具体描述
本书是由一位成功的数学家用传统的表述形式编写的教材，除了主要介绍傅里叶级数和傅里叶分析外，还讨论了另一类可以通过傅里叶方法进行研究的正交级数，即小波分析。
本书可以作为理工类专业高年级本科生和低年级研究生的教材。
本书特点：
·书中的所有命题、引理、定理和推论都给出了严格的证明
·某些情况下，对于一个给定的命题，给出几种证明方法，使学生可以对不同方法进行比较
·不属于傅里叶分析主流题材但与本书有特定联系的论题，在带有*号的小节中讲述
·理论和实践相结合
经典力学导论：从牛顿定律到现代场论的基石 内容简介 本书旨在为物理学、工程学以及应用数学领域的初学者和进阶学习者提供一套全面且严谨的经典力学理论框架。我们深知，经典力学是理解物质世界宏观运动规律的基石，其思想和方法深刻地影响了后续所有物理学分支的发展。因此，本书的撰写遵循了从基本概念的清晰界定到复杂系统分析的递进路线，力求内容翔实、推导细致，同时又不失对物理图像的深刻洞察。 全书共分为六大部分，涵盖了从牛顿力学的基础概念到拉格朗日、哈密顿体系的建立，并适度触及了连续介质力学和经典场论的初步概念。 --- 第一部分：牛顿力学的再审视与运动学基础 本部分着重于对经典力学的基本公设进行严格的梳理和审视。我们首先回顾了牛顿三大定律的物理内涵及其在惯性参考系中的适用性，并对“力”、“质量”和“惯性”等核心概念进行了操作性定义。 运动学分析是本部分的核心内容。我们详细讨论了质点在不同坐标系（笛卡尔、柱面、球坐标系）下的运动描述，特别强调了速度、加速度的张量表示法及其在非惯性系中的修正项——科里奥利力和离心力。通过大量的二维和三维实例，读者将能够熟练掌握如何利用微积分工具精确描述复杂的运动轨迹，为后续引入动力学概念奠定坚实基础。 相对性初步：在讨论完经典伽利略变换后，本部分会简要引入对伽利略相对性原理的批判性思考，为过渡到更高级的理论形式做好铺垫。 --- 第二部分：动力学与守恒定律的深度剖析 在明确了运动学的框架后，第二部分深入探讨了力的作用如何引起物体的运动变化——即动力学。 功与能的概念：我们超越了简单的$W = Fd$定义，引入了变力做功的积分形式，并详尽推导了动能定理。势能的概念被系统地引入，重点讨论了保守力场的定义及其势能函数的梯度关系。由此，机械能守恒定律作为最基本、应用最广泛的动力学原理被确立。 动量与角动量：冲量-动量定理的推导及其在碰撞问题中的应用是本节的重点。角动量守恒定律的几何意义和其与绕轴转动的紧密联系，通过考察刚体运动的特性得以展现。我们对中心力问题（如开普勒问题）的解析解进行了详细的几何和代数推导，揭示了轨道形状的本质决定因素。 多体系统与约束：本部分也涵盖了变质量系统（如火箭运动）的动力学分析，并首次引入了描述运动系统时不可或缺的“约束”概念。对光滑、无滑动的理想约束力的处理方式，为向解析力学过渡做了必要的概念储备。 --- 第三部分：解析力学的建立——拉格朗日力学 解析力学是物理学中实现从牛顿框架向更优雅、更具普适性形式转变的关键一步。本部分聚焦于拉格朗日力学的构造。 变分原理与最小作用量：本书从达朗贝尔原理出发，自然而然地导出了虚拟功原理，进而引出了牛顿方程的变分等价形式——欧拉-拉格朗日方程。我们对泛函的求导——泛函导数（泛函微元）进行了必要的数学介绍，确保读者能够理解最小作用量原理的严格性。 拉格朗日量与广义坐标：本书详细解释了引入广义坐标的必要性，及其如何有效地“消除”约束力，从而简化问题的求解过程。通过对自由度、约束方程的分析，读者将学会如何构建复杂系统的拉格朗日量。 守恒量的发现：诺特定理的优雅之处在于其揭示了系统对称性与守恒量之间的深刻联系。本部分将对诺特定理的数学表述及其在经典力学中的具体应用（如能量、动量、角动量守恒）进行透彻讲解。 --- 第四部分：哈密顿力学与相空间 哈密顿力学是连接经典力学与量子力学的桥梁。本部分侧重于从拉格朗日形式向哈密顿形式的正则变换。 哈密顿量与正则方程：通过勒让德变换，我们定义了哈密顿量，并推导出一组一阶微分方程——哈密顿正则方程。本书会详细分析相空间的几何结构，以及相轨迹的演化特性。 泊松括号：泊松括号被引入作为描述物理量时间演化的核心工具。我们展示了泊松括号如何封装了哈密顿正则方程的结构，并讨论了守恒量在泊松括号表示下的表现。 正则变换：本节详细介绍了保持哈密顿体系形式不变的正则变换的生成函数方法。通过引入生成函数，我们可以更灵活地寻找新的、更容易求解的正则坐标系。 --- 第五部分：刚体动力学与微扰理论入门 在掌握了抽象的解析力学工具后，本书回归到具体的宏观物理系统——刚体。 刚体的运动描述：刚体运动被分解为质心平动和绕质心的转动。详细分析了转动惯量张量、惯性主轴的概念，以及欧拉角在描述任意刚体定向上的应用。本书对欧拉方程的推导和其在陀螺仪等问题中的应用给出了详尽的分析。 微扰论基础：对于难以精确求解的系统，微扰论提供了强大的近似工具。本部分首先介绍了定态微扰论的一阶和二阶修正，以及非简并和简并情况下的处理方法。这些工具的应用实例将集中在简单的力学模型上，旨在展示如何处理微小、缓慢变化的系统。 --- 第六部分：连续介质力学的初步概念 本部分的目的是将离散质点体系的分析方法推广到具有无限多自由度的连续介质。 流体静力学与流体力学：我们从流体的基本性质（密度、压强）出发，推导了静力学平衡条件（欧拉方程的静力学形式）。在流体力学部分，本书引入了连续性方程（质量守恒）和牛顿第二定律的流体形式——纳维-斯托克斯方程的推导。 弹性形变基础：最后，我们简要介绍了弹性体的应力与应变张量的基本概念，阐述了胡克定律在三维空间中的张量形式，为理解材料在受力下的形变行为提供了力学基础。 全书贯穿了严谨的数学推导和清晰的物理图像，辅以丰富的例题解析，旨在培养读者运用场论观点思考宏观物理问题的能力，为后续进入热力学、电动力学或量子力学学习打下坚实的数学和理论基础。
作者简介
目录信息
1 FOURIER SERIES ON THE CIRCLE
1.1 Motivation and Heuristics
1.1.1 Motivation from Physics
1.1.1.1 The Vibrating String
1.1.1.2 Heat Flow in Solids
1.1.2 Absolutely Convergent Trigonometric Series
1.1.3 *Examples of Factorial and Bessel Functions
1.1.4 Poisson Kernel Example
1.1.5 *Proof of Laplace's Method
1.1.6 *Nonabsolutely Convergent Trigonometric Series
1.2 Formulation of Fourier Series
