A Duchon framework for the sphere
type: Item
instance of: scholarly article
title: A Duchon framework for the sphere
zbMATH Open document ID: 1051.41005
author: Simon Hubbert
author: Tanya M. Morton
published in: Journal of Approximation Theory
publication date: 2004-08-20T00:00:00Z
review text: The paper is motivated by finding error bounds for radial basis functions interpolation in the case when the data points lie on the surface \(S^{d-1}\) of the unit ball of the Euclidean space \({\mathbb R}^d\). For a given \(z \in S^{d-1}\) and \(\theta \in (0,2\pi]\), let \(G(z, \theta)\) denote the set of all points \(\zeta \in S^{d-1}\) whose geodesic distance from \(z\) is \(<\theta\). The authors study relations, in terms of appropriate Sobolev spaces \(W_2^\beta\), between the local and global smoothness of functions \(f \in S^{d-1}\to {\mathbb R}\). They show that for every \(\theta\) it is possible to cover \(S^{d-1}\) with a finite collection of geodesic balls \(G_i=G(z_i, \theta)\), each of the same radius \(\theta\), so that for every \(f \in W_2^\beta(S^{d-1})\) one has \[ \sum_{G_i} \| f | _{G_i}\| ^2_{W_2^\beta (G_i)} \leq Q \| f\| ^2_{W_2^\beta (S^{d-1})}\;, \] where the constant \(Q\) is independent of \(\theta\). The second purpose of the paper is to construct a bounded linear extension operator from \(W_2^\beta (G(z, \theta))\) to \(W_2^\beta (S^{d-1})\). It is shown that the norm of such an operator necessarily depends on \(\theta\). However the dependence on \(\theta\) can be eliminated if one takes only those functions \(f \in W_2^\beta (G(z, \theta))\) that vanish on a certain finite subset of \(G(z, \theta)\).
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 2093758
zbMATH Keywords: interpolation
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: radial basis functions
reviewed by: Yuly Makovoz
MaRDI profile type: Publication
full work available at URL: j.jat.2004.04.005
OpenAlex ID: W2016969219
cites work: Q4073917
cites work: Q5661106
cites work: Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^m$-splines
cites work: \(L_{p}\)-error estimates for radial basis function interpolation on the sphere
cites work: Q5634306
cites work: Q4314293
cites work: Q4169288
cites work: Q4938102
DOI: 10.1016/J.JAT.2004.04.005
description: scientific article; zbMATH DE number 2093758