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An Adaptive Compression Algorithm in Besov Spaces

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Constructive Approximation Aims and scope

Abstract.

Given a function f on [0,1] and a wavelet-type expansion of f , we introduce a new algorithm providing an approximation $\tilde f of f with a prescribed number D of nonzero coefficients in its expansion. This algorithm depends only on the number of coefficients to be kept and not on any smoothness assumption on f . Nevertheless it provides the optimal rate D of approximation with respect to the L q -norm when f belongs to some Besov space B α p,∈fty whenever α>(1/p-1/q) + . These results extend to more general expansions including splines and piecewise polynomials and to multivariate functions. Moreover, this construction allows us to compute easily the metric entropy of Besov balls.

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June 21, 1996. Dates revised: April 9, 1998; October 14, 1998. Date accepted: October 20, 1998.

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Birgé, L., Massart, P. An Adaptive Compression Algorithm in Besov Spaces. Constr. Approx. 16, 1–36 (2000). https://doi.org/10.1007/s003659910001

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  • DOI: https://doi.org/10.1007/s003659910001

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