Volume 7, March 2003
|Page(s)||239 - 250|
|Published online||15 May 2003|
Replicant compression coding in Besov spaces
Laboratoire de Probabilités et Modèles
Aléatoires, UMR 7599 du CNRS, Université Paris VI et Université
Paris VII, 16 rue de Clisson, 75013 Paris, France.
2 Université Paris X – Nanterre, 200 avenue de la République, 92001 Nanterre Cedex, France; email@example.com.
Revised: 20 January 2003
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 on a regular domain of 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.
Mathematics Subject Classification: 41A25 / 41A46 / 65F99 / 65N12 / 65N55
Key words: Entropy / coding / Besov spaces / wavelet bases / replication.
© EDP Sciences, SMAI, 2003
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