作者：吴晓丽
页数：444
出版社：高等教育出版社
出版日期：2024
ISBN：9787040612011
电子书格式：pdf/epub/txt
内容简介
在过去的200年中，调和分析一直是数学思想中最具影响力的主体之一，在其理论含义和在整个数学、科学和工程中的巨大适用范围方面都具有非凡的意义。在本书中，作者们传达了从傅里叶理论发展而来的思想所具有的显著的美和适用性。他们为高年级本科生和低年级研究生读者阐述了调和分析的基础知识，从傅里叶对热方程的研究以及将函数分解为余弦和正弦的和（频率分析），到二进制调和分析和将函数分解为哈尔基函数的和（时间局部化）。尽管主要讨论了傅里叶和哈尔情形，但本书也涉及介于这两种不同函数分解方式之间的领域：时频分析（小波分析）。书中同时呈现了有限和连续两种视角，引入离散傅里叶和哈尔变换以及快速算法，如快速傅里叶变换（FFT）及其小波模拟。 本书的方法结合了严谨的证明、引人入胜的动机和众多的应用。书中包含250多个练习题。每章结束时都会提供一些调和分析的专题研究，学生可以独立完成。
目录
List of figures
List of tables
IAS/Park City Mathematics Institute
Preface
Suggestions for instructors
Acknowledgements
Chapter 1.Fourier series: Some motivation
§1.1.An example: Amanda calls her mother
§1.2.The main questions
§1.3.Fourier series and Fourier coefficients
§1.4.History, and motivation from the physical world
§1.5.Project: Other physical models
Chapter 2.Interlude: Analysis concepts
§2.1.Nested classes of functions on bounded intervals
§2.2.Modes of convergence
§2.3.Interchanging limit operations
§2.4.Density
§2.5.Project: Monsters, Take I
Chapter 3.Pointwise convergence of Fourier series
§3.1.Pointwise convergence: Why do we care?
§3.2.Smoothness vs. convergence
§3.3.A suite of convergence theorems
§3.4.Project: The Gibbs phenomenon
§3.5.Project: Monsters, Take II
Chapter 4.Summability methods
§4.1.Partial Fourier sums and the Dirichlet kernel
§4.2.Convolution
§4.3.Good kernels, or approximations of the identity
§4.4.Fejer kernels and CesAro means
§4.5.Poisson kernels and Abel means
§4.6.Excursion into LP(T)
§4.7.Project: Weyl’s Equidistribution Theorem
§4.8.Project: Averaging and summability methods
Chapter 5.Mean-square convergence of Fourier series
§5.1.Basic Fourier theorems in L2(T)
§5.2.Geometry of the Hilbert space L2(T)
§5.3.Completeness of the trigonometric system
§5.4.Equivalent conditions for completeness
§5.5.Project: The isoperimetric problem
Chapter 6.A tour of discrete Fourier and Haar analysis
§6.1.Fourier series vs. discrete Fourier basis
§6.2.Short digression on dual bases in CN
§6.3.The Discrete Fourier Transform and its inverse
§6.4.The Fast Fourier Transform (FFT)
§6.5.The discrete Haar basis
§6.6.The Discrete Haar Transform
§6.7.The Fast Haar Transform
§6.8.Project: Two discrete Hilbert transforms
§6.9.Project: Fourier analysis on finite groups
Chapter 7.The Fourier transform in paradise
List of tables
IAS/Park City Mathematics Institute
Preface
Suggestions for instructors
Acknowledgements
Chapter 1.Fourier series: Some motivation
§1.1.An example: Amanda calls her mother
§1.2.The main questions
§1.3.Fourier series and Fourier coefficients
§1.4.History, and motivation from the physical world
§1.5.Project: Other physical models
Chapter 2.Interlude: Analysis concepts
§2.1.Nested classes of functions on bounded intervals
§2.2.Modes of convergence
§2.3.Interchanging limit operations
§2.4.Density
§2.5.Project: Monsters, Take I
Chapter 3.Pointwise convergence of Fourier series
§3.1.Pointwise convergence: Why do we care?
§3.2.Smoothness vs. convergence
§3.3.A suite of convergence theorems
§3.4.Project: The Gibbs phenomenon
§3.5.Project: Monsters, Take II
Chapter 4.Summability methods
§4.1.Partial Fourier sums and the Dirichlet kernel
§4.2.Convolution
§4.3.Good kernels, or approximations of the identity
§4.4.Fejer kernels and CesAro means
§4.5.Poisson kernels and Abel means
§4.6.Excursion into LP(T)
§4.7.Project: Weyl’s Equidistribution Theorem
§4.8.Project: Averaging and summability methods
Chapter 5.Mean-square convergence of Fourier series
§5.1.Basic Fourier theorems in L2(T)
§5.2.Geometry of the Hilbert space L2(T)
§5.3.Completeness of the trigonometric system
§5.4.Equivalent conditions for completeness
§5.5.Project: The isoperimetric problem
Chapter 6.A tour of discrete Fourier and Haar analysis
§6.1.Fourier series vs. discrete Fourier basis
§6.2.Short digression on dual bases in CN
§6.3.The Discrete Fourier Transform and its inverse
§6.4.The Fast Fourier Transform (FFT)
§6.5.The discrete Haar basis
§6.6.The Discrete Haar Transform
§6.7.The Fast Haar Transform
§6.8.Project: Two discrete Hilbert transforms
§6.9.Project: Fourier analysis on finite groups
Chapter 7.The Fourier transform in paradise
