Table of Contents

1 Wavelets.- 1.1 What can wavelets offer?.- 1.2 General remarks.- 1.3 Data compression.- 1.4 Local adaptivity.- 1.5 Nonlinear smoothing properties.- 1.6 Synopsis.- 2 The Haar basis wavelet system.- 3 The idea of multiresolution analysis.- 3.1 Multiresolution analysis.- 3.2 Wavelet system construction.- 3.3 An example.- 4 Some facts from Fourier analysis.- 5 Basic relations of wavelet theory.- 5.1 When do we have a wavelet expansion?.- 5.2 How to construct mothers from a father.- 5.3 Additional remarks.- 6 Construction of wavelet bases.- 6.1 Construction starting from Riesz bases.- 6.2 Construction starting from m0.- 7 Compactly supported wavelets.- 7.1 Daubechies’ construction.- 7.2 Coiflets.- 7.3 Symmlets.- 8 Wavelets and Approximation.- 8.1 Introduction.- 8.2 Sobolev Spaces.- 8.3 Approximation kernels.- 8.4 Approximation theorem in Sobolev spaces.- 8.5 Periodic kernels and projection operators.- 8.6 Moment condition for projection kernels.- 8.7 Moment condition in the wavelet case.-9 Wavelets and Besov Spaces.- 9.1 Introduction.- 9.2 Besov spaces.- 9.3 Littlewood-Paley decomposition.- 9.4 Approximation theorem in Besov spaces.- 9.5 Wavelets and approximation in Besov spaces.- 10 Statistical estimation using wavelets.- 10.1 Introduction.- 10.2 Linear wavelet density estimation.- 10.3 Soft and hard thresholding.- 10.4 Linear versus nonlinear wavelet density estimation.- 10.5 Asymptotic properties of wavelet thresholding estimates.- 10.6 Some real data examples.- 10.7 Comparison with kernel estimates.- 10.8 Regression estimation.- 10.9 Other statistical models.- 11 Wavelet thresholding and adaptation.- 11.1 Introduction.- 11.2 Different forms of wavelet thresholding.- 11.3 Adaptivity properties of wavelet estimates.- 11.4 Thresholding in sequence space.- 11.5 Adaptive thresholding and Stein’s principle.- 11.6 Oracle inequalities.- 11.7 Bibliographic remarks.- 12 Computational aspects and software.- 12.1 Introduction.- 12.2 The cascade algorithm.- 12.3 Discrete wavelet transform.- 12.4 Statistical implementation of the DWT.- 12.5 Translation invariant wavelet estimation.- 12.6 Main wavelet commands in XploRe.- A Tables.- A.1 Wavelet Coefficients.- A.2.- B Software Availability.- C Bernstein and Rosenthal inequalities.- D A Lemma on the Riesz basis.- Author Index.

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