Approximation orders for natural splines in arbitrary dimensions
type: Item
instance of: scholarly article
zbMATH Open document ID: 0967.41009
DOI: 10.1090/S0025-5718-00-01299-0
author: Tim Gutzmer
author: Jens Marcus Melenk
published in: Mathematics of Computation
publication date: 2001-02-19T00:00:00Z
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1566127
zbMATH Keywords: multivariate interpolation
MaRDI profile type: Publication
cites work: Q4073917
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cites work: Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^m$-splines
cites work: Multivariate interpolation at arbitrary points made simple
cites work: The uniform convergence of thin plate spline interpolation in two dimensions
cites work: Q4379958
cites work: Improved error bounds for scattered data interpolation by radial basis functions
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title: Approximation orders for natural splines in arbitrary dimensions
review text: Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain with Lipschitz boundary, and let \(m\), \(k\in\mathbb{N}_0\), \(p\in [2,\infty]\) be such that \(m>n/2\) and \(H^m(\Omega)\subset W^{k,p}(\Omega)\) hold. The aim of the authors is to improve a result by \textit{J. Duchon} [RAIRO, Anal. Numer. 12, 325-334 (1978; Zbl 0403.41003)] concerning the minimal norm interpolant of a function belonging to the Beppo Levi space \(BL^m(\Omega)\). It is shown that there exist constants \(C\) and \(h_0>0\) having the following property: for any collection \(X=\{x_1, \dots, x_N\} \subset\Omega\) of interpolation points containing a \({\mathcal P}^n_{m-1}\)-unisolvent subset and satisfying NEWLINE\[NEWLINEh=\sup_{x\in\Omega} \inf_{x_i\in X} \|x-x_i\|_2\leq h_0,NEWLINE\]NEWLINE and for all \(f\in BL^{2m} (\Omega)\) with \(\partial^\alpha f=0\) on \(\partial\Omega\) for \(|\alpha |=m, \dots,2m-1\), the function \(s_f \in BL^m (\mathbb{R}^n)\) that minimizes \(\|s_f\|_{BL^m (\mathbb{R}^n)}\) and interpolates \(s_f(x_i)= f(x_i)\), \(i=1,\dots,N\), satisfies \(\sum_{|\alpha |=k} \|\partial^\alpha (f-s_f) \|_{L^p (\Omega)}\leq Ch^{2m-k-n/2+n/p} \|f\|_{BL^{2m} (\Omega)}\).
reviewed by: Wolfgang W. Breckner
description: scientific article