Quasi-interpolation in shift invariant spaces
type: Item
instance of: scholarly article
title: Quasi-interpolation in shift invariant spaces
zbMATH Open document ID: 0981.41002
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-03-13T00:00:00Z
review text: Let \(\phi:\mathbb{R}^s\to\mathbb{R}\) be a compactly supported function, and \(S(\phi)\) denote the linear span of \(\{\phi(\cdot -{\mathbf k}) :{\mathbf k}\in\mathbb{Z}^s\}\), \(s\in\mathbb{N}.\) The authors consider the problem of approximating a continuous function \(f : \mathbb{R}^s\to \mathbb{R}\) on compact subsets of \(\mathbb{R}^s\) from the classes \(S(\phi(h\cdot))\), \(h > 0\). A construction of quasi-linear operators based on scattered data, with no conditions on their location, is described. The degree of polynomials which are preserved by these operators depends upon the density of the data. Rather than solving a system of equations, the authors solve a quadratic (or linear) programming problem of a size having the same order of magnitude. It is demonstrated how classical Markov inequalities for polynomials lead to the constructions of local, quasi-interpolatory operators.
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 65D07
zbMATH DE Number: 1545050
zbMATH Keywords: quasi-interpolatory operators
zbMATH Keywords: shift invariant spaces
zbMATH Keywords: quasi-linear operators
zbMATH Keywords: local functional based on scattered data
zbMATH Keywords: Markov inequalities for polynomials
author: H. N. Mhaskar
author: F. J. Narcowich
author: J. D. Ward
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
OpenAlex ID: W2071376846
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DOI: 10.1006/JMAA.2000.7051
description: scientific article; zbMATH DE number 1545050