Author | Hรคrdle, Wolfgang. author |
---|---|
Title | Wavelets, Approximation, and Statistical Applications [electronic resource] / by Wolfgang Hรคrdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov |
Imprint | New York, NY : Springer New York, 1998 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-2222-4 |
Descript | XVIII, 265 p. online resource |
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 Daubechiesโ 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 Steinโs 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