Fourier Series / Edition 1

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Overview

Suitable as a text in a course in analysis, mathematical methods, or mathematical physics.

This concise introduction to Fourier series covers history, major themes, theorems, examples and applications. It can be used to learn the 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 theory. 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 of difficulty are scattered throughout the book. These will help readers test their understanding of the material.

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Editorial Reviews

Brandy J. Lipton
"Overall, Bhatia is able to derive the important results of introductory Fourier Analysis in a readable way. His presentation is interesting, and his many short detours into related issues inspire the reader to delve deeper into analysis...Bhatia's text could certainly serve as a complimentary undergraduate resource or as the main textbook in a preparatory graduate course."
MAA Online
Kenneth A. Ross
"This short book is a very readable introduction to Fourier series suitable for scientists and engineers. It is sprinkled with hints about recent developments and has a lot of nice historical comments that will intrigue the best students and math majors. The author almost talks to the readers and skillfully highlights what is important. A fair amount of the material is in the extensive set of exercises. If this very nice text had been available when I was teaching, I would have used it for a junior-senior level course for science and math majors."
University of Oregon, Eugene
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Product Details

  • ISBN-13: 9780883857403
  • Publisher: Mathematical Association of America
  • Publication date: 12/31/2004
  • Series: Classroom Resource Material Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 140
  • Product dimensions: 10.10 (w) x 7.10 (h) x 0.50 (d)

Meet the Author

Rajendra Bhatia is a Professor in Statistical Mathematics at the Indian Statistical Institute.

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Table of Contents

1. History of Fourier Series
2. Heat Conduction and Fourier Series:
 a. The Laplace equation in two dimensions
 b. Solutions of the Laplace equation.
 c. The complete solution of the Laplace equation
3. Convergence of Fourier Series:
 a.  Abel summability and Cesaro summability
 b. The Dirichlet and Fejér kernels
 c. Pointwise Convergence of Fourier series
 d.  Term by term integration and differentiation
 e. Divergence of Fourier series
4.  Odds and Ends
 a. Since and cosine series
 b. Functions with arbitrary periods
 c. Some simple examples
 d. Infinite products
 e. p and infinite series
 f. Bernoulli numbers
 g. sin x/x
 h.  The Gibbs phenomenon
 i. Exercises
 j. A Historical digression
5. Convergence in L2 and L1
 a. L2-convergence of Fourier series
 b. Fourier coefficients of L1 functions
6. Some Applications
 a. An ergodic theorem and number theory
 b. the isoperimetric problem
 c. The vibrating string
 d. Band matrices
7. A Note on Normalisation
8. A Brief Bibliography
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