Appliedand Numerical Harmonic Analysis Series Editor John J. Benedetto UniversityofMaryland EditorialAdvisoryBoard AkramAldroubi DouglasCochran VanderbiltUniversity ArizonaStateUniversity IngridDaubechies HansG. Feichtinger Princeton University UniversityofVienna ChristopherHeil MuratKunt GeorgiaInstituteofTechnology SwissFederalInstitute ofTechnology, Lausanne JamesMcClellan WimSweldens GeorgiaInstituteofTechnology LucentTechnologies Bell Laboratories MichaelUnser Martin Vetterli SwissFederalInstitute SwissFederalInstitute ofTechnology, Lausanne ofTechnology, Lausanne VictorWickerhauser Washington University, St. Louis Jayakumar Ramanathan Methods of Applied Fourier Analysis Springer Science+Business Media, LLC Jayakumar Ramanathan Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197 Library of Congress Cataloging-in-Publication Data Ramanathan, Jayakumar, 1958- Methods of applied fourier analyis / Jayakumar Ramanathan. p. cm. --(Applied and numerical harmonic analysis) ISBN 978-1-4612-7267-0 ISBN 978-1-4612-1756-5 (eBook) DOI 10.1007/978-1-4612-1756-5 1. Fourier analysis. 1. Title. II. Series. QA403.5.R33 1998 515'.2433--dc21 98-4738 CIP Printed on acid-free paper © 1998 Springer Science+Business Media New York Originally published by Birkhliuser Boston in 1998 Softcover reprint ofthe hardcover Ist edition 1998 Copyright is not c1aimed for works of U.S. Government employees. AII rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific c1ients, is granted by Springer Science+Business Media, LLC, provided that the appropriate fee is paid direclly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, USA (Telephone: (978) 750-8400), stating the ISBN, the tille ofthe book, and the first and last page numbers of each article copied. The copyright owner's consent does not include copying for general distribution, promotion, new works, or resale. In these cases, specific written permission must first be obtained from the publisher. ISBN 978-1-4612-7267-0 Typeset by the Author in LArEX. 9 8 765 432 1 Dedicated to Beth, Lauren, and Nicole. Contents Preface xi 1 Periodic Functions 1 1.1 The Characters . ..... 1 1.2 Some Tools ofthe Trade . 4 1.3 Fourier Series: LP Theory 8 1.4 Fourier Series: L2 Theory 15 1.5 Fourier Analysis ofMeasures 19 1.6 Smoothness and Decay ofFourier Series 22 1.7 Translation Invariant Operators . 23 1.8 Problems ............ 27 2 Hardy Spaces 31 2.1 Hardy Spaces and Invariant Subspaces 31 2.2 Boundary Values ofHarmonic FUnctions . 36 2.3 Hardy Spaces and Analytic FUnctions 42 2.4 The Structure ofInner FUnctions 45 2.5 The HI Case ........... 50 2.6 The Szego-Kolmogorov Theorem 53 2.7 Problems .... . . . . . . . . . 59 3 Prediction Theory 63 3.1 Introduction to Stationary Random Processes 63 3.2 Examples ofStationary Processes . . 68 3.3 The Reproducing Kernel ....... 71 3.4 Spectral Estimation and Prediction . 76 3.5 Problems ............... 84 4 Discrete Systems and Control Theory 87 4.1 Introduction to System Theory 87 4.2 Translation Invariant Operators . 90 4.3 Hoc Control Theory . . . . . . . 93 4.4 The Nehari Problem . . . . . . . 103 4.5 Commutant Lifting and Interpolation 108 4.6 Proofofthe Commutant Lifting Theorem 112 4.7 Problems . 119 5 Harmonic Analysis in Euclidean Space 123 5.1 Function Spaces on Rn . 123 5.2 The Fourier Transform on £1 . 127 5.3 Convolution and Approximation 134 5.4 The Fourier Transform: £2 Theory 138 5.5 Fourier Transform of Measures 145 5.6 Bochner's Theorem . 152 5.7 Problems . 156 6 Distributions 159 6.1 General Distributions . 159 6.2 Two Theorems on Distributions. 164 6.3 Schwartz Space . . . . . 172 6.4 Tempered Distributions 175 6.5 Sobolev Spaces 179 6.6 Problems . 184 7 Functions with Restricted Transforms 187 7.1 General Definitions and the Sampling Formula . . . . . . . 187 7.2 The Paley-Wiener Theorem .... 192 7.3 Sampling Band-Limited Functions 199 7.4 Band-Limited Functions and Information 203 7.5 Problems . 215 8 Phase Space 219 8.1 The Uncertainty Principle . 