Fourier Series and Integral Transforms

Allan Pinkus and Samy Zafrany

Now Available from Cambridge University Press.

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TABLE OF CONTENTS

Chapter 0: Notation and Terminology

1. Basic Concepts in Set Theory
2. Calculus Notation
3. Useful Trigonometric Formulae

Chapter 1: Background: Inner Product Spaces

1. Introduction
2. Linear and Inner Product Spaces
3. The Norm
4. Orthogonal and Orthonormal Systems
5. Orthogonal Projections and Approximation in the Mean
6. Infinite Orthonormal Systems
7. Review Exercises

Chapter 2 Fourier Series

1. Introduction
2. Definitions
3. Evenness, Oddness, and Additional Examples
4. Complex Fourier Series
5. Pointwise Convergence and Dirichlet's Theorem
6. Uniform Convergence
7. Parseval's Identity
8. The Gibbs Phenomenon
9. Sine and Cosine Series
10. Differentiation and Integration of Fourier Series
11. Fourier Series on Other Intervals
12. Applications to Partial Differential Equations
13. Review Exercises

Chapter 3 The Fourier Transform

1. Introduction
2. Definitions and Basic Properties
3. Examples
4. Properties and Formulae
5. The Inverse Fourier Transform and Plancherel's Identity
6. Convolution
7. Applications of the Residue Theorem
8. Applications to Partial Differential Equations
9. Applications to Signal Processing
10. Review Exercises

Chapter 4: The Laplace Transform

1. Introduction
2. Definition and Examples
3. More Formulae and Examples
4. Applications to Ordinary Differential Equations
5. The Heaviside and Dirac-Delta Functions
6. Convolution
7. More Examples and Applications
8. The Inverse Transform Formula
9. Applications of the Inverse Transform
10. Review Exercises

Appendix A: The Residue Theorem and Related Results

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