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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.

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目录

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

　　要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。
　　从出版方面来讲，除了较好较快地出版我们自己的成果外，引进国外的先进出版物无疑也是十分重要与必不可少的。从数学来说，施普林格（Springer）出版社至今仍然是世界上的出版社。科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动我国数学的科研与教学十分有益的事。
　　这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。
国外数学名著系列（影印版）80：傅里叶分析及其应用 电子书 下载 mobi epub pdf txt

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