Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

Replicant compression coding in Besov spaces

Auteur(s):

Code(s) de Classification MSC:

• 41A25 Rate of convergence, degree of approximation
• 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
• 65F99 None of the above but in this section
• 65N12 Stability and convergence of numerical methods
• 65N55 Multigrid methods; domain decomposition

Résumé: We present here a new proof of Birman and Solomyak theorem on the metric entropy of the unit ball of a Besov space $B^s_{\pi,q}$ on regular domain of $\bR^d.$ The result is well known : if $s-d(1/\pi-1/p)_+ > 0,$ then the Kolmogorov metric entropy verifies $H(\epsilon) \asymp \epsilon^{-d/s} .$ This proof takes advantage of the representation of such spaces on wavelet type bases. The lower bound is a consequence of a very simple probabilistic exponential inequalities. To prove the upper bounds, we use a replicant universal coding based on a thresholding-quantizing procedure.

Mots Clés: Compression ; m-term approximation ; coding ; Kolmogorov entropy ; wavelets bases ; replication

Date: 2001-09-06

Prépublication numéro: PMA-681