Introduction to Fourier Analysis and Wavelets (Brooks Cole Series in Advanced Mathematics)

Cover page......Page 1
Title page......Page 2
Preface......Page 4
Contents......Page 6
1.1.1.1  The Vibrating String......Page 12
1.1.1.2  Heat Flow in Solids......Page 13
1.1.2  Absolutely Convergent Trigonometric Series......Page 14
1.1.3  *Examples of Factorial and Bessel Functions......Page 17
1.1.4  Poisson Kernel Example......Page 18
1.1.5  *Proof of Laplace's Method......Page 20
1.1.6  *Nonabsolutely Convergent Trigonometric Series......Page 22
1.2.1  Fourier Coefficients and Their Basic Properties......Page 24
1.2.2  Fourier Series of Finite Measures......Page 30
1.2.3  *Rates of Decay of Fourier Coefficients......Page 31
1.2.3.1  Piecewise Smooth Functions......Page 32
1.2.3.2  Fourier Characterization of Analytic Functions......Page 33
1.2.4.1  Other Proofs That Si(oo)=1......Page 35
1.2.5  Pointwise Convergence Criteria......Page 36
1.2.6  *Integration of Fourier Series......Page 40
1.2.6.1  Convergence of Fourier Series of Measures......Page 41
1.2.8  Gibbs-Wilbraham Phenomenon......Page 42
1.2.8.1  The General Case......Page 45
1.3.1  Mean Square Approximation--Parseval's Theorem......Page 46
1.3.2  *Application to the Isoperimetric Inequality......Page 49
1.3.3  *Rates of Convergence in L 2......Page 50
1.3.3.1  Application lo Absolutely-Convergent Fourier Series......Page 54
1.4.1  Approximate Identities......Page 56
1.4.1.1  Almost-Everywhere Convergence of the Abel Means......Page 60
1.4.2  Summability Matrices......Page 62
1.4.3  Fejer Means of a Fourier Series......Page 65
1.4.4  *Equidistribution Modulo One......Page 68
1.4.5  *Hardy's Tauberian Theorem......Page 70
1.5.1  Rates of Convergence in C(T)......Page 72
1.5.2  Approximation with Fejer Means......Page 73
1.5.3  *Jackson's Theorem......Page 76
1,5.4  *Higher-Order Approximation......Page 77
1.5.5  *Converse Theorems of Bernstein......Page 81
1.6  Divergence of Fourier Series......Page 84
1.6.1  The Example of du Bois-Reymond......Page 85
1.6.2  Analysis via Lebesgue Constants......Page 86
1.6.3  Divergence in the Space L^1......Page 89
1.7.0.2  Second Variation on the Theme-Improved Error Estimate......Page 91
1 7.1  *Application to Bessel Functions......Page 92
1.7.2  *The Local Limit Theorem of DeMoivre-Laplace......Page 93
1.8  Appendix: Proof of the Uniform Boundedness Theorem......Page 95
1.9  *Appendix: Higher-Order Bessel functions......Page 96
1.10  Appendix: Cantor's Uniqueness Theorem......Page 97
2.1  Motivation and Heuristics......Page 100
2.2  Basic Properties of the Fourier Transform......Page 102
2.2.1  Riemann-Lebesgue Lemma......Page 105
2.2.2  Approximate Identities and Gaussian Summability......Page 108
2.2.2.1  Improved Approximate Identities for Pointwise Convergence......Page 111
2.2.2.2  Application to the Fourier Transform......Page 113
2.2.2.3  The n-Dimensional Poisson Kernel......Page 117
2.2.3  Fourier Transforms of Tempered Distributions......Page 119
2.2.4  *Characterization of the Gaussian Density......Page 120
2.2.5  *Wiener's Density Theorem......Page 121
2.3.1  Dirichlet Kernel and Symmetric Partial Sums......Page 123
2.3.2  Example of the Indicator Function......Page 125
2.3.4  Dini Convergence Theorem......Page 126
2.3.5  Smoothing Operations in R^1-Averaging and Summability......Page 128
2.3.6  Averaging and Weak Convergence......Page 129
2.3.7  Cesaro Summability......Page 130
