An interpolant defined by subdivision and the analysis of the error
type: Item
instance of: scholarly article
title: An interpolant defined by subdivision and the analysis of the error
zbMATH Open document ID: 1006.65009
DOI: 10.1016/S0377-0427(01)00536-2
author: A. A. Ligun
author: A. A. Shumeiko
author: Stefano De Marchi
author: S. V. Timchenko
published in: Journal of Computational and Applied Mathematics
publication date: 2002-08-22T00:00:00Z
review text: Given the equidistant interpolation points \((x_i,y_i)\), \(i=0,\ldots,n\) of a function \(y(x)\), an interpolating subdivision scheme estimates values at successive midpoints doubling the number of intervals at each iteration. This results in a function \(\hat{y}\) that approximates \(y\). The scheme used here is based on a cubic interpolating polynomial at 4 neighboring points, which is evaluated in its central point to find the new function value of \(\hat{y}\). At the boundaries, either cubic spline interpolation or asymmetric polynomial interpolation is used. This is closely related to the \(9/16\) scheme of \textit{S. Dubuc}'s interpolating subdivision [J. Math. Anal. Appl. 114, 185-204, 1986; Zbl 0615.65005)]. The main theorem gives error bounds for \(y\in{\mathcal C}^4\) in \(L^p\)-norm for \(p=1,2,\infty\). These bounds are \(O(h^4)\) with \(h\) the step size of the given \(x_i\) and it depends on the fourth derivative of \(y\) and on a function \(\alpha\) defined in the interpolation interval \([a,b]\) that is obtained by a subdivision-like scheme applied to the data \((a,0)\), \(((a+b)/2,1)\) and \((b,0)\). A numerical example shows that this scheme gives better results than the Shepard method, described by \textit{P. Lancaster} and \textit{K. Salkauskas} [Curve and surface fitting, Academic Press (1986; Zbl 0649.65012)].
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 41A80
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 1787675
zbMATH Keywords: interpolating subdivision
zbMATH Keywords: splines
zbMATH Keywords: Dubuc interpolation
zbMATH Keywords: cubic spline interpolation
zbMATH Keywords: error bounds
zbMATH Keywords: numerical example
zbMATH Keywords: Shepard method
MaRDI profile type: Publication
cites work: Interpolation through an iterative scheme
cites work: Q3795136
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description: scientific article; zbMATH DE number 1787675