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1 FOURIER SERIES ON THE CIRCLE 1

1.1 Motivation and Heuristics 1

1.1.1 Motivation from Physics 1

1.1.1.1 The Vibrating String 1

1.1.1.2 Heat Flow in Solids 2

1.1.2 Absolutely Convergent Trigonometric Series 3

1.1.3 Examples of Factorial and Bessel Functions 6

1.1.4 Poisson Kernel Example 7

1.1.5 Proof of Laplace s Method 9

1.1.6 Nonabsolutely Convergent Trigonometric Series 11

1.2 Formulation of Fourier Series 13

1.2.1 Fourier Coefficients and Their Basic Properties 13

1.2.2 Fourier Series of Finite Measures 19

1.2.3 Rates of Decay of Fourier Coefficients 20

1.2.3.1 Piecewise Smooth Functions 21

1.2.3.2 Fourier Characterization of Analytic Functions 22

1.2.4 Sine Integral 24

1.2.4.1 Other Proofs That Si(∞)＝1 24

1.2.5 Pointwise Convergence Criteria 25

1.2.6 Integration of Fourier Series 29

1.2.6.1 Convergence of Fourier Series of Measures 30

1.2.7 Riemann Localization Principle 31

1.2.8 Gibbs-Wilbraham Phenomenon 31

1.2.8.1 The General Case 34

1.3 Fourier Series in L2 35

1.3.1 Mean Square Approximation—Parseval s Theorem 35

1.3.2 Application to the Isoperimetric Inequality 38

1.3.3 Rates of Convergence in L2 39

1.3.3.1 Application to Absolutely—Convergent Fourier Series 43

1.4 Norm Convergence and Sumlllability 45

1.4.1 Approximate Identities 45

1.4.1.1 Almost-Eveterywhere Convergence of the Abel Means 49

1.4.2 Summability Matrices 51

1.4.3 Fejer Mealls of a Fourier Series 54

1 .4.3.1 Wiener s Closure Theorem on the Circle 57

1.4.4 Equidistribution Modulo One 57

1.4.5 Hardy s Tauberian Theorem 59

1.5 Improved Trigonometric Approximation 61

1.5.1 Rates of Convergence in C(T) 61

1.5.2 Approximation with Fejer Means 62

1.5.3 Jackson s Theorem 65

1.5.4 Higher-Order Approximation 66

1.5.5 Converse Theorems of Bernstein 70

1.6 Divergence of Fourier Series 73

1.6.1 The Example of du Bois-Reymond 74

1.6.2 Analysis via Lebesgue Constants 75

1.6.3 Divergence in the Space L1 78

1.7 Appendix: Complements on Laplace s Method 80

1.7.0.1 First Variation on the Theme-Gaussian Approximation 80

1.7.0.2 Second Variation on the Theme-Improved Error Estimate 80

1.7.1 Application to Bessel Functions 81

1.7.2 The Local Limit Theorem of DeMoivre-Laplace 82

1.8 Appendix: Proof of the Uniform Boundedness Theorem 84

1.9 Appendix: Higher-Order Bessel functions 85

1.10 Appendix: Cantor s Uniqueness Theorem 86

2.1 Motivation and Heuristics 89

2 FOURIER TRANSFORMS ON THE LINE AND SPACE 89

2.2 Basic Properties of the Fourier Transform 91

2.2.1 Riemann-Lebesgue Lemma 94

2.2.2 Approximate Identities and Gaussian Summability 97

2.2.2.1 Improved Approximate Identities for Pointwise Convergence 100

2.2.2.2 Application to the Fourier Transform 102

2.2.2.3 The n-Dimensional Poisson Kernel 106

2.2.3 Fourier Transforms of Tempered Distributions 108

2.2.4 Characterization of the Gaussian Density 109

2.2.5 Wiener s Density Theorem 110

2.3 Fourier Inversion in One Dimension 112

2.3.1 Dirichlet Kernel and Symmetric Partial Sums 112

2.3.2 Example of the Indicator Function 114

2.3.4 Dini Convergence Theorem 115

2.3.3 Gibbs-Wilbraham Phenomenon 115

2.3.4.1 Extension to Fourier s Single Integral 117

2.3.5 Smoothing Operations in R1-Averaging and Summability 117

