September  2007, 6(3): 569-585. doi: 10.3934/cpaa.2007.6.569

Limits of radial basis function interpolants

1. 

Justus–Liebig University, Mathematics Institut, Arndtstr. 2, 35392 Giessen, Germany

2. 

Jagiellonian University, Mathematics Institute, ul. Remonta 4, 30–059 Krakow, Poland

Received  February 2006 Revised  April 2006 Published  June 2007

We solve some open problems posed by Fornberg et al. in [6], [9] and [12], related to radial basis functions with parameters. They concern the limits of interpolants using these radial basis functions when the aforementioned parameters tend to zero--which makes them "increasingly flat" in a term coined by Fornberg. These aspects of radial basis function interpolation are useful because they concern the numerical problems with ill-conditioned matrices for small parameters and how to solve the interpolation problems efficiently in the face of this ill-conditioning. Finally, there are some interesting links between radial basis function interpolation and polynomial interpolation coming out of this research. While answering several such conjectures, we also develop a number of new techniques--some of them with number-theoretic arguments--for attacking similar problems.
Citation: Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure and Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
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