Fourier Series and Integral Transforms

The aim of this book is to provide the reader with a basic understanding of Fourier series, Fourier transforms and Laplace transforms. The book is an expanded and polished version of the authors' notes for a one semester course, for students of mathematics, electrical engineering, physics and computer science. Prerequisites for readers of this book are a basic course in both calculus and linear algebra. Otherwise the material is self-contained with numerous exercises and various examples of applications.

Preface

vii

Notation and Terminology

1

(4)

Basic Concepts in Set Theory

1

(1)

Calculus Notation

2

(2)

Useful Trigonometric Formulae

4

(1)

Background: Inner Product Spaces

5

(27)

Introduction

5

(1)

Linear and Inner Product Spaces

5

(5)

The Norm

10

(5)

Orthogonal and Orthonormal Systems

15

(4)

Orthogonal Projections and Approximation in the Mean

19

(5)

Infinite Orthonormal Systems

24

(8)

Review Exercises

30

(2)

Fourier Series

32

(61)

Introduction

32

(1)

Definitions

32

(8)

Evenness, Oddness, and Additional Examples

40

(2)

Complex Fourier Series

42

(4)

Pointwise Convergence and Dirichlet's Theorem

46

(10)

Uniform Convergence

56

(7)

Parseval's Identity

63

(5)

The Gibbs Phenomenon

68

(4)

Sine and Cosine Series

72

(4)

Differentiation and Integration of Fourier Series

76

(5)

Fourier Series on Other Intervals

81

(4)

Applications to Partial Differential Equations

85

(8)

Review Exercises

89

(4)

The Fourier Transform

93

(47)

Introduction

93

(1)

Definitions and Basic Properties

93

(5)

Examples

98

(4)

Properties and Formulae

102

(6)

The Inverse Fourier Transform and Plancherel's Identity

108

(8)

Convolution

116

(3)

Applications of the Residue Theorem

119

(6)

Applications to Partial Differential Equations

125

(5)

Applications to Signal Processing

130

(10)

Review Exercises

137

(3)

The Laplace Transform

140

(42)

Introduction

140

(1)

Definition and Examples

140

(3)

More Formulae and Examples

143

(6)

Applications to Ordinary Differential Equations

149

(6)

The Heaviside and Dirac-Delta Functions

155

(7)

Convolution

162

(6)

More Examples and Applications

168

(5)

The Inverse Transform Formula

173

(2)

Applications of the Inverse Transform

175

(7)

Review Exercises

180

(2)

Appendix A: The Residue Theorem and Related Results

182

(4)

Appendix B: Leibniz's Rule and Fubini's Theorem

186

(2)

Index

188

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Fourier Series and Integral Transforms

Summary

Table of Contents

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