2.3.7.1  Approximation Properties of the Fejer Kernel......Page 132
2.3.8  Bernstein's Inequality......Page 133
2.3.9.1  Fourier Cosine Transform......Page 135
2.3.9.3  Generalized h-Transform......Page 136
2.4.1  Planchette's Theorem......Page 139
2.4.2  *Bernstein's Theorem for Fourier Transforms......Page 140
2.4.3  The Uncertainty Principle......Page 142
2.4.3.1  Uncertainty Principle on the Circle......Page 144
2.4.4.1  Hermite Polynomials......Page 145
2.4.4.2  Eigenfunction of the Fourier Transform......Page 147
2.4.4.3  Orthogonality Properties......Page 148
2.4.4.4  Completeness......Page 149
2.5.1  Bochner's Approach......Page 150
2.5.2  Piecewise Smooth Viewpoint......Page 156
2.5.3  Relations with the Wave Equation......Page 157
2.5.3.1  The Method of Brandolini and Colzani......Page 160
2.5.4  Bochner-Riesz Summability......Page 163
2.5.4.1  A General Theorem on Almost-Everywhere Summability......Page 164
2.6  Bessel Functions......Page 165
2.6.1  Fourier Transforms of Radial Functions......Page 168
2.6.2  L^2-Restriction Theorems for the Fourier Transform......Page 169
2.6.2.1  An Improved Result......Page 170
2.6.2.2  Limitations on the Range of p......Page 172
2.7  The Method of Stationary Phase......Page 173
2.7.1  Statement of the Result......Page 174
2.7.2  Application to Bessel Functions......Page 175
2.7.3  Proof of the Method of Stationary Phase......Page 176
2.7.4  Abel's Lemma......Page 178
3.2  The M. Riesz-Thorin Interpolation Theorem......Page 180
3.2.0.2  The Hausdorff-Young Inequality......Page 185
3.2.1  Stein's Complex Interpolation Theorem......Page 186
3.3  The Conjugate Function or Discrete Hilbert Transform......Page 187
3.3.1  L^p Theory of the Conjugate Function......Page 188
3.3.2  L^1 Theory of the Conjugate Function......Page 190
3.3.2.1  Identification as a Singular Integral......Page 194
3.4  The Hilbert Transform on R......Page 195
3.4.1  L^2 Theory of the Hilbert Transform......Page 196
3.4.2  L^p Theory of the Hilbert Transform, 1 < p < oo......Page 197
3.4.2.1  Applications to Convergence of Fourier Integrals......Page 198
3.4.3  L^1 Theory of the Hilbert Transform and Extensions......Page 199
3.4.3.1  Kolmogorov's Inequality for the Hilbert Transform......Page 203
3.4.4  Application to Singular Integrals with Odd Kernels......Page 205
3.5  Hardy-Littlewood Maximal Function......Page 208
3.5.1  Application to the Lebesgue Differentiation Theorem......Page 211
3.5.2  Application to Radial Convolution Operators......Page 213
3.5.3  Maximal Inequalities for Spherical Averages......Page 214
3.6  The Marcinkiewicz Interpolation Theorem......Page 217
3.7  Calderon-Zygmund Decomposition......Page 220
3.8  A Class of Singular Integrals......Page 221
3.9.1  General Properties......Page 223
3.9.2  Representation Theorems in the Disk......Page 225
3.9.3  Representation Theorems in the Upper Half-Plane......Page 227
3.9.4  Herglotz/Bochner Theorems and Positive Definite Functions......Page 230
4.1  Motivation and Heuristics......Page 233
4.2.1  Periodization of a Function......Page 234
4.2.2  Statement and Proof......Page 236
4.2.3  Shannon Sampling......Page 239
4.3  Multiple Fourier Series......Page 241
4.3.1  Basic L^1 Theory......Page 242
4.3.1.2  Representation of Spherical Partial Sums......Page 244
4.3.2  Basic L 2 Theory......Page 246
4.3.3  Restriction Theorems for Fourier Coefficients......Page 247
