Replicant compression coding in Besov spaces

Kerkyacharian, Gérard; Picard, Dominique

doi:10.1051/ps:2003011

Kerkyacharian, Gérard ; Picard, Dominique

ESAIM: Probability and Statistics, Volume 7 (2003), pp. 239-250

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 $B_{π, q}^{s}$ on a regular domain of $ℝ^{d} .$ The result is: if $s - d {(1 / π - 1 / p)}_{+}$ $> 0,$ then the Kolmogorov metric entropy satisfies $H (ϵ) \sim ϵ^{- d / s}$ . 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.

MR Zbl

DOI: 10.1051/ps:2003011

Classification: 41A25, 41A46, 65F99, 65N12, 65N55
Keywords: entropy, coding, Besov spaces, wavelet bases, replication

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     publisher = {EDP Sciences},
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     url = {https://www.numdam.org/articles/10.1051/ps:2003011/}
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Kerkyacharian, Gérard; Picard, Dominique. Replicant compression coding in Besov spaces. ESAIM: Probability and Statistics, Volume 7 (2003), pp. 239-250. doi: 10.1051/ps:2003011

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