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Replicant compression coding in Besov spaces
Published online by Cambridge University Press: 15 May 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 $B^s_{\pi,q}$ 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.
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- © EDP Sciences, SMAI, 2003
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