傅里叶分析及其应用+2

内容简介

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 equationr />1．2 Well-posed problemr />1．3 The one-dimensional wave equation
1．4 Fourier's method

2 Preparationr />2．1 Complex exponentialr />2．2 Complex-valued functions of a real variable
2．3 Cesaro summation of serier />2．4 Positive summation kernelr />2．5 The riemann-lebesgue lemma
2．6 *Some simple distributionr />2．7 *Computing with δ

3 Laplace and z transformr />3．1 The laplace transform
3．2 Operationr />3．3 Applications to differential equationr />3．4 Convolution
3．5 *Laplace transforms of distributionr />3．6 The z transform
3．7 Applications in control theory
Summary of chapter 3

4 Fourier serier />4．1 Definitionr />4．2 Dirichlet's and fejer's kernels; uniquener />4．3 Differentiable functionr />4．4 Pointwise convergence
4．5 Formulae for other periodr />4．6 Some worked exampler />4．7 The gibbs phenomenon
4．8 *Fourier series for distributionr />Summary of chapter 4

5 L2 theory
5．1 Linear spaces over the complex numberr />5．2 Orthogonal projectionr />5．3 Some exampler />5．4 The fourier system is complete
5．5 Legendre polynomialr />5．6 Other classical orthogonal polynomialr />Summary of chapter 5

6 Separation of variabler />6．1 The solution of fourier's problem
6．2 Variations on fourier's theme
6．3 The dirichlet problem in the unit dir />6．4 Sturm-liouville problemr />6．5 Some singular sturm-liouville problemr />Summary of chapter 6

7 Fourier transformr />7．1 Introduction
7．2 Definition of the fourier transform
7．3 Propertier />7．4 The inversion theorem．
7．5 The convolution theorem
7．6 Plancherel's formula
7．7 Application i
7．8 Application 2
7．9 Application 3： the sampling theorem
7．10 *Connection with the laplace transform
7．11 *Cistributions and fourier transformr />Summary of chapter 7

8 Distributionr />8．1 History
8．2 Fuzzy points - test functionr />8．3 Distributionr />8．4 Propertier />8．5 Fourier transformation
8．6 Convolution
8．7 Periodic distributions and fourier serier />8．8 Fundamental solutionr />8．9 Back to the starting point
Summary of chapter 8

9 Multi-dimensional fourier analysir />9．1 Rearranging serier />9．2 Double serier />9．3 Multi-dimensional fourier serier />9．4 Multi-dimensional fourier transformr />
Appendicer />A The ubiquitous convolution
B The discrete fourier transform
C Formulae
C．1 Laplace transformr />C．2 Z transformr />C．3 Fourier serier />C．4 Fourier transformr />C．5 Orthogonal polynomialr />D Answers to selected exerciser />E Lterature
Index