You are here

Fourier Analysis: An Introduction

Elias M. Stein and Rami Shakarchi
Princeton University Press
Publication Date: 
Number of Pages: 
Princeton Lectures in Analysis I
[Reviewed by
Fernando Q. Gouvêa
, on

Every once in a while, I am struck by how often mathematics textbooks sound just like each other. Glance at the table of contents of your typical analysis or algebra textbook, and you can be 90% sure of seing exactly the same sequence of topics each time. There are honorable exceptions of course, and I'm always glad to see one, because they indicate that (at least some) mathematicians are still actively thinking about what should be taught, in what order, and how.

So here is the first volume in the Princeton Lectures on Analysis, entitled Fourier Analysis: an Introduction and written by Elias M. Stein and Rami Shakarchi. The series wants to serve as an integrated introduction to "the core areas in analysis." The following volumes will treat complex analysis (volume 2), measure theory, integration, and hilbert spaces (volume 3), and selected other topics (volume 4). The basic pre-requisite for the series seems to be a standard undergraduate introduction to analysis covering the basic theory of convergence, derivatives, and the Riemann integral. Some basic familiarity with the complex numbers and elementary functions (e.g., complex exponentials) is assumed. So the book is aimed at graduate students and maybe advanced undergraduates.

The new series begins with Fourier analysis because the authors feel that this subject plays a central role in modern analysis and because it has played an important historical role. It is also much more concrete than abstract measure theory or functional analysis. Finally, the authors plan to use results from volume one in the following volumes, emphasizing that analysis is a coherent whole rather than a collection of disjointed topics.

The first book covers the basic theory of Fourier series, Fourier transforms in one and more dimensions, and finite Fourier analysis. The last topic allows the authors to present, as an application, the proof of Dirichlet's theorem on primes in arithmetic progressions. The result would make a great book for independent study courses with advanced undergraduates, and, I think, would also be useful for graduate courses. It's definitely worth a look.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College, where he occasionally gets to teach some analysis.

Foreword vii
Preface xi
Chapter 1. The Genesis of Fourier Analysis 1
Chapter 2. Basic Properties of Fourier Series 29
Chapter 3. Convergence of Fourier Series 69
Chapter 4. Some Applications of Fourier Series 100
Chapter 5. The Fourier Transform on R 129
Chapter 6. The Fourier Transform on R d 175
Chapter 7. Finite Fourier Analysis 218
Chapter 8. Dirichlet's Theorem 241
Appendix: Integration 281
Notes and References 299
Bibliography 301
Symbol Glossary 305