Fourier Series and Integral Transforms

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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.
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Editorial Reviews

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"With its frequent examples and exercises, the present book is eminently suitable for both self-study and a one-semester course." Choice
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Product Details

  • ISBN-13: 9780521592093
  • Publisher: Cambridge University Press
  • Publication date: 9/1/1997
  • Pages: 200
  • Product dimensions: 5.98 (w) x 8.98 (h) x 0.63 (d)

Table of Contents

0 Notation and Terminology 1
1 Basic Concepts in Set Theory 1
2 Calculus Notation 2
3 Useful Trigonometric Formulae 4
1 Background: Inner Product Spaces 5
1 Linear and Inner Product Spaces 5
2 The Norm 10
3 Orthogonal and Orthonormal Systems 15
4 Orthogonal Projections and Approximation in the Mean 19
5 Infinite Orthonormal Systems 24
2 Fourier Series 32
1 Definitions 32
2 Evenness, Oddness, and Additional Examples 40
3 Complex Fourier Series 42
4 Pointwise Convergence and Dirichlet's Theorem 46
5 Uniform Convergence 56
6 Parseval's Identity 63
7 The Gibbs Phenomenon 68
8 Sine and Cosine Series 72
9 Differentiation and Integration of Fourier Series 76
10 Fourier Series on Other Intervals 81
11 Applications to Partial Differential Equations 85
3 The Fourier Transform 93
1 Definitions and Basic Properties 93
2 Examples 98
3 Properties and Formulae 102
4 The Inverse Fourier Transform and Plancherel's Identity 108
5 Convolution 116
6 Applications of the Residue Theorem 119
7 Applications to Partial Differential Equations 125
8 Applications to Signal Processing 130
4 The Laplace Transform 140
1 Definition and Examples 140
2 More Formulae and Examples 143
3 Applications to Ordinary Differential Equations 149
4 The Heaviside and Dirac-Delta Functions 155
5 Convolution 162
6 More Examples and Applications 168
7 More Inverse Transform Formula 173
8 Applications of the Inverse Transform 175
App. A The Residue Theorem and Related Results 182
App. B Leibniz's Rule and Fubini's Theorem 186
Index 188
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