H.E. Cbrestenson. 1. A class of generalized Walsh fUlictions, Pacific J. Math. 5 (1955), 17-32.

J.E. Coury. 1. A class of Walsh AI-sets of measure zero, J. Math. Anal. Appl. 31 (1970), 318-320. 2. A class of Walsh M-sets of measure zero (Corrigendum), J. Math. Anal. Appl. 52 (1975), 669-671.

R.D. Crittenden and V.L. Shapiro. 1. Sets of,m.queness on the group 2"', Annals of Math. 81 (1965), 550-564.

A.V. Efimov. 1. 011 some approximation properties of penodic multiplicative orthollormal systems, Mat. Sbornik 69 (1966), 354-370.

2. On upper bounds for Walsh-Fourier coefficients, Mat. Zametki 6 (1969), 725-730. 3. On summability of the discrete Fourier transform by the Vallee-Pousin method, in "Proc. Symp. Theory

Approx. Funet. Complex Plane," Ufa. "BF", Akad. Nauk SSSR, 1976, p. 33. 4. 0 .. approximation properties of discrete Fourier transforms, in "Methods of Digital Information Pro

cessing," Sbornik Nauc Trud., Moscow "MIET", 1980, pp. 33-45. 5. Approximation properties of general multiplicati,'e systems, in "Proc. Inter. Congress Math.," VII, Sect.

9, Part II, Warsaw, 1983, p. 49.

A.V. Efimov and V.S. Kanygin. 1. An algorithm to compute the discrete transform of Hadamard type, in "Sbornik Nauc. Trud. Prob. Microelect.," Ser. Fiz.- Mat., Moscow "MIET", 1976, pp. 161-174.

A.V. Efimov and A.F. Karakulin. 1. A continual analogue of periodic multiplicative orthonormal systems, Dokl. Akad. Nauk SSSR 218 (1974), 268- 271.

A.V. Efimov and V.V. Lesin. 1. A soilition to·an integral eqllation using the Chrestenson-Levy functions, in "Applications of the Methods of the Theory of Functions and Functional Analysis to Problems of Mathematical Physics," Trud. SOy. Novosibirsk, 1979, pp. 44-47.

A.V. Efimov and A.S. Pospelov. 1. Integral transforms in connection with locally compact groups, in "Proc. Internat. Conf. Theory Approx. Funct.," Kiev, 1983.

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SUBJECT INDEX

Abelian group, 340 Abel's transformation, 153 Absolute convergence of Walsh series, 62,

64, 65 A-integral, 161 Algorithm

fast DWT, 270 fast DFT, 310

Approximation by Walsh and Haar polynomials, 213-259

Arithmetic means, 94

Banach space, 349 Best approximation

by Haar polynomials, 214 by polynomials in multiplicative

systems, 255 by Walsh polynomials in IJ', 219 by Walsh polynomials in

uniform norm, 213 Binary net, 3

(C, I)-means, 94 (C,l)-summability, 94-111 Capacity spectrum, 314 Carleson's theorem, 198 Characteristic, 320, 321 Characters of the group, 8 Closed

orthonormal system, 58, 351 set, 341

Coefficient of compression of information, 281 Complete orthonormal system, 351 Completeness of the Walsh system, 56 Compression of information, 281 Continual analogues of

multiplicative systems, 21, 27 Convergence of Walsh-Fourier series

at a point, 53 of L2 functions, 198

365

uniform, 45-50 Convex coefficients, 163 Convolution, 37

P-adic non-symmetric, 218 Cross product system, 29

Derivative with respect to the binary net, 55 lower, 75 upper, 74

Diagram of directions, 323, 332 Digital

holography, 313 filter, 310 filtering, 310

Dini-Lipschitz test for uniform convergence, 46, 47

Dini's test for convergence at a point, 53 Dirichlet kernel

for the Haar system, 223 for multiplicative systems, 26, 30 for the Walsh system, 19

Discrete Fourier transform (DFT), 281 Hadamard (Walsh) transform (DHT), 260 Holographic transform, 321 multiplicative transform, 269, 270,

275, 321 radiator, 331

Discretization of the multiplicative transform, 260

Distance on [0,1), 13 on the group G, 13

Distribution function, 112 Divergent Walsh-Fourier series, 194 Dyadic

expansion, 6 interval, 3 maximal function, 117 net, 3

366

Egoroff's theorem, 344 Equivalence classes of the group G, 340 Estimate of Walsh-Fourier coefficients, 60 Expansion

dyadic, 2 P-adic,22

Factor group, 310 Fast algorithm

DMT,270 DFT, :110

Filter, 310 Filtered signal, 310 Formal product of a Walsh series

with a Walsh polynomial, 86 Fourier series in multiplicative systems, 66 Fubini's theorem, 348

General Walsh series, 70 Generalized

Fourier-Stieltjes series, 79 Minkowski inequality, 350 multiplicative transform, 127 Riemann-Stieltjes integral, 78

Group, 340 Abelian (commutative), 340 G of dyadic seqnmces, 6 G(P),22 G(P),27 G(P'),28 metric, 342 operations Ell, e, 6, 22

Haar system, 17 Hardy-Littlewood maximal operator, 117 Hausdorff-Young-F. Riesz theorem, 352 Hilbert space, 351 Holder's inequality, 350 Holographic transform, 321 Hologram of an information vector, 321

Inequality Cauchy-Schwarz, 350 Chebyshev, 113 Holder's, 350

Minkowski, 350 Information vector, 321 Integral

A,161

SUBJECT INDEX

generalized Riemann-Stieltjes, 78 Kolmogorov, 354 Lebesgue, 345 Lebesgue-Stieltjes, 347

