The Wadsworth & Brooks/Cole Mathematics Series

Transcription

1 Harmonic Analysis

2 The Wadsworth & Brooks/Cole Mathematics Series Series Editors Raoul H. Bott, Harvard University David Eisenbud, Brandeis University Hugh L. Montgomery, University of Michigan Paul J. Sally, University of Chicago Barry Simon, California Institute of Technology Richard P. Stanley, Massachusetts Institute of Technology M. Adams, V. Guillemin, Measure Theory and Probability w. Beckner, A. Calderon, R. Fefferman, P. Jones, Conference on Harmonic Analysis in Honor of Antoni Zygmund G. Chartrand, L. Lesniak, Graphs & Digraphs, Second Edition J. Cochran, Applied Mathematics: Principles, Techniques, and Applications W. Derrick, Complex Analysis and Applications, Second Edition J. Dieudonne, History of Algebraic Geometry R. Dudley, Real Analysis and Probability R. Durrett, Brownian Motion and Martingales in Analysis R. Epstein, W. Carnielli, Computability: Computable Functions, Logic, and the Foundations of Mathematics S. Fisher, Complex Variables, Second Edition P. Garrett, Holomorphic Hilbert Modular Forms A. Garsia, Topics in Almost Everywhere Convergence R. Gunning, Introduction to Holomorphic Functions of Several Variables Volume I: Function Theory Volume II: Local Theory Volume III: Homological Theory H. Helson, Harmonic Analysis J. Kevorkian, Partial Differential Equations: Analytical Solution Techniques R. McKenzie, G. McNulty, W. Taylor, Algebras, Lattices, Varieties, Volume I E. Mendelson, Introduction to Mathematical Logic, Third Edition B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions R. Salem, Algebraic Numbers and Fourier Analysis, and L. Carleson, Selected Problems on Exceptional Sets R. Stanley, Enumerative Combinatorics, Volume I J. Strikwerda, Finite Difference Schemes and Partial Differential Equations K. Stromberg, An Introduction to Classical Real Analysis

3 Harmonic Analysis Henry Helson University of California, Berkeley Wadsworth & Brooks/Cole Advanced Books & Software Pacific Grove, California

4 Wadsworth & Brooks/Cole Advanced Books & Software A Division of Wadsworth, Inc by Addison~Wesley Publishing Company, Inc by Wadsworth, Inc., Belmont, California All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transcribed, in any form or by any means--electronic, mechanical, photocopying, recording, or otherwise-without the prior written permission of the publisher, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, California 93950, a division of Wadsworth, Inc Library of Congress Cataloging-in-Publication Data Helson, Henry. Harmonic analysis / Henry Helson. p. cm. -- (The Wadsworth & Brooks/Cole mathematics series) Reprint. Originally published: Reading, Mass. : Addison-Wesley, Includes bibliographical references and index. ISBN ISBN (ebook) DOI / I. Harmonic analysis. I. Title. II. Series. QA403.H '.2433-dc CIP Sponsoring Editor: John Kimmel Editorial Assistant: Jennifer Kehr Production Coordinator: Dorothy Bell Cover Design: Vernon T. Boes

5 Contents

6 Preface... yix Chapter 1 Fourier Series and Integrals Definitions and easy results The Fourier transform Convolution; approximate identities; Fejer's theorem Unicity theorem; Parseva1 relation; Fourier-Stieltjes coefficients The classical kernels Summabi1ity; metric theorems Pointwise summabi1ity Positive definite sequences; Herg10tz' theorem The inequality of Hausdorff and Young * 1.10 Multiple Fourier series; Minkowski's theorem *1.11 Measures with bounded powers; homomorphisms of II...44 Chapter 2. The Fourier Integral Introduction Kernels on R The P1anchere1 theorem Another convergence theorem; the Poisson summation formula *2.5. Finite cyclic groups; Gaussian sums *Starred sections present material that is less fundamental. vii

7 viii CONTENTS Chapter 3. Hardy Spaces HP(T) Invariant subspaces, factoring, proof of the theorem of F. and M. Riesz Theorems of Beurling and Szego Structure of inner functions Theorem of Hardy and Littlewood; Hilbert's inequality Hardy spaces on the line Chapter 4. Conjugate Functions Conjugate series and functions Theorems of Kolmogorov and Zygmund Theorems of M. Riesz and Zygmund The conjugate function as a singular integral The Hilbert transform Maximal functions Rademacher functions; absolute Fourier multipliers Chapter 5. Translation Theorems of Wiener and Beurling; the Titchmarsh convolution theorem The Tauberian theorem Spectral sets of bounded functions *5.4. A theorem of Szego; theorem of Grui.ewska and Rajchman; idempotent measures Chapter 6. Distribution Equidistribution of sequences Distribution of (nku) Dynamical systems; (k 2 u) Appendix. Integration by parts Bibliographic Notes Index

8 Preface

9 Fourier series and complex function theory used to be the central topics in the curriculum of any analyst, which was to say, of most mathematicians. Now there is more to learn and mathematics has become more specialized. Even within analysis, a student writing a dissertation on partial differential equations or ergodic theory may never have heard of a conjugate function. That is a pity, because the basic examples and classical results of all parts of analysis, and beyond, are to be found in the beautiful subject that used to be called Fourier series, and now goes by the more exalted name of Harmonic Analysis. Even if the classical techniques are not sufficient to solve modern problems, a knowledge of them is indispensable for a proper view of the methods that have descended from them. The subject has not been neglected by gifted expositors, but it is large and much of it seems specialized. My aim is to present concisely in each part the basic definitions, methods, and theorems, with as little interdependence among sections as the subject allows. It is hoped that proofs are as simple as possible. Care has been taken to make the index serviceable. Special notation is avoided. If these objectives have been achieved, then a reader should be able to find a desired piece of information and follow the argument without a lot of preliminary work. This goal of simplicity is inconsistent with other goals. This is not a treatise; if the topics that are here are useful and interesting, no apology is offered for others that are missing. It is also not oriented towards the vigorous modern subject, in which function theory has largely been displaced by powerful new real methods. A book along those lines could not at this stage be so elementary as this one. xi

10 xii PREFACE The reader is assumed to know the elementary part of complex function theory, general topology, integration, and linear spaces. All the needed information is contained in a usual first-year graduate course on analysis. These prerequisites are modest but essential. To be sure there is a big gap between learning the Banach-Steinhaus theorem, for example, and applying it to a real problem. Filling that gap is one of the objectives of this book. It is a natural objective, because integration theory and functional analysis to a great extent developed in response to the problems of Fourier series! The exposition has been condensed somewhat by relegating proofs of some technical points to the problem sets. Other problems give results that are needed in subsequent sections; and many problems simply present interesting results of the subject that are not otherwise covered. Problems range in difficulty from very simple to very hard. The system of numeration is simple: Sec. 3.2 is the second section of Chapter 3. The second section of the current chapter is Sec. 2. Formula (3.2) is the second formula of Sec. 3, of the current chapter unless otherwise mentioned. With pleasure I record the debt to my notes from a course on Real Variables given by R. Salem in I wish to thank R. Fefferman, Y. Katznelson, and A. 6 Cairbre for sympathetic criticism of the manuscript. Mr. Carl Harris of the Addison-Wesley Publishing Company has been most helpful in bringing the book to publication. H.H. Berkeley, California November, 1982

11 Harmonic Analysis