On the Interpolation of Smooth Functions via Radial Basis Functions

Abstract

We consider the interpolatory theory of bandlimited functions at both the integer lattice and at more general point sets in Rd by forming interpolants which lie in the linear span of translates of a single Radial Basis Function (RBF). Asymptotic behavior of the interpolants in terms of a given parameter associated with the RBF is considered; in these instances the original bandlimited function can be recovered in L2 and uniformly by a limiting process.

Additionally, multivariate interpolation of nonuniform data is considered, and sufficient conditions are given on a family of RBFs which allow for recovery of multidimensional bandlimited functions. We also consider the rate of approximation that can be obtained in different cases. Sometimes, we may say something about the rate in terms of the RBF parameter mentioned above, while other times, we achieve rates based on a shrinking mesh size. The latter technique allows us to consider interpolation of Sobolev functions and their associated approximation rates as well.