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Limits of radial basis function interpolants

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  • 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.

    Mathematics Subject Classification: Primary: 41A05, 41A15 ; Secondary: 41A10, 65D05, 65D07, 41A30.

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