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IMA Journal of Numerical Analysis Advance Access published online on February 26, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn051
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A note on radial basis function interpolant limits

Martin D. Buhmann{dagger}

Mathematics Institute, Justus-Liebig University, Arndtstrasse 2, D-35392 Giessen, Germany

Slawomir Dinew{ddagger}

Mathematics Department, Jagiellonian University, ul. Reymonta 4, PL-30-059 Krakow, Poland

Elisabeth Larsson§

Scientific Computing, Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden

{dagger} Corresponding author. Email: buhmann{at}math.uni-giessen.de

{ddagger} Email: slawomir.dinew{at}im.uj.edu.pl

§ Email: bette{at}it.uu.se

Received on 1 November 2007. Revised on 26 May 2008.


  Abstract

Radial basis functions (RBFs) are very useful in multivariate interpolation because of their ability to produce highly accurate results for scattered data. Many of them, especially the Gaussian RBF and the multiquadric RBF, contain parameters that need to be adjusted in order to improve the approximations. In fact, it is often of interest to let the parameters tend to certain limits. Here we study if and when the limits of RBF interpolants with parameters exist. Mainly, the dependence of the limit on the properties of the radial functions and on the geometries of the data points is investigated, and some examples are provided.

Key Words: radial basis function; interpolation; limit; divergence


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