傅里叶分析及其应用 : 第2版

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 Fourier 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

7.11 *Distributions and Fourier transforms

Summary of Chapter 7

《傅里叶分析及其应用(影印版)》是国外数学名著系列之一。《傅里叶分析及其应用(影印版)》内容简介：A carefully prepared account of the basic ideas in Fourier analysis and its applications to the study of partial differential equations. The author succeeds to make his exposition accessible to readers with a limited background, for example, those not acquainted with the Lebesgue integral. Readers should be familiar with calculus, linear algebra, and complex numbers. At the same time, the author has managed to include discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability and multi-dimensional Fourier analysis, topics which one usually does not find in books at this level. A variety of worked examples and exercises will help the readers to apply their newly acquired knowledge.【作者简介】作者：（瑞典）弗雷特布拉德（Anders Vretblad）