本书介绍了调和分析中的一些主题，适合于低年级研究生或高年级本科生阅读。学习本书的必备先修知识是实数轴上Lebesgue测度和积分的基础知识。本书适合对调和分析及相关知识感兴趣的本科生、研究生以及数学研究人员阅读参考。 This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary prerequisites to using the text are rudiments of the Lebesgue measure and integration on the real line. It begins with a thorough treatment of Fourier series on the circle and their applications to approximation theory, probability, and plane geometry (the isoperimetric theorem). Frequently, more than one proof is offered for a given theorem to illustrate the multiplicity of approaches. The second chapter treats the Fourier transform on Euclidean spaces, especially the author's results in the three-dimensional piecewise smooth case, which is distinct from the classical Gibbs–Wilbraham phenomenon of one-dimensional Fourier analysis. The Poisson summation formula treated in Chapter 3 provides an elegant connection between Fourier series on the circle and Fourier transforms on the real line, culminating in Landau's asymptotic formulas for lattice points on a large sphere. Much of modern harmonic analysis is concerned with the behavior of various linear operators on the Lebesgue spaces L^p(R^n). Chapter 4 gives a gentle introduction to these results, using the Riesz–Thorin theorem and the Marcinkiewicz interpolation formula. One of the long-time users of Fourier analysis is probability theory. In Chapter 5 the central limit theorem, iterated log theorem, and Berry–Esseen theorems are developed using the suitable Fourier-analytic tools. The final chapter furnishes a gentle introduction to wavelet theory, depending only on the L^2 theory of the Fourier transform (the Plancherel theorem). The basic notions of scale and location parameters demonstrate the flexibility of the wavelet approach to harmonic analysis. The text contains numerous examples and more than 200...

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1 FOURIER SERIES ON THE CIRCLE
  1.1 Motivation and Heuristics
   1.1.1 Motivation from Physics
   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
   1.2.4 Sine Integral
   1.2.5 Pointwise Convergence Criteria
   1.2.6 *Integration of Fourier Series
   1.2.7 Riemann Localization Principle
   1.2.8 Gibbs-Wilbraham Phenomenon
  1.3 Fourier Series in L2
   1.3.1 Mean Square Approximation—Parseval's Theorem
   1.3.2 *Application to the Isoperimetric Inequality
   1.3.3 *Rates of Convergence in L2
  1.4 Norm Convergence and SummabiHty
   1.4.1 Approximate Identities
   1.4.2 Summability Matrices
   1.4.3 Fejér Means of a Fourier Series
   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 Fejér 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
   1.7.0.1 First Variation on the Theme-Gaussian Approximation
   1.7.0.2 Second Variation on the Theme-Improved Error Estimate
   1.7.1 *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
   2.2.1 Riemann-Lebesgue Lemma
   2.2.2 Approximate Identities and Gaussian Summability
   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
   2.3.5 Smoothing Operations in R1-Averaging and Summability
   2.3.6 Averaging and Weak Convergence
   2.3.7 Cesàro Summability
   2.3.8 Bernstein's Inequality
   2.3.9 *One-Sided Fourier Integral Representation
  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
   2.4.4 Spectral Analysis of the Fourier Transform
  2.5 Spherical Fourier Inversion in Rn
   2.5.1 Bochner's Approach
   2.5.2 Piecewise Smooth Viewpoint
   2.5.3 Relations with the Wave Equation
   2.5.4 Bochner-Riesz Summability
  2.6 Bessel Functions
   2.6.1 Fourier Transforms of Radial Functions
   2.6.2 L2-Restriction Theorems for the Fourier Transform
  2.7 The Method of Stationary Phase
   2.7.1 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
  3.1 Motivation and Heuristics
  3.2 The M. Riesz-Thorin Interpolation Theorem
   3.2.0.1 Generalized Young's Inequality
   3.2.0.2 The Hausdorff-Young Inequality
   3.2.1 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
  3.4 The Hilbert Transform on R
   3.4.1 L2 Theory of the Hilbert Transform
   3.4.2 LP Theory of the Hilbert Transform, 1 <∞
   3.4.3 L1 Theory of the Hilbert Transform and Extensions
   3.4.4 Application to Singular Integrals with Odd Kernels
  3.5 Hardy-Littlewood Maximal Function
   3.5.1 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 Calderón-Zygmund Decomposition
  3.8 A Class of Singular Integrals
  3.9 Properties of Harmonic Functions
   3.9.1 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
  4.1 Motivation and Heuristics
  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
   4.3.1 Basic Ll Theory
   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
   4.5.1 Kendall's Mean Square Error
   4.5.2 Landau's Asymptotic Formula
   4.5.3 Application to Multiple Fourier Series
  4.6 Schrödinger Equation and Gauss Sums
   4.6.1 Distributions on the Circle
   4.6.2 The Schrödinger Equation on the Circle
  4.7 Recurrence of Random Walk
5 APPLICATIONS TO PROBABILITY THEORY
  5.1 Motivation and Heuristics
  5.2 Basic Definitions
   5.2.1 The Central Limit Theorem
  5.3 Extension to Gap Series
   5.3.1 Extension to Abel Sums
  5.4 Weak Convergence of Measures
   5.4.1 An Improved Continuity Theorem
  5.5 Convolution Semigroups
  5.6 The Berry-Esséen Theorem
   5.6.1 Extension to Different Distributions
  5.7 The Law of the Iterated Logarithm
6 INTRODUCTION TO WAVELETS
  6.1 Motivation and Heuristics
   6.1.1 Heuristic Treatment of the Wavelet Transform
  6.2 Wavelet Transform
   6.2.0.1 Wavelet Characterization of Smoothness
  6.3 Haar Wavelet Expansion
   6.3.1 Haar Functions and Haar Series
   6.3.2 Haar Sums and Dyadic Projections
   6.3.3 Completeness of the Haar Functions
   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 *Lévy's Modulus of Continuity
  6.4 Multiresolution Analysis
   6.4.1 Orthonormal Systems and Riesz Systems
   6.4.2 Scaling Equations and Structure Constants
   6.4.3 From Scaling Function to MRA
   6.4.4 Meyer Wavelets
   6.4.5 From Scaling Function to Orthonormal Wavelet
  6.5 Wavelets with Compact Support
   6.5.1 From Scaling Filter to Scaling Function
   6.5.2 Explicit Construction of Compact Wavelets
   6.5.3 Smoothness of Wavelets
   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
   6.6.2 Jackson and Bernstein Approximation Theorems
  6.7 Wavelets in Several Variables
   6.7.1 Two Important Examples
   6.7.2 General Formulation of MRA and Wavelets in Rd
   6.7.3 Examples of Wavelets in Rd
References
Notations
Index