Wavelet approximation of distributions with bounded variation derivatives
type: Item
instance of: scholarly article
title: Wavelet approximation of distributions with bounded variation derivatives
zbMATH Open document ID: 1166.42018
author: Paweł Bechler
published in: The Journal of Fourier Analysis and Applications
publication date: 2009-06-08T00:00:00Z
review text: The topic of this paper is a wavelet approximation of distributions from the spaces \(\text{BV}^{r}\), which are the spaces of distributions \(f\) such that the distributional derivatives \(D^{\alpha} f\) with \(|\alpha |\leq r\) exist as measures of bounded variation. These spaces are a generalization of the classical \(\text{BV}\) spaces to higher orders of smoothness. The main aim of this paper is to generalize as many of known results about the Haar approximations as possible to higher orders of smoothness of wavelet approximations, where smoothness is measured in the sense of \(\text{BV}\). This paper discusses estimates for wavelet coefficients of \(\text{BV}^{r}\) distributions, direct (Jackson) and inverse (Bernstein) inequalities for \(n\)-term approximation of elements of \(\text{BV}^{r}\) in the \(L_{p}\) spaces using compactly supported wavelets. The optimal rates of approximation are established. The linear approximation in similar contexts is considered for comparison.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 5562680
zbMATH Keywords: wavelet approximation
zbMATH Keywords: compactly supported wavelets
zbMATH Keywords: bounded variation
zbMATH Keywords: smoothness
zbMATH Keywords: distributional derivatives
zbMATH Keywords: Bernstein-type inequalities
reviewed by: V. L. Leontiev
MaRDI profile type: Publication
full work available at URL: s00041-008-9052-y
OpenAlex ID: W2076032315
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DOI: 10.1007/S00041-008-9052-Y
description: scientific article; zbMATH DE number 5562680