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傅里叶分析和小波分析导论 英文版PDF|Epub|txt|kindle电子书版本下载
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- (美)平斯基(Pinsky,M.A.)著 著
- 出版社: 北京:机械工业出版社
- ISBN:7111119142
- 出版时间:2003
- 标注页数:376页
- 文件大小:12MB
- 文件页数:390页
- 主题词:傅里叶分析-高等学校-教材-英文;小波分析-高等学校-教材-英文
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图书目录
1 FOURIER SERIES ON THE CIRCLE1
1.1 Motivation and Heuristics1
1.1.1 Motivation from Physics1
1.1.1.1 The Vibrating String1
1.1.1.2 Heat Flow in Solids2
1.1.2 Absolutely Convergent Trigonometric Series3
1.1.3 Examples of Factorial and Bessel Functions6
1.1.4 Poisson Kernel Example7
1.1.5 Proof of Laplace s Method9
1.1.6 Nonabsolutely Convergent Trigonometric Series11
1.2 Formulation of Fourier Series13
1.2.1 Fourier Coefficients and Their Basic Properties13
1.2.2 Fourier Series of Finite Measures19
1.2.3 Rates of Decay of Fourier Coefficients20
1.2.3.1 Piecewise Smooth Functions21
1.2.3.2 Fourier Characterization of Analytic Functions22
1.2.4 Sine Integral24
1.2.4.1 Other Proofs That Si(∞)=124
1.2.5 Pointwise Convergence Criteria25
1.2.6 Integration of Fourier Series29
1.2.6.1 Convergence of Fourier Series of Measures30
1.2.7 Riemann Localization Principle31
1.2.8 Gibbs-Wilbraham Phenomenon31
1.2.8.1 The General Case34
1.3 Fourier Series in L235
1.3.1 Mean Square Approximation—Parseval s Theorem35
1.3.2 Application to the Isoperimetric Inequality38
1.3.3 Rates of Convergence in L239
1.3.3.1 Application to Absolutely—Convergent Fourier Series43
1.4 Norm Convergence and Sumlllability45
1.4.1 Approximate Identities45
1.4.1.1 Almost-Eveterywhere Convergence of the Abel Means49
1.4.2 Summability Matrices51
1.4.3 Fejer Mealls of a Fourier Series54
1 .4.3.1 Wiener s Closure Theorem on the Circle57
1.4.4 Equidistribution Modulo One57
1.4.5 Hardy s Tauberian Theorem59
1.5 Improved Trigonometric Approximation61
1.5.1 Rates of Convergence in C(T)61
1.5.2 Approximation with Fejer Means62
1.5.3 Jackson s Theorem65
1.5.4 Higher-Order Approximation66
1.5.5 Converse Theorems of Bernstein70
1.6 Divergence of Fourier Series73
1.6.1 The Example of du Bois-Reymond74
1.6.2 Analysis via Lebesgue Constants75
1.6.3 Divergence in the Space L178
1.7 Appendix: Complements on Laplace s Method80
1.7.0.1 First Variation on the Theme-Gaussian Approximation80
1.7.0.2 Second Variation on the Theme-Improved Error Estimate80
1.7.1 Application to Bessel Functions81
1.7.2 The Local Limit Theorem of DeMoivre-Laplace82
1.8 Appendix: Proof of the Uniform Boundedness Theorem84
1.9 Appendix: Higher-Order Bessel functions85
1.10 Appendix: Cantor s Uniqueness Theorem86
2.1 Motivation and Heuristics89
2 FOURIER TRANSFORMS ON THE LINE AND SPACE89
2.2 Basic Properties of the Fourier Transform91
2.2.1 Riemann-Lebesgue Lemma94
2.2.2 Approximate Identities and Gaussian Summability97
2.2.2.1 Improved Approximate Identities for Pointwise Convergence100
2.2.2.2 Application to the Fourier Transform102
2.2.2.3 The n-Dimensional Poisson Kernel106
2.2.3 Fourier Transforms of Tempered Distributions108
2.2.4 Characterization of the Gaussian Density109
2.2.5 Wiener s Density Theorem110
2.3 Fourier Inversion in One Dimension112
2.3.1 Dirichlet Kernel and Symmetric Partial Sums112
2.3.2 Example of the Indicator Function114
