Replicant compression coding in Besov spaces

Gérard Kerkyacharian; Dominique Picard

ESAIM: Probability and Statistics (2010)

  • Volume: 7, page 239-250
  • ISSN: 1292-8100

Abstract

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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) ~ 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.

How to cite

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Kerkyacharian, Gérard, and Picard, Dominique. "Replicant compression coding in Besov spaces." ESAIM: Probability and Statistics 7 (2010): 239-250. <http://eudml.org/doc/104306>.

@article{Kerkyacharian2010,
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 $B^s_\{\pi,q\}$ on a regular domain of $\{\mathbb R\}^d.$ 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. },
author = {Kerkyacharian, Gérard, Picard, Dominique},
journal = {ESAIM: Probability and Statistics},
keywords = {Entropy; coding; Besov spaces; wavelet bases; replication.; entropy; Besov spaces; replication},
language = {eng},
month = {3},
pages = {239-250},
publisher = {EDP Sciences},
title = {Replicant compression coding in Besov spaces},
url = {http://eudml.org/doc/104306},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Kerkyacharian, Gérard
AU - Picard, Dominique
TI - Replicant compression coding in Besov spaces
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 239
EP - 250
AB - 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 $B^s_{\pi,q}$ on a regular domain of ${\mathbb R}^d.$ 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.
LA - eng
KW - Entropy; coding; Besov spaces; wavelet bases; replication.; entropy; Besov spaces; replication
UR - http://eudml.org/doc/104306
ER -

References

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