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內容簡介

　　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.

內頁插圖

目錄

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 Cesaro 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
3．1 The laplace transform
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 fejer'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
Summary of chapter 4

5 L2 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-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 i
7．8 Application 2
7．9 Application 3： the sampling theorem
7．10 *Connection with the laplace transform
7．11 *Cistributions and fourier transforms
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 Laplace transforms
C．2 Z transforms
C．3 Fourier series
C．4 Fourier transforms
C．5 Orthogonal polynomials
D Answers to selected exercises
E Lterature
Index

前言/序言

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　　從齣版方麵來講，除瞭較好較快地齣版我們自己的成果外，引進國外的先進齣版物無疑也是十分重要與必不可少的。從數學來說，施普林格（Springer）齣版社至今仍然是世界上的齣版社。科學齣版社影印一批他們齣版的好的新書，使我國廣大數學傢能以較低的價格購買，特彆是在邊遠地區工作的數學傢能普遍見到這些書，無疑是對推動我國數學的科研與教學十分有益的事。
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國外數學名著係列（影印版）80：傅裏葉分析及其應用 下載 mobi epub pdf txt 電子書

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