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The Wavelet Digest
   -> Volume 11, Issue 1

Book: Intro. to Fourier Analysis and Wavelets
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Mark Pinsky (

PostPosted: Fri Dec 28, 2001 12:42 pm    
Subject: Book: Intro. to Fourier Analysis and Wavelets
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Dear Sir/Madam,

As a long-time subscriber to the Wavelet Digest, I would like to offer a contribution to this excellent Newsletter.

Brooks-Cole, Inc. has now published my "Introduction to Fourier Analysis and Wavelets", ISBN no. 0-534-37660-6. This 375 page book is unique in combining a mathematically rigorous treatment of classical Fourier Analysis with a definitve final chapter on one-dimensional wavelets.

Here is an abbreviated table of contents:
Chapter 1: Fourier Series on the Circle
Chapter 2: Fourier Transforms on the Line and Space
Chapter 3: Fourier Analysis in Lebesgue Spaces
Chapter 4: Poisson Summation Formula and Multiple Fourier Series
Chapter 5: Applications to Probability Theory
Chapter 6: Introduction to Wavelets

From the preface:
This book provides a self-contained treatment of classical Fourier analysis at the upper undergraduate or begining graduate level. We assume that the reader is familiar with the rudiments of Lebesgue measure and integral on the real line. Our viewpoint is mostly classical and concrete, preferring explicit calculations to existential arguments. In some cases, several different proofs are offered for a given proposition, to compare different methods.

We owe a debt of gratitude to Paul Sally Jr., who encouraged this project from the beginning. Gary Ostedt gave us full editorial support at the initial stages, followed by Bob Pirtle and his efficient staff. Further thanks are due to Robert Fefferman, whose lectures provided much of the inspiration for the basic parts of the book. Further assistance and feedback was provided by Marshall Ash, William Beckner, Miron Bekker, Leonardo Colzani, Galia Dafni, George Gasper, Umberto Neri, Cora Sadosky, Aurel Stan, and Michael Taylor. Needless to say, the writing of Chapter 1 was strongly influenced by the classical treatise of Zygmund and the elegant text of Katznelson. The latter chapters were influenced in many ways by the books of Stein and and Stein/Weiss. The final chapter on wavelets owes much to the texts of Hernandez/Weiss and Wojtaszczyk.

The book contains more than 175 exercises, which are an integral part of the text. It can be expected that a careful reader will be able to complete all of these exercises. Starred sections contain material that may be considered supplementary to the main themes of Fourier analysis. In this connection, it is fitting to comment on the role of Fourier analysis, which plays the dual role of queen and servant of mathematics. Fourier-analytic ideas have an inner harmony and beauty--- quite apart from any applications to number theory, approximation theory, partial differential equations or probability theory. In writing this book it has been difficult to resist the temptation to develop some of these applications, as a testimonial of the power and flexibility of the subject. The following list of ``extra topics'' are included in the starred sections: Stirling's formula, Laplace asymptotic method, the isoperimetric inequality, equidistribution modulo one, Jackson/Bernstein theorems, Wiener's density theorem, one-sided heat equation with Robin boundary condition, the uncertainty principle, Landau's asymptotic lattice point formula, Gaussian sums and the Schrodinger equation, the central limit theorem, the Berry Esseen theorem and the law of the iterated logarithm. While none of these topics is ``mainstream Fourier analysis'', each of them has a definite relation to some part of the subject.

A word about the organization of the first two chapters, which are essentially independent of one another. Readers with some sophistication but little previous knowledge of Fourier series can begin with Chapter 2 and anticipate a self-contained treatment of the $n$-dimensional Fourier transform and many of its applications.
By contrast, readers who wish an introductory treatment of Fourier series should begin with Chapter 1, which provides a reasonably complete introduction to Fourier analysis on the circle. In both cases we emphasize the Riesz-Fischer and Plancherel theorems, which demonstrate the natural harmony of Fourier analysis with the Hilbert spaces $L^2(\T)$ and $L^2(symb_Re.gif^n)$. However much of modern harmonic analysis is carried out in the $L^p$ spaces for $psymb_ne.gif 2$, which is the subject of Chapter 3. Here we find the interpolation theorems of Riesz-Thorin and Marcinkiewiecz, which are applied to discuss the boundedness of the Hilbert transform and its application to the $L^p$ convergence of Fourier series and integrals. In Chapter 4 we merge the subjects of Fourier series and Fourier transforms by means of the Poisson summation formula in one and several dimensions. This also has applications to number theory and multiple Fourier series, as noted above.

Chapter 5 explores the application of Fourier methods to probability theory. Limit theorems for sums of independent random variables are equivalent to the study of iterated convolutions of a probability measure on the line, leading to the central limit theorem for convergence and the Berry-Esseen theorems for error estimates. These are then applied to prove the law of the iterated logarithm.

The final Chapter 6 deals with wavelets, which form a class of orthogonal expansions which can be studied by means of Fourier analysis--specifically the Plancherel theorem from Chapter 2. In contrast to Fourier series and integral expansions, which require one parameter (the frequency), wavelet expansions involve two indices---the scale and the location parameter. This allows additional freedom and leads to improved convergence proerties of wavelet expansions in contrast with Fourier expansions. We briefly include an application to Brownian motion, where the wavelet approach furnishes an easy access to the precise modulus of continuity of the standard Brownian motion.

Many of the topics in this book have been ``class-tested'' to a group of graduate students and faculty members at Northwestern University during the academic years 1998-2000. We are grateful to this audience for the opportunity to develop and improve our original efforts.


I do hope that you might be able to list my book in the next issue
of the Wavelet Digest. Many thanks for your careful consideration.

Mark A. Pinsky
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
Fax: 847-491-8906
All times are GMT + 1 Hour
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