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Replicant compression coding in Besov spaces

Published online by Cambridge University Press:  15 May 2003

Gérard Kerkyacharian
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;
Dominique Picard
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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We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B^s_{\pi,q}$ on a regular domain of ${\mathbb R}^d.$ The result is: if s - d(1/π - 1/p)+> 0, then the Kolmogorov metric entropy satisfies H(ε) ~ ε-d/s. This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound, we provide a new universal coding based on a thresholding-quantizing procedure using replication.

Research Article
© EDP Sciences, SMAI, 2003

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