1.2.1 Fourier Coefficients and Their Basic Properties
1.2.2 Fourier Series of Finite Measures
1.2.3 *Rates of Decay of Fourier Coefficients
1.2.3.1 Piecewise Smooth Functions
1.2.3.2 Fourier Characterization of Analytic Functions
1.2.4 Sine Integral
1.2.4.1 Other Proofs That Si( ) =l
1.2.5 Pointwise Convergence Criteria
1.2.6 *Integration of Fourier Series
1.2.6.1 Convergence of Fourier Series of Measures
1.2.7 Riemann Localization Principle
1.2.8 Gibbs-Wilbraham Phenomenon
1.2.8.l The General Case
1.3 Fourier Series in L2
1.3.l Mean Square Approximation-Parseval's Theorem
1.3.2 *Application to the Isoperimetric Inequality
1.3.3 *Rates of Convergence in L2
1.3.3.1 Application to Absolutely-Convergent Fourier
Series
1.4 Norm Convergence and Summability
1.4.1 Approximate Identities
1.4.1.1 Almost-Everywhere Convergence of the Abel
Means
1.4.2 Summability Matrices
l.4.3 Fejer Means of a Fourier Series
1.4.3.1 Wiener's Closure Theorem on the Circle
1.4.4 *Equidistribution Modulo One
1.4.5 *Hardy's Tauberian Theorem
1.5 Improved Trigonometric Approximation
1.5.1 Rates of Convergence in C(T)
1.5.2 Approximation with Fejer Means
1.5.3 *Jackson's Theorem
1.5.4 *Higher-Order Approximation
1.5.5 *Converse Theorems of Bernstein
1.6 Divergence of Fourier Series
1.6.1 The Example of du Bois-Reymond
1.6.2 Analysis via Lebesgue Constants
1.6.3 Divergence in the Space L1
1.7 *Appendix: Complements on Laplace's Method
1.7.O.l First Variation on the Theme-Gaussian
Approximation
1.7.0.2 Second Variation on the Theme-Improved Error
Estimate
1.7.l *Application to Bessel Functions
1.7.2 *The Local Limit Theorem of DeMoivre-Laplace
1.8 Appendix: Proof of the Uniform Boundedness Theorem
1.9 *Appendix: Higher-Order Bessel functions
1.10 Appendix: Cantor's Uniqueness Theorem
2 FOURIER TRANSFORMS ON THE LINE AND SPACE
2.1 Motivation and Heuristics
2.2 Basic Properties of the Fourier Transform
2.2.l Riemann-Lebesgue Lemma
2.2.2 Approximate Identities and Gaussian Summability
2.2.2.1 Improved Approximate Identities for Pointwise
Convergence
2.2.2.2 Application to the Fourier Transform
2.2.2.3 The n-Dimensional Poisson Kernel
2.2.3 Fourier Transforms of Tempered Distributions
2.2.4 *Characterization of the Gaussian Density
2.2.5 *Wiener's Density Theorem
2.3 Fourier Inversion in One Dimension
2.3.1 Dirichlet Kernel and Symmetric Partial Sums
2.3.2 Example of the Indicator Function
2.3.3 Gibbs-Wilbraham Phenomenon
2.3.4 Dini Convergence Theorem
2.3.4.1 Extension to Fourier's Single Integral
2.3.5 Smoothing Operations in R1-Averaging and Summability
2.3.6 Averaging and Weak Convergence
2.3.7 Cesaro Summability
2.3.7.1 Approximation Propenies of the Fejer Kernel
2.3.8 Bemstein's Inequality
2.3.9 *One-Sided Fourier Integral Representation
2.3.9.1 Fourier Cosine Transform
2.3.9.2 Fourier Sine Transform
2.3.9.3 Generalized h-Transform
2.4 L2 Theory in Rn