2.3.6 Averaging and Weak Convergence 118

2.3.7 Cesaro Summability 119

2.3.7.1 Approximation Properties of the Fejer Kernel 121

2.3.8 Bernstein s Inequality 122

2.3.9 One-Sided Fourier Integral Representation 124

2.3.9.1 Fourier Cosine Transform 124

2.3.9.2 Fourier Sine Transform 125

2.3.9.3 Generalized h-Transform 125

2.4 L2 Theory in Rn 128

2.4.1 Plancherel s Theorem 128

2.4.2 Bernstein s Theorem for Fourier Transforms 129

2.4.3 The Uncertainty Principle 131

2.4.3.1 Uncertainty Principle on the Circle 133

2.4.4 Spectral Analysis of the Fourier Transform 134

2.4.4.1 Hermite Polynomials 134

2.4.4.2 Eigenfunction of the Fourier Transform 136

2.4.4.3 Orthogonality Properties 137

2.4.4.4 Completeness 138

2.5 Spherical Fourier Inversion in Rn 139

2.5.1 Bochner s Approach 139

2.5.2 Piecewise Smooth Viewpoint 145

2.5.3 Relations with the Wave Equation 146

2.5.3.1 The Method of Brandolini and Colzani 149

2.5.4 Bochner-Riesz Summability 152

2.5.4.1 A General Theorem on Almost-Everywhere Summability 153

2.6 Bessel Functions 154

2.6.1 Fourier Transforms of Radial Functions 157

2.6.2 L2-Restriction Theorems for the Fourier Transform 158

2.6.2.1 An Improved Result 159

2.6.2.2 Limitations on the Range of p 161

2.7 The Method of Stationary Phase 162

2.7.1 Statement of the Result 163

2.7.2 Application to Bessel Functions 164

2.7.3 Proof of the Method of Stationary Phase 165

2.7.4 Abel s Lemma 167

3 FOURIER ANALYSIS IN Lp SPACES 169

3.1 Motivation and Heuristics 169

3.2 The M. Riesz-Thorin Interpolation Theorem 169

3.2.0.1 Generalized Young s Inequality 174

3.2.0.2 The Hausdorff-Young Inequality 174

3.2.1 Stein s Complex Interpolation Theorem 175

3.3 The Conjugate Function or Discrete Hilbert Transform 176

3.3.1 Lp Theory of the Conjugate Function 177

3.3.2 L1 Theory of the Conjugate Function 179

3.3.2.1 Identification as a Singular Integral 183

3.4 The Hilbcrt Transform on R 184

3.4.1 L2 Theory of the Hilbert Transform 185

3.4.2 Lp Theory of the Hilbert Transform, 1＜p＜∞ 186

3.4.2.1 Applications to Convergence of Fourier Integrals 187

3.4.3 L1 Theory of the Hilbert Transform and Extensions 188

3.4.3.1 Kolmogorov s Inequality for the Hilbert Transform 192

3.4.4 Application to Singular Integrals with Odd Kernels 194

3.5 Hardy-Littlewood Maximal Function 197

3.5.1 Application to the Lebesgue Differentiation Theorem 200

3.5.2 Application to Radial Convolution Operators 202

3.5.3 Maximal Inequalities for Spherical Averages 203

3.6 The Marcinkiewicz Interpolation Theorem 206

3.7 Calderon-Zygmund Decomposition 209

3.8 A Class of Singular Integrals 210

3.9 Properties of Harmonic Functions 212

3.9.1 General Properties 212

3.9.2 Representation Theorems in the Disk 214

3.9.3 Representation Theorems in the Upper Half-Plane 216

3.9.4 Herglotz/Bochner Theorems and Positive Definite Functions 219

4 POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES 222

4.1 Motivation and Heuristics 222

4.2 The Poisson Summation Formula in R1 223

4.2.1 Periodization of a Function 223

4.2.2 Statement and Proof 225

4.2.3 Shannon Sampling 228

4.3 Multiple Fourier Series 230

4.3.1 Basic L1 Theory 231

4.3.1.1 Pointwise Convergence for Smooth Functions 233

4.3.1.2 Representation of Spherical Partial Sums 233