2.3.4 Dini Convergence Theorem115
2.3.3 Gibbs-Wilbraham Phenomenon115
2.3.4.1 Extension to Fourier s Single Integral117
2.3.5 Smoothing Operations in R1-Averaging and Summability117
2.3.6 Averaging and Weak Convergence118
2.3.7 Cesaro Summability119
2.3.7.1 Approximation Properties of the Fejer Kernel121
2.3.8 Bernstein s Inequality122
2.3.9 One-Sided Fourier Integral Representation124
2.3.9.1 Fourier Cosine Transform124
2.3.9.2 Fourier Sine Transform125
2.3.9.3 Generalized h-Transform125
2.4 L2 Theory in Rn128
2.4.1 Plancherel s Theorem128
2.4.2 Bernstein s Theorem for Fourier Transforms129
2.4.3 The Uncertainty Principle131
2.4.3.1 Uncertainty Principle on the Circle133
2.4.4 Spectral Analysis of the Fourier Transform134
2.4.4.1 Hermite Polynomials134
2.4.4.2 Eigenfunction of the Fourier Transform136
2.4.4.3 Orthogonality Properties137
2.4.4.4 Completeness138
2.5 Spherical Fourier Inversion in Rn139
2.5.1 Bochner s Approach139
2.5.2 Piecewise Smooth Viewpoint145
2.5.3 Relations with the Wave Equation146
2.5.3.1 The Method of Brandolini and Colzani149
2.5.4 Bochner-Riesz Summability152
2.5.4.1 A General Theorem on Almost-Everywhere Summability153
2.6 Bessel Functions154
2.6.1 Fourier Transforms of Radial Functions157
2.6.2 L2-Restriction Theorems for the Fourier Transform158
2.6.2.1 An Improved Result159
2.6.2.2 Limitations on the Range of p161
2.7 The Method of Stationary Phase162
2.7.1 Statement of the Result163
2.7.2 Application to Bessel Functions164
2.7.3 Proof of the Method of Stationary Phase165
2.7.4 Abel s Lemma167
3 FOURIER ANALYSIS IN Lp SPACES169
3.1 Motivation and Heuristics169
3.2 The M. Riesz-Thorin Interpolation Theorem169
3.2.0.1 Generalized Young s Inequality174
3.2.0.2 The Hausdorff-Young Inequality174
3.2.1 Stein s Complex Interpolation Theorem175
3.3 The Conjugate Function or Discrete Hilbert Transform176
3.3.1 Lp Theory of the Conjugate Function177
3.3.2 L1 Theory of the Conjugate Function179
3.3.2.1 Identification as a Singular Integral183
3.4 The Hilbcrt Transform on R184
3.4.1 L2 Theory of the Hilbert Transform185
3.4.2 Lp Theory of the Hilbert Transform, 1<p<∞186
3.4.2.1 Applications to Convergence of Fourier Integrals187
3.4.3 L1 Theory of the Hilbert Transform and Extensions188
3.4.3.1 Kolmogorov s Inequality for the Hilbert Transform192
3.4.4 Application to Singular Integrals with Odd Kernels194
3.5 Hardy-Littlewood Maximal Function197
3.5.1 Application to the Lebesgue Differentiation Theorem200
3.5.2 Application to Radial Convolution Operators202
3.5.3 Maximal Inequalities for Spherical Averages203
3.6 The Marcinkiewicz Interpolation Theorem206
3.7 Calderon-Zygmund Decomposition209
3.8 A Class of Singular Integrals210
3.9 Properties of Harmonic Functions212
3.9.1 General Properties212
3.9.2 Representation Theorems in the Disk214
3.9.3 Representation Theorems in the Upper Half-Plane216
3.9.4 Herglotz/Bochner Theorems and Positive Definite Functions219
4 POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES222
4.1 Motivation and Heuristics222
4.2 The Poisson Summation Formula in R1223
4.2.1 Periodization of a Function223
4.2.2 Statement and Proof225
4.2.3 Shannon Sampling228
4.3 Multiple Fourier Series230
4.3.1 Basic L1 Theory231
