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Fourier分析与小波分析引论（影印版）

样章
作者：
Mark A. Pinsky
定价：
169.00 元
版面字数：
630千字
开本：
特殊
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2025年
ISBN：
978-7-04-063097-8
物料号：
63097-00
出版时间：
2025-02-13
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
分析
暂无
  • 目录
    • 前辅文
      • 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(∞) = 1
                                  • 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
                                          • 1.2.8.1 The General Case
                                        • 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
                                                • 1.3.3.1 Application to Absolutely-Convergent Fourier Series
                                              • 1.4 Norm Convergence and SummabiHty
                                                • 1.4.1 Approximate Identities
                                                  • 1.4.1.1 Almost-Everywhere Convergence of the Abel Means
                                                  • 1.4.2 Summability Matrices
                                                    • 1.4.3 Fejér 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 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
                                                                                              • 2.2.2.1 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 Cesàro Summability
                                                                                                                      • 2.3.7.1 Approximation Properties of the Fejér Kernel
                                                                                                                      • 2.3.8 Bernstein'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
                                                                                                                                    • 2.4.3.1 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
                                                                                                                                            • 2.5.1 Bochner's Approach
                                                                                                                                              • 2.5.2 Piecewise Smooth Viewpoint
                                                                                                                                                • 2.5.3 Relations with the Wave Equation
                                                                                                                                                  • 2.5.3.1 The Method of Brandolini and Colzani
                                                                                                                                                  • 2.5.4 Bochner-Riesz Summability
                                                                                                                                                    • 2.5.4.1 A General Theorem on Almost-Everywhere Summability
                                                                                                                                                  • 2.6 Bessel Functions
                                                                                                                                                    • 2.6.1 Fourier Transforms of Radial Functions
                                                                                                                                                      • 2.6.2 L2-Restriction Theorems for the Fourier Transform
                                                                                                                                                        • 2.6.2.1 An Improved Result
                                                                                                                                                          • 2.6.2.2 Limitations on the Range of p
                                                                                                                                                        • 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
                                                                                                                                                                              • 3.3.2.1 Identification as a Singular Integral
                                                                                                                                                                            • 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<p<∞
                                                                                                                                                                                  • 3.4.2.1 Applications to Convergence of Fourier Integrals
                                                                                                                                                                                  • 3.4.3 L1 Theory of the Hilbert Transform and Extensions
                                                                                                                                                                                    • 3.4.3.1 Kolmogorov's Inequality for the Hilbert Transform
                                                                                                                                                                                    • 3.4.4 Application to Singular Integrals with Odd Kernels
                                                                                                                                                                                    • 3.5 Hardy-Littlewood Maximal Function
                                                                                                                                                                                      • 3.5.1 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
                                                                                                                                                                                                                    • 4.3.1.1 Pointwise Convergence for Smooth 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
                                                                                                                                                                                                                            • 4.5.1 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 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
                                                                                                                                                                                                                                              • 5.2.1.1 Restatement in Terms of Independent Random Variables
                                                                                                                                                                                                                                            • 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
                                                                                                                                                                                                                                                  • 5.4.1.1 Another Proof of Bochner's 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
                                                                                                                                                                                                                                                                  • 6.3.3.1 Haar Series in C0 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 *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
                                                                                                                                                                                                                                                                                  • 6.4.3.1 Additional Remarks
                                                                                                                                                                                                                                                                                  • 6.4.4 Meyer Wavelets
                                                                                                                                                                                                                                                                                    • 6.4.5 From Scaling Function to Orthonormal Wavelet
                                                                                                                                                                                                                                                                                      • 6.4.5.1 Direct Proof that V1ΘV0 Is Spanned by {Ψ(t-k)}k∈Z
                                                                                                                                                                                                                                                                                        • 6.4.5.2 Null Integrability of Wavelets Without Scaling Functions
                                                                                                                                                                                                                                                                                      • 6.5 Wavelets with Compact Support
                                                                                                                                                                                                                                                                                        • 6.5.1 From Scaling Filter to Scaling Function
                                                                                                                                                                                                                                                                                          • 6.5.2 Explicit Construction of Compact Wavelets
                                                                                                                                                                                                                                                                                            • 6.5.2.1 Daubechies Recipe
                                                                                                                                                                                                                                                                                              • 6.5.2.2 Hernandez-Weiss Recipe
                                                                                                                                                                                                                                                                                              • 6.5.3 Smoothness of Wavelets
                                                                                                                                                                                                                                                                                                • 6.5.3.1 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
                                                                                                                                                                                                                                                                                                            • 6.7.1.1 Tensor Product of Wavelets
                                                                                                                                                                                                                                                                                                            • 6.7.2 General Formulation of MRA and Wavelets in Rd
                                                                                                                                                                                                                                                                                                              • 6.7.2.1 Notations for Subgroups and Cosets
                                                                                                                                                                                                                                                                                                                • 6.7.2.2 Riesz Systems and Orthonormal 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 V1ΘV0
                                                                                                                                                                                                                                                                                                                        • 6.7.2.6 Cohen's Theorem in Rd
                                                                                                                                                                                                                                                                                                                        • 6.7.3 Examples of Wavelets in Rd
                                                                                                                                                                                                                                                                                                                      • References
                                                                                                                                                                                                                                                                                                                        • Notations
                                                                                                                                                                                                                                                                                                                          • Index

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