2.4.1 Plancherel's Theorem
2.4.2 *Bernstein's Theorem for Fourier Transforms
2.4.3 The Uncertainty Principle
2.4.3.l Uncertainty Principle on the Circle
2.4.4 Spectral Analysis of the Fourier Transform
2.4.4.1 Hermite Polynomials
2.4.4.2 Eigenfunction of the Fourier Transform
2.4.4.3 Orthogonality Properties
2.4.4.4 Completeness
2.5 Spherical Fourier Inversion in Rn
2.5.l Bochner's Approach
2.5.2 Piecewise Smooth Viewpoint
2.5.3 Relations with the Wave Equation
2.5.3.l The Method of Brandolini and Colzani
2.5.4 Bochner-Riesz Summability
2.5.4.l A General Theorem on Almost-Everywhere
Summability
2.6 Bessel Functions
2.6.l Fourier Transforms of Radial Functions
2.6.2 L2-Restriction Theorems for ate Fourier Transform
2.6.2.l An Improved Result
2.6.2.2 Limitations on the Range of p
2.7 The Method of Stationary Phase
2.7.l Statement of the Result
2.7.2 Application to Bessel Functions
2.7.3 Proof of the Method of Stationary Phase
2.7.4 Abel's Lemma
3 FOURIER ANALYSIS IN LP SPACES
3.l Motivation and Heuristics
3.2 The M.Riesz-Thorin Interpolation Theorem
3.2.0.l Generalized Young's Inequality
3.2.0.2 The Hausdorff-Young Inequality
3.2.l Stein's Complex Interpolation Theorem
3.3 The Conjugate Function or Discrete Hilbert Transform
3.3.1 LP Theory of the Conjugate Function
3.3.2 L1 Theory of the Conjugate Function
3.3.2.l Identification as a Singular Integral
3.4 The Hilbert Transform on R
3.4.l L2 Theory of the Hilbert Transform
3.4.2 Lp Theory of the Hilbert Transform.l [ P [
3.4.2.l Applications to Convergence of Fourier Integrals
3.4.3 L1 Theory of the Hilbert Transform and Extensions
3.4.3.l Kolmogorov's Inequality for the Hilbert
Transform
3.4.4 Application to Singular Integrals with Odd Kernels
3.5 Hardy-Littlewood Maximal Function
3.5.l Application to the Lebesgue Differentiation Theorem
3.5.2 Application to Radial Convolution Operators
3.5.3 Maximal Inequalities for Spherical Averages
3.6 The Marcinkiewicz Interpolation Theorem
3.7 Calderon-Zygmund Decomposition
3.8 A Class of Singular Integrals
3.9 Propenies of Harmonic Functions
3.9.l General Properties
3.9.2 Representation Theorems in the Disk
3.9.3 Representation Theorems in the Upper Half-Plane
3.9.4 Herglotz/Bochner Theorems and Positive Definite
Functions
4 POISSON SUMMATION FORMULA AND
MULTIPLE FOURIER SERIES
4.l Motivation and Heudstics
4.2 The Poisson Summation Formula in R1
4.2.1 Periodization of a Function
4.2.2 Statement and Proof
4.2.3 Shannon Sampling
4.3 Multiple Fourier Series
4.3.l Basic L1 Theory
4.3.1.1 Pointwise Convergence for Smooath Functions
4.3.1.2 Representation of Spherical Partial Sums
4.3.2 Basic L2 Theory
4.3.3 Restriction Theorems for Fourier Coefficients
4.4 Poisson Summation Formula in Rd
4.4.1 *Simultaneous Nonlocalization
4.5 Application to Lattice Points
4.5.l Kendall's Mean Square Error
4.5.2 Landau's Asymptotic Formula
