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 電子書 2024

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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。
　　這次科學齣版社購買瞭版權，一次影印瞭23本施普林格齣版社齣版的數學書，就是一件好事，也是值得繼續做下去的事情。大體上分一下，這23本書中，包括基礎數學書5本，應用數學書6本與計算數學書12本，其中有些書也具有交叉性質。這些書都是很新的，2000年以後齣版的占絕大部分，共計16本，其餘的也是1990年以後齣版的。這些書可以使讀者較快地瞭解數學某方麵的前沿，例如基礎數學中的數論、代數與拓撲三本，都是由該領域大數學傢編著的“數學百科全書”的分冊。對從事這方麵研究的數學傢瞭解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數學類的書以“經典”為主，應用和計算數學類的書以“前沿”為主。這些書的作者多數是國際知名的大數學傢，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲奬”和“沃爾夫數學奬”。這些大數學傢的著作無疑將會對我國的科研人員起到非常好的指導作用。
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