4.3.1.1 Pointwise Convergence for Smooth Functions233
4.3.1.2 Representation of Spherical Partial Sums233
4.3.2 Basic L2 Theory235
4.3.3 Restriction Theorems for Fourier Coefficients236
4.4 Poisson Summation Formula in Rd238
4.4.1 Simultaneous Nonlocalization239
4.5 Application to Lattice Points241
4.5.1 Kendall s Mean Square Error241
4.5.2 Landau s Asymptotic Formula243
4.5.3 Application to Multiple Fourier Series244
4.5.3.1 Three-Dimensional Case245
4.5.3.2 Higher-Dimensional Case247
4.6 Schrodinger Equation and Gauss Sums247
4.6.1 Distributions on the Circle248
4.6.2 The Schrodinger Equation on the Circle250
4.7 Recurrence of Random Walk252
5 APPLICATIONS TO PROBABILITY THEORY256
5.1 Motivation and Heuristics256
5.2 Basic Definitions256
5.2.1 The Central Limit Theorem260
5.2.1.1 Restatement in Terms of Independent Random Variables261
5.3 Extension to Gap Series262
5.3.1 Extension to Abel Sums266
5.4 Weak Convergence of Measures268
5.4.1 An Improved Continuity Theorem269
5.4.1.1 Another Proof of Bochner s Theorem270
5.5 Convolution Semigroups272
5.6 The Berry-Esseen Theorem276
5.6.1 Extension to Different Distributions279
5.7 The Law of the Iterated Logarithm280
6 INTRODUCTION TO WAVELETS284
6.1 Motivation and Heuristics284
6.1.1 Heuristic Treatment of the Wavelet Transform285
6.2 Wavelet Transform286
6.2.0.1 Wavelet Characterization of Smoothness290
6.3 Haar Wavelet Expansion291
6.3.1 Haar Functions and Haar Series291
6.3.2 Haar Sums and Dyadic Projections292
6.3.3 Completeness of the Haar Functions295
6.3.3.1 Haar Series in Co and Lp Spaces296
6.3.3.2 Pointwise Convergence of Haar Series298
6.3.4 Construction of Standard Brownian Motion299
6.3.5 Haar Function Representation of Brownian Motion301
6.3.6 Proof of Continuity301
6.3.7 Levy s Modulus of Continnity302
6.4 Multiresolution Analysis303
6.4.1 Orthonormal Systems and Riesz Systems304
6.4.2 Scaling Equations and Structure Constants310
6.4.3 From Scaling Function to MRA313
6.4.3.1 Additional Remarks315
6.4.4 Meyer Wavelets318
6.4.5 From Scaling Function to Orthonormal Wavelet319
6.4.5.1 Direct Proof that V1θV0 Is Spanned by {ψ(t—k)}?324
6.4.5.2 Null Integrability of Wavelets Without Scaling Functions325
6.5 Wavelets with Compact Support326
6.5.1 From Scaling Filter to Scaling Function327
6.5.2 Explicit Construction of Compact Wavelets330
6.5.2.1 Daubechies Recipe331
6.5.2.2 Hernandez-Weiss Recipe333
6.5.3 Smoothness of Wavelets334
6.5.3.1 A Negative Result336
6.5.4 Cohcn s Extension of Theorem 6.5.1338
6.6 Convergence Properties of Wavelet Expansions341
6.6.1 Wavelet Series in Lp Spaces341
6.6.1.1 Large Scale Analysis345
6.6.1.2 Almost-Everywhere Convergence346
6.6.2 Jackson and Bernstein Approximation Theorems347
6.6.1.3 Convergence at a Preassigned Point347
6.7 Wavelets in Several Variables352
6.7.1 Two Important Examples352
6.7.1.1 Tensor Product of Wavelets354
6.7.2 General Formulation of MRA and Wavelets in Rd354
6.7.2.1 Notations for Subgroups and Cosets355
6.7.2.2 Riesz Systems and Orthonormal Systems in Rd356
6.7.2.3 Scaling Equation and Structure Constants357
6.7.2.4 Existence of the Wavelet Set358
6.7.2.5 Proof That the Wavelet Set Spans V1θV0361
6.7.2.6 Cohen s Theorem in Rd362
6.7.3 Examples of Wavelets in Rd362
References365
Notations369
Index373