Fourier Series and Integral Transforms

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Cambridge University Press, Jul 10, 1997 - Mathematics - 189 pages
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This volume provides 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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Contents

Notation and Terminology
1
2 Calculus Notation
2
3 Useful Trigonometric Formulae
4
Background 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
The Fourier Transform
93
2 Examples
98
3 Properties and Formulae
102
4 The Inverse Fourier Transform and Plancherels Identity
108
6 Applications of the Residue Theorem
119
7 Applications to Partial Differential Equations
125
8 Applications to Signal Processing
130
The Laplace Transform
140

Fourier Series
32
2 Evenness Oddness and Additional Examples
40
3 Complex Fourier Series
42
4 Pointwise Convergence and Dirichlets Theorem
46
5 Uniform Convergence
56
6 Parsevals 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
2 More Formulae and Examples
143
3 Applications to Ordinary Differential Equations
149
4 The Heaviside and DiracDelta Functions
155
5 Convolution
162
6 More Examples and Applications
168
7 The Inverse Transform Formula
173
8 Applications of the Inverse Transform
175
The Residue Theorem and Related Results
182
Leibnizs Rule and Fubinis Theorem
186
Index
188
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About the author (1997)

Allan Pinkus has been in the Department of Mathematics at Technion since 1977, and became a full Professor in 1987. He is the author of three research monographs, one textbook, over 100 research articles, and he has edited six proceedings. His main research interests center around approximation theory. Pinkus is a member of various editorial boards and served for ten years as editor-in-chief of the Journal of Approximation Theory. He has held numerous visiting appointments, and has lectured extensively at international conferences.

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