# Analysis Fourier Analysis and Its Applications by Gerald B. Folland

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1. Feb 22, 2013

### Astronuc

Staff Emeritus

Code (Text):
1 Overture
1.1 Some equations of mathematical physics
1.2 Linear differential operators
1.3 Separation of variable

2 Fourier Series
2.1 The Fourier series of a periodic function
2.2 A convergence theorem
2.3 Derivatives, integrals, and uniform convergence
2.4 Fourier series on interval
2.5 Some applications
2.6 Further remarks on Fourier series

3 Orthogonal Sets of Functions
3.1 Vectors and inner products
3.2 Functions and inner products
3.3 Convergence and completeness
3.4 More about L2 spaces; the dominated convergence theorem
3.5 Regular Sturm-Liouville problems
3.6 Singular Sturm-Liouville problems

4. Some Boundary Value Problems
4.1 Some useful techniques
4.2 One-dimensional heat flow
4.3 One-dimensional wave motion
4.4 The Dirichlet problem
4.5 Multiple Fourier series and applications

5 Bessel Functions
5.1 Solutions of Bessel's equation
5.2 Bessel function identities
5.3 Asymptotics and zeros of Bessel functions
5.4 Orthogonal sets of Bessel functions
5.5 Applications of Bessel functions
5.6 Variants of Bessel functions

6 Orthogonal Polynomials
6.1 Introduction
6.2 Legendre polynomials
6.3 Spherical coordinates and Legendre functions
6.4 Hermite polynomials
6.5 Laguerre polynomials
6.6 Other orthogonal bases

7 The Fourier Transform
7.1 Convolutions
7.2 The Fourier Transform
7.3 Some applications
7.4 Fourier transforms and Sturm-Liouville problems
7.5 Multivariable convolutions and Fourier transforms
7.6 Transforms related to the Fourier transform

The Laplace Transform
8.1 The Laplace Transform
8.2 The inversion formula
8.3 Applications: Ordinary differential equations
8.4 Applications: Partial differential equations
8.5 Applications: Integral equations
8.6 Asymptotics of Laplace transforms

9 Generalized Functions
9.1 Distributions
9.2 Convergence, convolution, and approximation
9.3 More examples: Periodic distributions and finite parts
9.4 Tempered distributions and Fourier transforms
9.5 Weak solutions of differential equations

10 Green's Functions
10.1 Green's functions for ordinary differential operators
10.2 Green's functions for partial differential operators
10.3 Green's functions and regular Sturm-Liouville problems
10.4 Green's functions and singular Sturm-Liouville problems

Appendices
1 Some physical derivations
2 Summary of complex variable theory
3 The gamma function
4 Calculations in polar coordinates
5 The fundamental theorem of ordinary differential equations