Approximate approximations from scattered data
type: Item
instance of: scholarly article
title: Approximate approximations from scattered data
zbMATH Open document ID: 1119.41015
published in: Journal of Approximation Theory
publication date: 2007-05-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0508241
review text: In this interesting paper, the authors extend the approximate multivariate quasi-inter\-po\-lation on a uniform grid by dilated shifts of a smooth, rapidly decaying function [see e.g. \textit{V. Maz'ya} and \textit{G. Schmidt}, IMA J. Numer. Anal. 16, 13--29 (1996; Zbl 0838.65005)] to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions, the authors present a constructive method for an approximate partition of unity and apply this method to the cubature of an integral operator with radial kernel. Numerical tests are given for scattered data quasi-interpolation of univariate functions and for approximate partitions of unity with Gaussians in the univariate/bivariate case.
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 65D15
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 5151552
zbMATH Keywords: quasi-interpolation
zbMATH Keywords: approximate approximation
zbMATH Keywords: scattered data
zbMATH Keywords: scattered data quasi-interpolation
zbMATH Keywords: multivariate approximation
zbMATH Keywords: error estimates
zbMATH Keywords: approximate partition of unity
zbMATH Keywords: Gaussian function
zbMATH Keywords: cubature of integral operators
author: F. Lanzara
author: V. Maz'ya
author: Gunther Schmidt
reviewed by: Manfred Tasche
MaRDI profile type: Publication
OpenAlex ID: W2107833185
arXiv ID: math/0508241
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DOI: 10.1016/J.JAT.2006.08.003
description: scientific article