Fourier series

Rajendra Bhatia0883857405, 9780883857403

This is a concise introduction to Fourier series covering history, major themes, theorems, examples and applications. It can be used to learn this subject, and also to supplement, enhance and embellish undergraduate courses on mathematical analysis.
The book begins with a brief summary of the rich history of the subject over three centuries. The subject is presented in a way that enables the reader to appreciate how a mathematical theory develops in stages from a practical problem (such as conduction of heat) to an abstract theory dealing with concepts such as sets, functions, infinity and convergence. The abstract theory will provide unforeseen applications in diverse areas.
The book begins with a description of the problem that led Fourier to introduce his famous series. The mathematical problems this leads to are discussed rigorously. Examples, exercises and directions for further reading and research are provided, along with a chapter that provides material at a more advanced level suitable for graduate students. The author demonstrates applications of the theory as well as a broad range of problems.
Exercises of varying levels if difficulty are scattered throughout the book. These will help readers test their understanding of the material.

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Table of contents :
Front cover……Page 1
Date-line……Page 2
Title page……Page 3
CLASSROOM RESOURCE MATERIALS……Page 4
Contents……Page 7
Preface……Page 9
0 A History of Fourier Series……Page 11
1.1 The Laplace equation in two dimensions……Page 23
1.2 Solutions of the Laplace equation……Page 25
1.3 The complete solution of the Laplace equation……Page 29
2 Convergence of Fourier Series……Page 37
2.1 Abel summability and Cesaro summability……Page 38
2.2 The Dirichlet and the Fejer kernels……Page 39
2.3 Pointwise convergence of Fourier series……Page 45
2.4 Term by term integration and differentiation……Page 54
2.5 Divergence of Fourier series……Page 56
3.1 Sine and cosine series……Page 61
3.2 Functions with arbitrary periods……Page 63
3.3 Some simple examples……Page 65
3.4 Infinite products……Page 71
3.5 $pi$ and infinite series……Page 73
3.6 Bernoulli numbers……Page 75
3.7 $sin{x}/x$……Page 77
3.8 The Gibbs phenomenon……Page 80
3.9 Exercises……Page 82
3.10 A historical digression……Page 85
4.1 $L_2$ convergence of Fourier series……Page 89
4.2 Fourier coefficients of $L_1$ functions……Page 95
5.1 An ergodic theorem and number theory……Page 105
5.2 The isoperimctric problem……Page 108
5.3 The vibrating string……Page 111
5.4 Band matrices……Page 115
A A Note on Normalisation……Page 121
B A Brief Bibliography……Page 123
Index……Page 127
Notation……Page 129
About the Author……Page 130
Back cover……Page 131