4.5.3 Application to Multiple Fourier Series
4.5.3.1 Three-Dimensional Case
4.5.3.2 Higher-Dimensional Case
4.6 Schrodinger Equation and Gauss Sums
4.6.l Distibutions on the Circle
4.6.2 The Schrodinger Equation on the Circle
4.7 Recurrence of Random Walk
5 APPLICATIONS TO PROBABILITY THEORY
5.l Motivation and Heuristics
5.2 Basic Definitions
5.2.1 The Central Limit Theorem
5.2.1.1 Restatement in Terms of Independent
Random Variables
5.3 Extension to Gap Series
5.3.l Extension to Abel Sums
5.4 Weak Convergence of Measures
5.4.l An Improved Continuity Theorem
5.4.l.l Another Proof of Bochner's Theorem
5.5 Convolution Semigroups
5.6 The Berry-Esseen Theorem
5.6.l Extension to Different Distributions
5.7 The Law of the Iterated Logarithm
6 INTRODUCTION TO WAVELETS
6.l Motivation and Heuristics
6.l.1 Heuristic Treatment of the Wavelet Transform
6.2 Wavelet Transform
6.2.0.l Wavelet Characterization of Smoothness
6.3 Haar Wavelet Expansion
6.3.l Haar Functions and Haar Series
6.3.2 Haar Sums and Dyadic Projections
6.3.3 Completeness of the Haar Functions
6.3.3.1 Haar Series in Co and Lp Spaces
6.3.3.2 Pointwise Convergence of Haar Series
6.3.4 *Construction of Standard Brownian Motion
6.3.5 *Haar Function Representation of Brownian Motion
6.3.6 *Proof of Continuity
6.3.7 *Levy's Modulus of Continuity
6.4 Multiresolution Analysis
6.4.l Orthonormal Systems and Riesz Systems
6.4.2 Scaling Equations and Structure Constants
6.4.3 From Scaling Function to MRA
6.4.3.l Additional Remarks
6.4.4 Meyer Wavelets
6.4.5 From Scaling Function to Orthonormal Wavelet
6.4.5.1 Direct Proof that V1 V0 Is Spanned by
{ (t-k)}k
6.4.5.2 Null Integrability of Wavelets Without-
Scaling Functions
6.5 Wavelets with Compact Support
6.5.l From Scaling Filter to Scaling Function
6.5.2 Explicit Construction of Compact Wavelets
6.5.2.l Daubechies Recipe
6.5.2.2 Hemandez-Weiss Recipe
6.5.3 Smoothness of Wavelets
6.5.3.l A Negative Result
6.5.4 Cohen's Extension of Theorem 6.5.1
6.6 Convergence Properties of Wavelet Expansions
6.6.1 Wavelet Series in Lp Spaces
6.6.1.1 Large Scale Analysis
6.6.1.2 Almost-Everywhere Convergence
6.6.1.3 Convergence at a Preassigned Point
6.6.2 Jackson and Bernstein Approximation Theorems
6.7 Wavelets in Several Variables
6.7.1 Two Important Examples
6.7.l.l Tensor Product of Wavelets
6.7.2 General Formulation of MRA and Wavelets in Rd
6.7.2.l Notations for Subgroups and Cosets
6.7.2.2 Riesz Systems and Orhonormal Systems in Rd
6.7.2.3 Scaling Equation and Structure Constants
6.7.2.4 Existence of the Wavelet Set
6.7.2.5 Proof That the Wavelet Set Spans V1 V0
6.7.2.6 Cohen's Theorem in Rd
6.7.3 Examples of Wavelets in Rd
References
Notations
Index
· · · · · · (收起)
1.1 Motivation and Heuristics
1.1.1 Motivation from Physics
1.1.1.1 The Vibrating String
1.1.1.2 Heat Flow in Solids
1.1.2 Absolutely Convergent Trigonometric Series
1.1.3 *Examples of Factorial and Bessel Functions
