Multilevel Characterizations of Anisotropic Function Spaces

We consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces H a p for p > 0, which are known to be equal to Lp for p > 1. In particular, we characterize those functions that provide a specific convergence rate by intrinsic smoothness properties. To this end we apply restricted nonlinear approximation, see [3], which is a generalization of n-term approximation in which a weight function is used to control the terms of the approximations.

Section: Anisotropic Wavelet Basesmentioning

confidence: 99%

“…Typically, there exists p 0 ∈ (0, 1), which may depend on the chosen wavelet basis, such that p ∈ (p 0 , ∞) are admissible, see [17]. Furthermore, for p > 1 it is well-known that H a p R d = L p R d with equivalent norms, see [21].…”

Section: Anisotropic Wavelet Basesmentioning

confidence: 99%

Anisotropic wavelet bases and thresholding

Hochmuth

2007

“…They considered one‐parameter group of diagonal dilations of the form where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$t\in \mathbf {R}$\end{document} and a = ( a 1 , a 2 , …, a n ) is a given anisotropy. The theory of these anisotropic Triebel‐Lizorkin and Besov spaces has been developed in a way parallel to the theory of corresponding isotropic spaces in a series of, some of them recent, papers (see, for instance, 18, 20, 21, 24, and 26).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Characterizations of anisotropic Besov spaces

Barrios

2011

“…Besov and Triebel-Lizorkin sequence spaces associated to the wavelet transform are very well-known and contain Lipschitz, Hardy, Lebesgue and Sobolev spaces, see for example [7,13,[19][20][21] to name just a few in the isotropic cases. In the anisotropic case the reader is referred to [3,4,15] to name just a few.…”

Section: Waveletsmentioning

confidence: 99%

Restricted thresholding: recovering smoothness and preserving edges

Vera

2018

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 L2‐norm is not necessarily the best norm to measure the approximation error.