4.3.2 Basic L2 Theory 235

4.3.3 Restriction Theorems for Fourier Coefficients 236

4.4 Poisson Summation Formula in Rd 238

4.4.1 Simultaneous Nonlocalization 239

4.5 Application to Lattice Points 241

4.5.1 Kendall s Mean Square Error 241

4.5.2 Landau s Asymptotic Formula 243

4.5.3 Application to Multiple Fourier Series 244

4.5.3.1 Three-Dimensional Case 245

4.5.3.2 Higher-Dimensional Case 247

4.6 Schrodinger Equation and Gauss Sums 247

4.6.1 Distributions on the Circle 248

4.6.2 The Schrodinger Equation on the Circle 250

4.7 Recurrence of Random Walk 252

5 APPLICATIONS TO PROBABILITY THEORY 256

5.1 Motivation and Heuristics 256

5.2 Basic Definitions 256

5.2.1 The Central Limit Theorem 260

5.2.1.1 Restatement in Terms of Independent Random Variables 261

5.3 Extension to Gap Series 262

5.3.1 Extension to Abel Sums 266

5.4 Weak Convergence of Measures 268

5.4.1 An Improved Continuity Theorem 269

5.4.1.1 Another Proof of Bochner s Theorem 270

5.5 Convolution Semigroups 272

5.6 The Berry-Esseen Theorem 276

5.6.1 Extension to Different Distributions 279

5.7 The Law of the Iterated Logarithm 280

6 INTRODUCTION TO WAVELETS 284

6.1 Motivation and Heuristics 284

6.1.1 Heuristic Treatment of the Wavelet Transform 285

6.2 Wavelet Transform 286

6.2.0.1 Wavelet Characterization of Smoothness 290

6.3 Haar Wavelet Expansion 291

6.3.1 Haar Functions and Haar Series 291

6.3.2 Haar Sums and Dyadic Projections 292

6.3.3 Completeness of the Haar Functions 295

6.3.3.1 Haar Series in Co and Lp Spaces 296

6.3.3.2 Pointwise Convergence of Haar Series 298

6.3.4 Construction of Standard Brownian Motion 299

6.3.5 Haar Function Representation of Brownian Motion 301

6.3.6 Proof of Continuity 301

6.3.7 Levy s Modulus of Continnity 302

6.4 Multiresolution Analysis 303

6.4.1 Orthonormal Systems and Riesz Systems 304

6.4.2 Scaling Equations and Structure Constants 310

6.4.3 From Scaling Function to MRA 313

6.4.3.1 Additional Remarks 315

6.4.4 Meyer Wavelets 318

6.4.5 From Scaling Function to Orthonormal Wavelet 319

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 Functions 325

6.5 Wavelets with Compact Support 326

6.5.1 From Scaling Filter to Scaling Function 327

6.5.2 Explicit Construction of Compact Wavelets 330

6.5.2.1 Daubechies Recipe 331

6.5.2.2 Hernandez-Weiss Recipe 333

6.5.3 Smoothness of Wavelets 334

6.5.3.1 A Negative Result 336

6.5.4 Cohcn s Extension of Theorem 6.5.1 338

6.6 Convergence Properties of Wavelet Expansions 341

6.6.1 Wavelet Series in Lp Spaces 341

6.6.1.1 Large Scale Analysis 345

6.6.1.2 Almost-Everywhere Convergence 346

6.6.2 Jackson and Bernstein Approximation Theorems 347

6.6.1.3 Convergence at a Preassigned Point 347

6.7 Wavelets in Several Variables 352

6.7.1 Two Important Examples 352

6.7.1.1 Tensor Product of Wavelets 354

6.7.2 General Formulation of MRA and Wavelets in Rd 354

6.7.2.1 Notations for Subgroups and Cosets 355

6.7.2.2 Riesz Systems and Orthonormal Systems in Rd 356

6.7.2.3 Scaling Equation and Structure Constants 357

6.7.2.4 Existence of the Wavelet Set 358

6.7.2.5 Proof That the Wavelet Set Spans V1θV0 361

6.7.2.6 Cohen s Theorem in Rd 362

6.7.3 Examples of Wavelets in Rd 362

References 365

Notations 369

Index 373