1.1.4 Poisson Kernel Example
1.1.5 *Proof of Laplace's Method
1.1.6 *Nonabsolutely Convergent Trigonometric Series
1.2 Formulation of Fourier Series
1.2.1 Fourier Coefficients and Their Basic Properties
1.2.2 Fourier Series of Finite Measures
1.2.3 *Rates of Decay of Fourier Coefficients
1.2.3.1 Piecewise Smooth Functions
1.2.3.2 Fourier Characterization of Analytic Functions
1.2.4 Sine Integral
1.2.4.1 Other Proofs That Si( ) =l
1.2.5 Pointwise Convergence Criteria
1.2.6 *Integration of Fourier Series
1.2.6.1 Convergence of Fourier Series of Measures
1.2.7 Riemann Localization Principle
1.2.8 Gibbs-Wilbraham Phenomenon
1.2.8.l The General Case
1.3 Fourier Series in L2
1.3.l Mean Square Approximation-Parseval's Theorem
1.3.2 *Application to the Isoperimetric Inequality
1.3.3 *Rates of Convergence in L2
1.3.3.1 Application to Absolutely-Convergent Fourier
Series
1.4 Norm Convergence and Summability
1.4.1 Approximate Identities
1.4.1.1 Almost-Everywhere Convergence of the Abel
Means
1.4.2 Summability Matrices
l.4.3 Fejer Means of a Fourier Series
1.4.3.1 Wiener's Closure Theorem on the Circle
1.4.4 *Equidistribution Modulo One
1.4.5 *Hardy's Tauberian Theorem
1.5 Improved Trigonometric Approximation
1.5.1 Rates of Convergence in C(T)
1.5.2 Approximation with Fejer Means
1.5.3 *Jackson's Theorem
1.5.4 *Higher-Order Approximation
1.5.5 *Converse Theorems of Bernstein
1.6 Divergence of Fourier Series
1.6.1 The Example of du Bois-Reymond
1.6.2 Analysis via Lebesgue Constants
1.6.3 Divergence in the Space L1
1.7 *Appendix: Complements on Laplace's Method
1.7.O.l First Variation on the Theme-Gaussian
Approximation
1.7.0.2 Second Variation on the Theme-Improved Error
Estimate
1.7.l *Application to Bessel Functions
1.7.2 *The Local Limit Theorem of DeMoivre-Laplace
1.8 Appendix: Proof of the Uniform Boundedness Theorem
1.9 *Appendix: Higher-Order Bessel functions
1.10 Appendix: Cantor's Uniqueness Theorem
2 FOURIER TRANSFORMS ON THE LINE AND SPACE
2.1 Motivation and Heuristics
2.2 Basic Properties of the Fourier Transform
2.2.l Riemann-Lebesgue Lemma
2.2.2 Approximate Identities and Gaussian Summability
2.2.2.1 Improved Approximate Identities for Pointwise
Convergence
2.2.2.2 Application to the Fourier Transform
2.2.2.3 The n-Dimensional Poisson Kernel
2.2.3 Fourier Transforms of Tempered Distributions
2.2.4 *Characterization of the Gaussian Density
2.2.5 *Wiener's Density Theorem
2.3 Fourier Inversion in One Dimension
2.3.1 Dirichlet Kernel and Symmetric Partial Sums
2.3.2 Example of the Indicator Function
2.3.3 Gibbs-Wilbraham Phenomenon
2.3.4 Dini Convergence Theorem
2.3.4.1 Extension to Fourier's Single Integral
2.3.5 Smoothing Operations in R1-Averaging and Summability
2.3.6 Averaging and Weak Convergence
2.3.7 Cesaro Summability
2.3.7.1 Approximation Propenies of the Fejer Kernel
2.3.8 Bemstein's Inequality
