XIV CONTENTS 6.3.4 ""Construction of Standard Brownian Motion 6.3.5 *Haar Function Representation of Brownian Motion 6.3.6 *Proof of Continuity 6.3.7 *Levy's Modulus of Continuity 6.4 Multiresolution Analysis 6.4.1 Orthonormal Systems and Riesz Systems 6.4.2 Scaling Equations and Structure Constants 6.4.3 From Scaling Function to MRA 6.4.3.1 Additional Remarks 6.4.4 Meyer Wavelets 6.4.5 From Scaling Function to Orthonormal Wavelet 6.4.5.1 Direct Proof that Vx 0 Vb Is Spanned by {V(t-k)}keZ 6.4.5.2 Null Integrability of Wavelets Without Scaling Functions 6.5 Wavelets with Compact Support 6.5.1 From Scaling Filter to Scaling Function 6.5.2 Explicit Construction of Compact Wavelets 6.5.2.1 Daubechies Recipe 6.5.2.2 Hernandez-Weiss Recipe 6.5.3 Smoothness of Wavelets 6.5.3.1 A Negative Result 6.5.4 Cohen's Extension of Theorem 6.5.1 6.6 Convergence Properties of Wavelet Expansions 6.6.1 Wavelet Series in U Spaces 6.6.1.1 Large Scale Analysis 6.6.1.2 Almost-Everywhere Convergence 6.6.1.3 Convergence at a Preassigned Point 6.6.2 Jackson and Bernstein Approximation Theorems 6.7 Wavelets in Several Variables 6.7.1 Two Important Examples 6.7.1.1 Tensor Product of Wavelets 6.7.2 General Formulation of MRA and Wavelets in Rd 6.7.2.1 Notations for Subgroups and Cosets 6.7.2.2 Riesz Systems and Orthonormal Systems in 6.7.2.3 Scaling Equation and Structure Constants 6.7.2.4 Existence of the Wavelet Set 6.7.2.5 Proof That the Wavelet Set Spans V{ 0 V0 6.7.2.6 Cohen's Theorem in R^ 6.7.3 Examples of Wavelets in M.d References Notations Index
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