Interpolation theorem, 113 Inverse theorems in the theory

of approximation, 217

Jackson's theorem, 217 Jordan test for uniform convergence, 49

Kolmogorov integral, 354 theorem on divergent Fourier series, 194

Kotel'nikov's theorem, 143

Lacunary sequence in the Hadamard sense, 177 subsystem of the Walsh system, 173 Walsh series, 177

Lebesgue constants, 40, 41, 42 dominated convergence theorem, 348 integral, 345 measure, 343

Levy's theorem, 348 Linear

functional, 350 method of summation, 94 space, 349

Lipschitz classes, 43 Local Fourier coefficients, 200 Lusin's theorem, 344

Matrix (p, q)-factorable, 299 (w~ mY, 14-18

Maxim;U Hardy-Littlewood operator, 117 dyadic function, 117

Measurable function, 344

SUBJECT INDEX

Measure Haar,344 Lebesgue, 343 Lebesgue-Stieltjes, 343 space, 342

Method of partial diagrams, 332 of summation, 94 (C, 1),94,95 regular, 95

Metric, 341 group, 342 space, 341

Modified Dirichlet kernel, 119 interval [0, W, 6 interval [0, 11p, 23 ray [O,oo)p, 28

Moduli of continuity in the V-norm, 222 w(6, 1),43 w'(6,J),44 wp(f) (1'-modulus), 130 w~p)(f), 256, 247

Monotone coefficients, 153 M-set, 93 Multiplicative system

on the group, 21 on the interval, 23

Norm, 213, 219, 349 Null series, 89

Open set, 342 Operation EEl

on the group G, 6 on the group G(P), 22 on the non-negative integers, 8 on the unit interval, 10

Operator Hardy-Littlewood maximal, 117 of strong type (p,p) of weak type (p, p) sublinear, 113 AI, 117 HI, 119

Orthogonality condition for the Walsh system, 4

Paley enumeration, 1, 353 Parseval's identity, 352

for the Walsh system, 58 for a multiplicative system, 69

Partial sums of Walsh series of order n, 18, 39 of order 2n, 39

367

Partial sums of series in a multiplicative system, 67

Phase spectrum, 314 Plancherel identity, 136 Principle of localization, 50 Product

of Walsh functions, 3 formal of a Walsh series with a

Walsh polynomial, 86 Pseudo-complex basis, 280

Quasiconvex coefficients, 163 Quasi-positivity of a kernel, 99

Rademacher functions, 1 system, 173

Radiator, 331 Rank (of an interval or a net), 3 Regular method of summation, 95 Riemann-Stieltjes generalized integral, 78 Riesz-Fischer theorem, 352

Set closed, 341 compact, 342 of type :Fq , 178 of uniqueness, 93 open, 341

Shift operation, 10 Simple function, 36 Space

Banach, 349 compact, 342 complete, 342 Hilbert, 351 locally compact, 342 measure, 342 metric, 341

368

normed linear, 349 of continuous functions, 349 of integrable functions, 349

Strong type (p,p) (operator), 113 Subgroup, 340 Sublinear operator, 113 System

Chrestenson-Levy, 324, 353 Haar, 17 multiplicative:

on the group, 21 on the interval, 23 continual analogues, 27

Price, 23 Rademacher, 1, 173 Walsh:

Test

on the group, 5 on the interval, 2

for uniform convergence, 46-50 Dini-Lipschitz, 46, 47

for convergence at a point, 50 Dini,53

Theorem Banach-Steinhaus, 351 Carleson's analogue, 198 central limit, 186 direct (approximation theory), 217 Egoroff's, 344 Fubini's, 348 Hausdorff-Young-F. Riesz, 352 Kolmogorov (divergent Fourier series), 194 Kotel'nikov's, 143 Lebesgue dominated convergence, 348 Levy's, 348 Lusin's, 344 Marcinkiewicz interpolation, 113 Mercer's, 352 Riesz-Fischer, 352 of uniqueness for the Walsh system, 57 of uniqueness for representation

by Walsh series, 85 Young's, 142

Toeplitz conditions for regularity of a summation method, 95, 96

Transform direct multiplicative, 127 discrete Fourier, 281

Hadamard, 260 holographic, 321

SUBJECT INDEX

multiplicative direct (DDMT), 269 multiplicative inverse (IDMT), 270

inverse multiplicative, 135 Transformation

Abel's, 153 discrete generated by a matrix A, 305

factorable, 305 Walsh-Fourier, 36 9 : [0,1) --+ G, 9 9p : [0,1) --+ G(P), 24 9'/> : [0,00) --+ G(P), 28 A: G--+ [0,1]' 7 Ap : G(P) --+ [0,1]' 23 A,/> : G(P) --+ 1/,00),28

Translation invariance of the Lebesgue integral, 34

Uniform convergence of Walsh-Fourier series, 46-50 norm, 213

Uniformly -continuous function, 44 equiconvergent series, 87

Uniqueness of Walsh series, 85 Uniqueness of the Walsh system, 57

Variation bounded,49 of a function, 49 total,346

Walsh polynomials, 15 series

general,70 with monotone coefficients, 153 with quasiconvex codTicients, 165

Walsh-Fourier series, 35 divergent, 194 of L2 functions, 198

Walsh-Fourier-Stieltjes series, 701 Weak type operator, 113

Zone coding, 281

walsh series and transforms: theory and applications

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