219 8.2 The Ambiguity Function . . . . . . . 227 8.3 Phase Space and Orthonormal Bases 236 8.4 The Zak Transform and the Wilson Basis 243 8.5 An Approximation Theorem. 256 8.6 Problems . 260 viii 9 Wavelet Analysis 263 9.1 Multiresolution Approximations. 263 9.2 Wavelet Bases .......... 267 9.3 Examples ............. 276 9.4 Compactly Supported Wavelets . 281 9.5 Compactly Supported Wavelets II 290 9.6 Problems ............ 299 A The Discrete Fourier Transform 301 A.l The Analysis ofPeriodic Sequences . 301 A.2 The Cooley-Thkey Algorithm . . 306 A.3 The Good-Winograd Algorithm. 310 A.4 Fast Computations ofIntegrals 313 B The Hermite Functions 317 Bibliography 323 Index 328 ix Preface From its inception, harmonic analysis has made fundamental connections with other scientific disciplines. The aimofthis book is to provide a math ematical introduction to this field with special emphasis on those topics that do find direct application in engineering and the sciences. It must be mentioned that, even with such a guideline, there is still great latitude in thechoiceofpossible topics; the particularchoices made hereare primarily a matter oftaste. The material should be accessible to graduate students in mathematics with a good background in analysis. The first chapter is an introduction to the basic ideas oftrigonometric series. The material includes a treatment ofthe £1 and £2 theory together with important ancillary topics such as the Fourier analysis of measures. Chapter 2 is an introduction to the theory ofHardy spaces. The structure of inner and outer functions is presented along with a proofofthe Szego Kolmogorov theorem. With the theory ofthe first two chapters as a foundation, the next two chaptersdevelop material that isdirectly relevant to applications. Chapter 3explores the prediction theory ofdiscretestationary stochastic processes. This includes a discussion ofthe spectral theory ofstationary processes as well as prediction theory (including the maximum entropy solution). The fourth chapter explores connections of Fourier series with discrete control theory. Ideasand theorems basicto Boo controltheoryaregiven, including Nehari's theorem and the commutant lifting theorem. Chapters 5 and 6 are again ofa foundational nature. The first ofthese mirrorschapter 1andexpositsthe basictheoryofharmonicanalysisinRn. Chapter 6 is an introduction to the theory ofdistributions. This includes the theory oftempered distributions as well as a rudimentary treatment of Sobolev spaces. The last threechaptersare devoted to application-oriented topics inthe Rn setting. Chapter 7 begins with the connection between functions with restricted Fourier transform and analytic function theory, the main result being the Paley-Wiener theorems. The chapter then turns to the analy- sis of band-limited functions with the aim of making rigorous connections with the ideasofinformationtheory. Chapter8isdevoted totheanalysisof functions using techniques where the spatial (time) and frequency domains are treatedonanequal footing. Thechapterbeginswithadiscussionofthe uncertainty principle and the ambiguity transform. The rest is devoted to various positive and negative results about the distribution oforthonormal bases within phase space. The subject of the final chapter is wavelet the ory. After presentingthe basic ideasofmultiresolution approximations and examples, the chapter covers the theory ofcompactly supported wavelets. There are two appendices covering the discrete Fourier transform and Hermite functions. Problems of varying difficulty are given at the end of each chapter. Some ask the reader to fill in details while others explore topics related to those presented in the text. Acknowledgment It isa pleasureto thank myfamily for theirencour agement throughout the duration of this project. Thanks also to Wayne Yuhaszand LaurenLaveryofBirkhauserfor theirpatientguidancethrough the publication process. Jayakumar Ramanathan Ann Arbor, MI 1998 xii