Error estimates for the cardinal spline interpolation
type: Item
instance of: scholarly article
title: Error estimates for the cardinal spline interpolation
zbMATH Open document ID: 1171.41005
DOI: 10.4171/ZAA/1381
author: G. Vainikko
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2009-05-05T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: This paper is devoted to a study of cardinal spline interpolation in the Sobolev space \(V^{m,\infty}(\mathbb R)\) of functions with a bounded \(m\)th derivative on the real line. Specifically, for the interpolation using cardinal splines of degree \(m-1\) with knots \(ih\) and interpolation nodes \((i+1/2)h\), \(i \in \mathbb Z\), with some \(h>0\), it is well known that the interpolation error is uniformly bounded by \(\Phi_{m+1} \pi^{-m} h^m \| f \|_\infty\), where \(\Phi_{m+1}\) is the \((m+1)\)st Favard constant, if \(f \in V^{m,\infty}(\mathbb R)\) and, additionally, \(f\) is 1-periodic. The paper under review demonstrates that the same error bound holds for all \(f \in V^{m,\infty}(\mathbb R)\), no matter whether they are periodic or not. Moreover it is shown in a constructive way that this error bound is unimprovable. Additional results presented in the paper include, among others, bounds for the difference between derivatives of the function to be interpolated and the corresponding derivatives of the interpolating splines.
reviewed by: Kai Diethelm
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A44
zbMATH DE Number: 5550895
zbMATH Keywords: cardinal splines
zbMATH Keywords: interpolation
zbMATH Keywords: error estimates
zbMATH Keywords: best constants
MaRDI profile type: Publication
cites work: A practical guide to splines.
cites work: QUASI-INTERPOLATION BY SPLINES ON THE UNIFORM KNOT SETS
cites work: Cardinal interpolation and spline functions
cites work: Cardinal interpolation and spline functions. II: Interpolation of data of power growth
cites work: Q5680530
cites work: Q3895980
cites work: Q5294847
description: scientific article; zbMATH DE number 5550895