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这本专著介绍了偏微分方程中用到的傅里叶分析及其应用的基本知识，作者以深入浅出的语言介绍此理论，即使基础知识较少的读者阅读此专著也不会觉得困难。其次，作者还介绍了更前沿的理论，例如，Gibbs现象，Sturm-Liouville定理，多维傅里叶分析等，而这些理论在其他此类专著中基本很难看到。而且此书中的一系列例题和习题可以帮助读者更好的理解书中的知识。
【目录】
- Preface
1 Introduction
1.1 The classical partial differential equations
1.2 Wall-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 Cesàro summation of series
2.4 Positive summation kernels
2.5 The Riemann-Lebesgue lemma
2.6 *Some simple distributions
2.7 *Computing with δ
3 Laplace and Z transforms
3 Laplace and Z transforms
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 Fourier series
4.1 Definitions
4.2 Dirichlet's and Fejér's kernels;uniqueness
4.3 Differentiable functions
4.4 Pointwise convergence
4.5 Formulae for other periods
4.6 Some worked examples
4.7 The Gibbs phenomenon
4.8 *Fourier series for distributions
Smnmary of Chapter 4
5 L^2 Theory
5.1 Linear spaces over the complex numbers
5.2 Orthogonal projections
5.3 Some examples
5.4 The Fourier 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 of Fourier's problem
6.2 Variations on Fourier's theme
6.3 The Dirichlet problem in the unit disk
6.4 Sturm-Liouvi]le problems
6.5 Some singular Sturm-Liouville problems
Summary of Chapter 6
7 Fourier transforms
7.1 Lntreduction
7.2 Definition of the Fourier transform
7.3 Properties
?,4 The inversion theorem
?.5 The convolution theorem
7.6 plancherel's formula
7.7 Application l
7.8 Application 2
7.9 Application 3:The sampling theorem
7.10 *Connection with the Laplace transform
7.11 *Distributions and Fourier trausforms
Summary of Chapter 7
8 Distributions
8.1 History
8.2 Fuzzy points-test functions
8.3 Distributions
8.4 Properties
8.5 Fourier transformation
8.6 Convolution
8.7 Periodic distributions and Fourier series
8.8 Fundamental solutions
8.9 Back to the starting point
Summary of Chapter 8
9 Multi-dimensional Fourier analysis
9.1 Rearranging series
9.2 Double series
9.3 Multi-dimensional Fourier series
9.4 Multi-dimensional Fourier transforms
Appendices
A The ubiquitous convolution
B The discrete Fourier transform
C Formulae
C.1 Lapisce transforms
C.2 Z transforms
C.3 Fourier series
C.4 Fourier transforms
C.5 Orthogonal polynomials
D Answers to selected exercises
E Literature
Index
