Wavelets, Approximation, and Statistical Applications

1. Front Matter

   Pages i-xviii

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2. Wavelets

   • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

   Pages 1-16

3. The Haar basis wavelet system

   • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

   Pages 17-23

4. The idea of multiresolution analysis

   • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

   Pages 25-29

5. Some facts from Fourier analysis

   • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

   Pages 31-34

6. Basic relations of wavelet theory

   • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

   Pages 35-45

7. Construction of wavelet bases

   • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

   Pages 47-58

8. Compactly supported wavelets

   • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

   Pages 59-69

9. Wavelets and Approximation

   • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

   Pages 71-100

10. Wavelets and Besov Spaces

    • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

    Pages 101-124

11. Statistical estimation using wavelets

    • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

    Pages 125-191

12. Wavelet thresholding and adaptation

    • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

    Pages 193-213

13. Computational aspects and statistical software implementations

    • Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

    Pages 215-235

14. Back Matter

    Pages 237-268

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The mathematical theory of ondelettes (wavelets) was developed by Yves Meyer and many collaborators about 10 years ago. It was designed for ap­ proximation of possibly irregular functions and surfaces and was successfully applied in data compression, turbulence analysis, image and signal process­ ing. Five years ago wavelet theory progressively appeared to be a power­ ful framework for nonparametric statistical problems. Efficient computa­ tional implementations are beginning to surface in this second lustrum of the nineties. This book brings together these three main streams of wavelet theory. It presents the theory, discusses approximations and gives a variety of statistical applications. It is the aim of this text to introduce the novice in this field into the various aspects of wavelets. Wavelets require a highly interactive computing interface. We present therefore all applications with software code from an interactive statistical computing environment. Readers interested in theory and construction of wavelets will find here in a condensed form results that are somewhat scattered around in the research literature. A practioner will be able to use wavelets via the available software code. We hope therefore to address both theory and practice with this book and thus help to construct bridges between the different groups of scientists. This te. xt grew out of a French-German cooperation (Seminaire Paris­ Berlin, Seminar Berlin-Paris). This seminar brings together theoretical and applied statisticians from Berlin and Paris. This work originates in the first of these seminars organized in Garchy, Burgundy in 1994.

• DEX
• Invariant
• Lemma
• Oracle
• Sequence space
• Sobolev space
• adaptation
• availability
• code
• construction
• functions
• kernel
• signal processing
• turbulence
• wavelet

• Wirtschaftswissenschaftliche Fakultät, Institut fär Statistik und Ökonometrie, Humboldt-Universität zu Berlin, Berlin, Germany

  Wolfgang Härdle

• URA CNRS 1321 Modal, Université Paris X, Nanterre Cedex, France

  Gerard Kerkyacharian

• UFR Mathématique, URA CNRS 1321, Université Paris VII, Paris cedex 5, France

  Dominique Picard

• Laboratoire de Probabilités, Université Paris VI, Paris cedex 5, France

  Alexander Tsybakov

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