2.3.9 *One-Sided Fourier Integral Representation
2.3.9.1 Fourier Cosine Transform
2.3.9.2 Fourier Sine Transform
2.3.9.3 Generalized h-Transform
2.4 L2 Theory in Rn
2.4.1 Plancherel's Theorem
2.4.2 *Bernstein's Theorem for Fourier Transforms
2.4.3 The Uncertainty Principle
2.4.3.l Uncertainty Principle on the Circle
2.4.4 Spectral Analysis of the Fourier Transform
2.4.4.1 Hermite Polynomials
2.4.4.2 Eigenfunction of the Fourier Transform
2.4.4.3 Orthogonality Properties
2.4.4.4 Completeness
2.5 Spherical Fourier Inversion in Rn
2.5.l Bochner's Approach
2.5.2 Piecewise Smooth Viewpoint
2.5.3 Relations with the Wave Equation
2.5.3.l The Method of Brandolini and Colzani
2.5.4 Bochner-Riesz Summability
2.5.4.l A General Theorem on Almost-Everywhere
Summability
2.6 Bessel Functions
2.6.l Fourier Transforms of Radial Functions
2.6.2 L2-Restriction Theorems for ate Fourier Transform
2.6.2.l An Improved Result
2.6.2.2 Limitations on the Range of p
2.7 The Method of Stationary Phase
2.7.l Statement of the Result
2.7.2 Application to Bessel Functions
2.7.3 Proof of the Method of Stationary Phase
2.7.4 Abel's Lemma
3 FOURIER ANALYSIS IN LP SPACES
3.l Motivation and Heuristics
3.2 The M.Riesz-Thorin Interpolation Theorem
3.2.0.l Generalized Young's Inequality
3.2.0.2 The Hausdorff-Young Inequality
3.2.l Stein's Complex Interpolation Theorem
3.3 The Conjugate Function or Discrete Hilbert Transform
3.3.1 LP Theory of the Conjugate Function
3.3.2 L1 Theory of the Conjugate Function
3.3.2.l Identification as a Singular Integral
3.4 The Hilbert Transform on R
3.4.l L2 Theory of the Hilbert Transform
3.4.2 Lp Theory of the Hilbert Transform.l [ P [
3.4.2.l Applications to Convergence of Fourier Integrals
3.4.3 L1 Theory of the Hilbert Transform and Extensions
3.4.3.l Kolmogorov's Inequality for the Hilbert
Transform
3.4.4 Application to Singular Integrals with Odd Kernels
3.5 Hardy-Littlewood Maximal Function
3.5.l Application to the Lebesgue Differentiation Theorem
3.5.2 Application to Radial Convolution Operators
3.5.3 Maximal Inequalities for Spherical Averages
3.6 The Marcinkiewicz Interpolation Theorem
3.7 Calderon-Zygmund Decomposition
3.8 A Class of Singular Integrals
3.9 Propenies of Harmonic Functions
3.9.l General Properties
3.9.2 Representation Theorems in the Disk
3.9.3 Representation Theorems in the Upper Half-Plane
3.9.4 Herglotz/Bochner Theorems and Positive Definite
Functions
4 POISSON SUMMATION FORMULA AND
MULTIPLE FOURIER SERIES
4.l Motivation and Heudstics
4.2 The Poisson Summation Formula in R1
4.2.1 Periodization of a Function
4.2.2 Statement and Proof
4.2.3 Shannon Sampling
4.3 Multiple Fourier Series
4.3.l Basic L1 Theory
4.3.1.1 Pointwise Convergence for Smooath Functions
4.3.1.2 Representation of Spherical Partial Sums
4.3.2 Basic L2 Theory
4.3.3 Restriction Theorems for Fourier Coefficients
4.4 Poisson Summation Formula in Rd
