Approximation by piecewise constants on convex partitions
type: Item
instance of: scholarly article
title: Approximation by piecewise constants on convex partitions
zbMATH Open document ID: 1238.41013
published in: Journal of Approximation Theory
publication date: 2012-03-22T00:00:00Z
review text: Let \(\Omega\subset\mathbb{R}^d\), \(d\in \mathbb{N}\), be a bounded domain and let \(\Delta\) be a convex partition of \(\Omega\). Denote by \(|\Delta|\) the number of convex cells in \(\Delta\). For \(\Omega := (0,1)^d\), the author shows that the order of approximation of an \(f\in W^2_p (\Omega)\) by piecewise constants can be improved to \(\mathcal{O}(|\Delta|^{-2/(d+1)})\) by using suitable anisotropic convex partitions obtained from a simple algorithm. Furthermore, he proves that the saturation order of piecewise constant approximation in the \(L_\infty\)-norm on convex partitions is \(|\Delta|^{-2/(d+1)}\). In addition, it is shown that the saturation order for linear approximation on convex partitions is \(|\Delta|^{-2/d}\), which is the same as for isotropic partitions.
Mathematics Subject Classification ID: 41A29
Mathematics Subject Classification ID: 41A40
zbMATH DE Number: 6017050
zbMATH Keywords: piecewise constant approximation
zbMATH Keywords: linear approximation
zbMATH Keywords: saturation
author: O. Davydov
reviewed by: Peter Massopust
MaRDI profile type: Publication
full work available at URL: j.jat.2011.11.001
OpenAlex ID: W2013427511
cites work: Q5864529
cites work: Algorithms and Error Bounds for Multivariate Piecewise Constant Approximation
cites work: The Bramble--Hilbert Lemma for Convex Domains
cites work: Q4884590
cites work: Optimal meshes for finite elements of arbitrary order
DOI: 10.1016/J.JAT.2011.11.001
description: scientific article; zbMATH DE number 6017050