Limit Problems for Interpolation by Analytic Radial Basis Functions

Interpolation by analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become “increasingly flat”, or the data points “coalesce” in the limit while the radial basis functions stays fixed. Both cases call for a careful regularization. If carried out explicitly, this yields a preconditioning technique for the degenerating linear systems behind such interpolation problems. This paper deals with both degeneration cases. For the “increasingly flat” limit, we recover results by Larsson and Fornberg together with Lee, Yoon, and Yoon concerning convergence of interpolants towards polynomials. With slight modifications, the same technique also allows to handle scenarios with coalescing data points for fixed radial basis functions. The results show that the degenerating local Lagrange interpolation problems converge towards certain Hermite-Birkhoff problems. This is an important prerequisite for dealing with approximation by radial basis functions adaptively, using freely varying data sites.