Preface
1 Introduction
1.1 The classical partial differential equations
1.2 Well-posed problems
1.3 The one-dimensional wave equation
1.4 Fourier'S method

2 Preparations
2.1 Complex exponentials
2.2 Complex-valued functions of a real variable
2.3 Ceshro summation of series
2.4 Positive summation kernels
2.5 The Riemann-Lebesgue lemma
2.6*Some simple distributions
2.7*Computing with ?

3 Laplace and Z transforms
3.1 The Laplace transfclrm
3.2 Operations
3.3 Applications to differential equations
3.4 Convolution
3.5*Laplace transforms of distributions
3.6　The Z transform
3.7 Applications in control theory
Summary of Chapter 3

4 FDurier series
4.1 Definitions
4.2 Dirichlet'S and Fejdr'S kernels；uniqueness
4.3 Diffrentiable functions
4.4 Pointwise convergence
4.5 Formulae for other periods
4.6 Some worked examples
4.7 The Gibbs phenomenon
4.8　*F0urier series for distributions
Summary of Chapter 4

5 L2 Theory
5.1 Linear SlClaces over the complex numbers
5.2 Orthogonal projections
5.3 Some examples
5.4 Tlle F0urier system iS complete
5.5 Legendre polynomials
5.6 Other classical orthogonal polynomials
Summary of Chapter 5

6 Separation of variables
6.1　The solution 0f F0urier'S problem
6.2 Vaiations on Fourier'S theme
6.3 The Dirichlet problem in the unit disk
6.4 Sturm-Liouville problems
6.5 Some singular Sturm-Liouville problems
Summary of Chapter 6

7 F0urier transforms
7.1 Introduction
7.2 Definition of the Fourier transform
7.3 Properties
7.4 The inversion theorem
7.5 The convolution theorem
7.6 Plancherel'S formula
7.7 Application 1
7.8 Application 2
7.9 Application 3：The sampling theorem
7.10 木Connection with the Laplace transfocrm
7.1 1*Distributions and Fourier transforms
Summary of Chapter 7