4.4.1 *Simultaneous Nonlocalization
4.5 Application to Lattice Points
4.5.l Kendall's Mean Square Error
4.5.2 Landau's Asymptotic Formula
4.5.3 Application to Multiple Fourier Series
4.5.3.1 Three-Dimensional Case
4.5.3.2 Higher-Dimensional Case
4.6 Schrodinger Equation and Gauss Sums
4.6.l Distibutions on the Circle
4.6.2 The Schrodinger Equation on the Circle
4.7 Recurrence of Random Walk
5 APPLICATIONS TO PROBABILITY THEORY
5.l Motivation and Heuristics
5.2 Basic Definitions
5.2.1 The Central Limit Theorem
5.2.1.1 Restatement in Terms of Independent
Random Variables
5.3 Extension to Gap Series
5.3.l Extension to Abel Sums
5.4 Weak Convergence of Measures
5.4.l An Improved Continuity Theorem
5.4.l.l Another Proof of Bochner's Theorem
5.5 Convolution Semigroups
5.6 The Berry-Esseen Theorem
5.6.l Extension to Different Distributions
5.7 The Law of the Iterated Logarithm
6 INTRODUCTION TO WAVELETS
6.l Motivation and Heuristics
6.l.1 Heuristic Treatment of the Wavelet Transform
6.2 Wavelet Transform
6.2.0.l Wavelet Characterization of Smoothness
6.3 Haar Wavelet Expansion
6.3.l Haar Functions and Haar Series
6.3.2 Haar Sums and Dyadic Projections
6.3.3 Completeness of the Haar Functions
6.3.3.1 Haar Series in Co and Lp Spaces
6.3.3.2 Pointwise Convergence of Haar Series
6.3.4 *Construction of Standard Brownian Motion
6.3.5 *Haar Function Representation of Brownian Motion
6.3.6 *Proof of Continuity
6.3.7 *Levy's Modulus of Continuity
6.4 Multiresolution Analysis
6.4.l Orthonormal Systems and Riesz Systems
6.4.2 Scaling Equations and Structure Constants
6.4.3 From Scaling Function to MRA
6.4.3.l Additional Remarks
6.4.4 Meyer Wavelets
6.4.5 From Scaling Function to Orthonormal Wavelet
6.4.5.1 Direct Proof that V1 V0 Is Spanned by
{ (t-k)}k
6.4.5.2 Null Integrability of Wavelets Without-
Scaling Functions
6.5 Wavelets with Compact Support
6.5.l From Scaling Filter to Scaling Function
6.5.2 Explicit Construction of Compact Wavelets
6.5.2.l Daubechies Recipe
6.5.2.2 Hemandez-Weiss Recipe
6.5.3 Smoothness of Wavelets
6.5.3.l A Negative Result
6.5.4 Cohen's Extension of Theorem 6.5.1
6.6 Convergence Properties of Wavelet Expansions
6.6.1 Wavelet Series in Lp Spaces
6.6.1.1 Large Scale Analysis
6.6.1.2 Almost-Everywhere Convergence
6.6.1.3 Convergence at a Preassigned Point
6.6.2 Jackson and Bernstein Approximation Theorems
6.7 Wavelets in Several Variables
6.7.1 Two Important Examples
6.7.l.l Tensor Product of Wavelets
6.7.2 General Formulation of MRA and Wavelets in Rd
6.7.2.l Notations for Subgroups and Cosets
6.7.2.2 Riesz Systems and Orhonormal Systems in Rd
6.7.2.3 Scaling Equation and Structure Constants
6.7.2.4 Existence of the Wavelet Set
6.7.2.5 Proof That the Wavelet Set Spans V1 V0
6.7.2.6 Cohen's Theorem in Rd
6.7.3 Examples of Wavelets in Rd
References
Notations
Index
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