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Restricted thresholding: recovering smoothness and preserving edges. (English) Zbl 1408.41009

Summary: Restricted non linear approximation is a generalization of the \(N\)-term approximation in which a measure on the index set of the approximants controls the type, instead of the number, of elements in the approximation. Thresholding is a well-known type of non linear approximation. We relate a generalized upper and lower Temlyakov property with the decreasing rate of the thresholding approximation. This relation is in the form of a characterization through some general discrete Lorentz spaces. Thus, not only we recover some results in the literature but find new ones. As an application of these results, we compress and reduce noise of some images with wavelets and shearlets and show, at least empirically, that the \(L^2\)-norm is not necessarily the best norm to measure the approximation error.

MSC:

41A25 Rate of convergence, degree of approximation
41A63 Multidimensional problems
42B35 Function spaces arising in harmonic analysis
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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