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Replicant compression coding in Besov spaces

Published online by Cambridge University Press:  15 May 2003

Gérard Kerkyacharian  and
Dominique Picard
Gérard Kerkyacharian
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 du CNRS, Université Paris VI et Université Paris VII, 16 rue de Clisson, 75013 Paris, France. Université Paris X – Nanterre, 200 avenue de la République, 92001 Nanterre Cedex, France; picard@math.jussieu.fr.
Dominique Picard
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 du CNRS, Université Paris VI et Université Paris VII, 16 rue de Clisson, 75013 Paris, France.
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Abstract

We present here a new proof of the theorem ofBirman and Solomyak on the metric entropy of the unit ball of aBesov space $B^s_{\pi,q}$ on a regular domain of ${\mathbb R}^d.$ Theresult is: if s - d(1/π - 1/p) +> 0, then the Kolmogorov metric entropy satisfies H(ε) ~ ε-d/s . This prooftakes advantage of the representation of such spaces on wavelet typebases and extends the result to more general spaces. The lower boundis a consequence of very simple probabilistic exponentialinequalities. To prove the upper bound, we provide a newuniversal coding based on a thresholding-quantizing procedure usingreplication.

Keywords

EntropycodingBesovspaceswavelet basesreplication.

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 239 - 250
Copyright
© EDP Sciences, SMAI, 2003

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