4.4  Poisson Summation Formula in R^d......Page 249
4.4.1  *Simultaneous Nonlocalization......Page 250
4.5.1  Kendall's Mean Square Error......Page 252
4.5.2  Landau's Asymptotic Formula......Page 254
4.5.3  Application to Multiple Fourier Series......Page 255
4.5.3.1  Three-Dimensional Case......Page 256
4.6  Schrodinger Equation and Gauss Sums......Page 258
4.6.1  Distributions on the Circle......Page 259
4.6.2  The Schrodinger Equation on the Circle......Page 261
4.7  Recurrence of Random Walk......Page 263
5.2  Basic Definitions......Page 267
5.2.1  The Central Limit Theorem......Page 271
5.2.1.1  Restatement in Terms of Independent Random Variables......Page 272
5.3  Extension to Gap Series......Page 273
5.3.1  Extension to Abel Sums......Page 277
5.4  Weak Convergence of Measures......Page 279
5.4.1  An Improved Continuity Theorem......Page 280
5.4.1.1  Another Proof of Bochner's Theorem......Page 281
5.5  Convolution Semigroups......Page 283
5.6  The Berry-Ess6en Theorem......Page 287
5.6.1  Extension to Different Distributions......Page 290
5.7  The Law of the Iterated Logarithm......Page 291
6.1  Motivation and Heuristics......Page 295
6.1.1  Heuristic Treatment of the Wavelet Transform......Page 296
6.2  Wavelet Transform......Page 297
6.2.0.1  Wavelet Characterization of Smoothness......Page 301
6.3.1  Haar Functions and Haar Series......Page 302
6.3.2  Haar Sums and Dyadic Projections......Page 303
6.3.3  Completeness of the Haar Functions......Page 306
6.3.3.1  Haar Series in Co and L t, Spaces......Page 307
6.3.3.2  Pointwise Convergence of Haar Series......Page 309
6.3.4  *Construction of Standard Brownian Motion......Page 310
6.3.6  *Proof of Continuity......Page 312
6.3.7  *L6vy's Modulus of Continuity......Page 313
6.4  Multiresolution Analysis......Page 314
6.4.1  Orthonormal Systems and Riesz Systems......Page 315
6.4.2  Scaling Equations and Structure Constants......Page 321
6.4.3  From Scaling Function to MRA......Page 324
6.4.3.1  Additional Remarks......Page 326
6.4.4  Meyer Wavelets......Page 329
6.4.5  From Scaling Function to Orthonormal Wavelet......Page 330
6.4.5.1  Direct Proof that V_1 \ominus V_0 Is Spanned by {\Phi{t-k)}_k......Page 335
6.4.5.2  Null lntegrability of Wavelets Without Scaling Functions......Page 336
6.5  Wavelets with Compact Support......Page 337
6.5.1  From Scaling Filter to Scaling Function......Page 338
6.5.2  Explicit Construction of Compact Wavelets......Page 341
6.5.2.1  Daubechies Recipe......Page 342
6.5.2.2  Hernandez-Weiss Recipe......Page 344
6.5.3  Smoothness of Wavelets......Page 345
6.5.3.1  A Negative Result......Page 347
6.5.4  Cohen's Extension of Theorem 6.5.1......Page 349
6.6.1  Wavelet Series in L^p Spaces......Page 352
6.6.1.1  Large Scale Analysis......Page 356
6.6.1.2  Almost-Everywhere Convergence......Page 357
6.6.2  Jackson and Bernstein Approximation Theorems......Page 358
6.7.1  Two Important Examples......Page 363
6.7.2  General Formulation of MRA and Wavelets in R^d......Page 365
6.7.2.1  Notations for Subgroups and Cosets......Page 366
6.7.2.2  Riesz Systems and Orthonormal Systems in R^d......Page 367
6.7.2.3  Scaling Equation and Structure Constants......Page 368
6.7.2.4  Existence of the Wavelet Set......Page 369
6.7.2.5  Proof That the Wavelet Set Spans V_1 \ominus V_0......Page 372
6.7.3  Examples of Wavelets in R^d......Page 373
References......Page 376
Notations......Page 380
Index......Page 384