# American Institute of Mathematical Sciences

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