CONTENTS Xiii 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
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