INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

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1 #. INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS Mark A. Pinsky Northwestern University BROOKS/COLE * THOMSON LEARNING Australia Canada Mexico Singapore Spain United Kingdom United States

2 1 FOURIER SERIES ON THE CIRCLE Motivation and Heuristics Motivation from Physics The Vibrating String Heat Flow in Solids Absolutely Convergent Trigonometrie Series *Examples of Factorial and Bessel Functions Poisson Kernel Example *Proof of Laplace's Method *Nonabsolutely Convergent Trigonometrie Series Formulation of Fourier Series Fourier Coefficients and Their Basic Properties Fourier Series of Finite Measures *Rates of Decay of Fourier Coefficients Piecewise Smooth Functions Fourier Characterization of Analytic Functions Sine Integral Öther Proofs That Si(oo) = Pointwise Convergence Criteria * Integration of Fourier Series Convergence of Fourier Series of Measures Riemann Localization Principle Gibbs-Wilbraham Phenomenon The General Case Fourier Series in L Mean Square Approximation Parseval's Theorem * Application to the Isoperimetric Inequality 38 ix

3 * *Rates of Convergence in L Application to Absolutely-Convergent Fourier Series Norm Convergence and Summability Approximate Identities Almost-Everywhere Convergence of the Abel Means Summability Matrices Fejer Means of a Fourier Series Wiener's Closure Theorem on the Circle *Equidistribution Modulo One *Hardy's Tauberian Theorem Improved Trigonometrie Approximation Rates of Convergence in C(T) Approximation with Fejer Means *Jackson's Theorem *Higher-Order Approximation *Converse Theorems of Bernstein Divergence of Fourier Series The Example of du Bois-Reymond Analysis via Lebesgue Constants Divergence in the Space L * Appendix: Cqmplements on Laplace's Method First Variation on the Theme-Gaussian Approximation Second Variation on the Theme-Improved Error Estimate * Application to Bessel Functions *The Local Limit Theorem of DeMoivre-Laplace Appendix: Proof of the Uniform Boundedness Theorem * Appendix: Higher-Order Bessel funetions Appendix: Cantor's Uniqueness Theorem 86 2 FOURIER TRANSFORMS ON THE LINE AND SPACE 2.1 Motivation and Heuristics 2.2 Basic Properties of the Fourier Transform Riemann-Lebesgue Lemma Approximate Identities and Gaussian Summability Improved Approximate Identities for Pointwise Convergence Application to the Fourier Transform The n-dimensional Poisson Kernel

4 fc XI Fourier Transforms of Tempered Distributions *Characterization of the Gaussian Density *Wiener's Density Theorem Fourier Inversion in One Dimension Dirichlet Kernel and Symmetrie Partial Sums Example of the Indicator Function Gibbs-Wilbraham Phenomenon Dini Convergence Theorem Extension to Fourier's Single Integral Smoothing Operations in R'-Averaging and Summability Averaging and Weak Convergence Cesäro Summability Approximation Properties of the Fejer Kernel Bernstein's Inequality *One-Sided Fourier Integral Representation Fourier Cosine Transform Fourier Sine Transform Generalized /z-transform L 2 Theory in K" Plancherel's Theorem *Bernstein's Theorem for Fourier Transforms The Uncertainty Principle Uncertainty Principle on the Circle Spectral Analysis of the Fourier Transform Hermite Polynomials Eigenfunction of the Fourier Transform Orthogonality Properties Completeness Spherical Fourier Inversion in E" Bochner's Approach Piecewise Smooth Viewpoint Relations with the Wave Equation The Method of Brandolini and Colzani Bochner-Riesz Summability A General Theorem on Almost-Everywhere Summability Bessel Functions Fourier Transforms of Radial Functions L 2 -Restriction Theorems for the Fourier Transform An Improved Result Limitations on the Range of p The Method of Stationary Phase Statement of the Result Application to Bessel Functions Proof of the Method of Stationary Phase Abel's Lemma 167

