Replicant compression coding in Besov spaces

¹ 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; picard@math.jussieu.fr.

Received: 9 April 2001
Revised: 20 January 2003

Abstract

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.