Replicant compression coding in Besov spaces

Kerkyacharian, Gérard; Picard, Dominique

doi:10.1051/ps:2003011

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Replicant compression coding in Besov spaces

Kerkyacharian, Gérard ; Picard, Dominique

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

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Résumé

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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     author = {Kerkyacharian, G\'erard and Picard, Dominique},
     title = {Replicant compression coding in {Besov} spaces},
     journal = {ESAIM: Probability and Statistics},
     pages = {239--250},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2003},
     doi = {10.1051/ps:2003011},
     mrnumber = {1987788},
     zbl = {1031.41014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2003011/}
}

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Kerkyacharian, Gérard; Picard, Dominique. Replicant compression coding in Besov spaces. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 239-250. doi: 10.1051/ps:2003011

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