Methods of Applied Fourier Analysis

333 Pages·1998·23.817 MB·English

by  Jayakumar Ramanathan (auth.)

Appliedand Numerical Harmonic Analysis
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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.
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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