This is a concise introduction to Fourier series covering history, major themes, theorems, examples, and applications. It can be used for self study, or to supplement undergraduate courses on mathematical analysis. Beginning with a brief summary of the rich history of the subject over three centuries, the reader will 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 then provides unforeseen applications in diverse areas. Exercises of varying difficulty are included throughout to test understanding. A broad range of applications are also covered, 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.
Author(s): Rajendra Bhatia
Series: Classroom Resource Materials
Publisher: Mathematical Association of America
Year: 2005
Language: English
Pages: 120
copyright page
title page
Contents
Preface
0 A History of Fourier Series
1. The motion of a vibrating string
2. J. D’Alembert
3. L. Euler
4. D. Bernoulli
5. J. Fourier
6. P. Dirichlet
7. B. Riemann
8. P. du Bois-Reymond
9. G. Cantor
10. L. Fejér
11. H. Lebesgue
12. A.N. Kolmogorov
13. L. Carleson
14. The L_2 theory and Hilbert spaces
15. Some modern developments—I
16. Some modern developments—II
17. Pure and applied mathematics
Chapt 1 Heat Conduction and Fourier Series
1.1 The Laplace equation in two dimensions
1.2 Solutions of the Laplace equation
1.3 The complete solution of the Laplace equation
2 Convergence of Fourier Series
2.1 Abel summability and Cesàro summability
2.2 The Dirichlet and the Fejér kernels
2.3 Pointwise convergence of Fourier series
2.4 Term by term integration and differentiation
2.5 Divergence of Fourier series
3 Odds and Ends
3.1 Sine and cosine series
3.2 Functions with arbitrary periods
3.3 Some simple examples
3.4 Infinite products
3.5 π and infinite series
3.6 Bernoulli numbers
3.7 sinx/x
3.8 The Gibbs phenomenon
3.9 Exercises
3.10 A historical digression
4 Convergence in L_2 and L_1
4.1 L_2 convergence of Fourier series
4.2 Fourier coefficients of L_1 functions
5 Some Applications
5.1 An ergodic theorem and number theory
5.2 The isoperimetric problem
5.3 The vibrating string
5.4 Band matrices
A A Note on Normalisation
B A Brief Bibliography
Analysis
Fourier series
General reading
History and biography
Index
Notation
About the Author