5 v Xll 3 FOURIER ANALYSIS IN Iß SPACES Motivation and Heuristics The M. Riesz-Thorin Interpolation Theorem Generalized Young's Inequality The Hausdorff-Young Inequality Stein's Complex Interpolation Theorem The Conjugate Function or Discrete Hilbert Transform LP Theory of the Conjugate Function L 1 Theory of the Conjugate Function Identification as a Singular Integral The Hilbert Transform on K L? Theory of the Hilbert Transform LP Theory of the Hilbert Transform, 1 < p < co Applications to Convergence of Fourier Integrals L 1 Theory of the Hilbert Transform and Extensions Kolmogorov's Inequality for the Hilbert Transform Application to Singular Integrals with Odd Kernels Hardy-Littlewood Maximal Function Application to the Lebesgue Differentiation Theorem Application to Radial Convolution Operators Maximal Inequalities for Spherical Averages The Marcinkiewicz Interpolation Theorem Calderön-Zygmund Decomposition A Class of Singular Integrals Properties of Harmonie Functions General Properties Representation Theorems in the Disk Representation Theorems in the Upper Half-Plane Herglotz/Bochner Theorems and Positive Definite Functions 219 POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES Motivation and Heuristics ThePoissonSummationFormulainM Periodization of a Function Statement and Proof Shannon Sampling Multiple Fourier Series Basic L 1 Theory Pointwise Convergence for Smooth Functions Representation of Spherical Partial Sums 233

6 % XÜi Basic L 2 Theory Restriction Theorems for Fourier Coefncients Poisson Summation Formula in R ä *Simultaneous Nonlocalization Application to Lattice Points Kendall's Mean Square Error Landau's Asymptotic Formula Application to Multiple Fourier Series Three-Dimensional Case Higher-Dimensional Case Schrödinger Equation and Gauss Sums Distributions on the Circle The Schrödinger Equation on the Circle Recurrence of Random Walk APPLICATIONS TO PROBABILITY THEORY Motivation and Heuristics Basic Definitions The Central Limit Theorem Restatement in Terms of Independent Random Variables Extension to Gap Series Extension to Abel Sums Weak Convergence of Measures An Improved Continuity Theorem Another Proof of Bochner's Theorem Convolution Semigroups The Berry-Esseen Theorem Extension to Different Distributions The Law of the Iterated Logarifhm INTRODUCTION TO WAVELETS Motivation and Heuristics Heuristic Treatment of the Wavelet Transform Wavelet Transform Wavelet Characterization of Smoothness Haar Wavelet Expansion Haar Functions and Haar Series Haar Sums and Dyadic Projections Completeness of the Haar Functions Haar Series in Co and L p Spaces Pointwise Convergence of Haar Series 298

7 XIV *Construction of Standard Brownian Motion *Haar Function Representation of Brownian Motion *Proof of Continuity *Levy's Modulus of Continuity Multiresolution Analysis Orthonormal Systems and Riesz Systems Scaling Equations and Structure Constants From Scaling Function to MRA Additional Remarks Meyer Wavelets From Scaling Function to Orthonormal Wavelet Direct Proof that V { G V 0 Is Spanned by mt-k)} kez Null Integrability of Wavelets Without Scaling Functions Wavelets with Compact Support From Scaling Filter to Scaling Function Explicit Construction of Compact Wavelets Daubechies Recipe Hernandez-Weiss Recipe Smoothness of Wavelets A Negative Result Cohen's Extension of Theorem Convergence Properties of Wavelet Expansions Wavelet Series in LP Spaces Large Scale Analysis Almost-Everywhere Convergence Convergence at a Preassigned Point Jackson and Bernstein Approximation Theorems Wavelets in Several Variables Two Important Examples Tensor Product of Wavelets General Formulation of MRA and Wavelets in W Notations for Subgroups and Cosets Riesz Systems and Orthonormal Systems in R d Scaling Equation and Structure Constants Existence of the Wavelet Set Proof That the Wavelet Set Spans V, 0 V Cohen's Theorem in R d Examples of Wavelets in W References 365 Notations 369 Index 373

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