Multivariate interpolation with increasingly flat radial basis functions of finite smoothness

Abstract

In this paper, we consider multivariate interpolation with radial basis functions of finite smoothness. In particular, we show that interpolants by radial basis functions in R d with finite smoothness of even order converge to a polyharmonic spline interpolant as the scale parameter of the radial basis functions goes to zero, i.e., the radial basis functions become increasingly flat.

Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines

“…Extreme scaling of kernels, i.e., "flat" limits are investigated and they provide a new connection between finitely smooth RBF kernels and piecewise polynomial or polyharmonic splines (see [38]). We also use the eigenfunction expansions to move Fornberg's RBF-QR algorithm onto a more standard theoretical foundation which provides at the same time an algorithm for Gaussians that is applicable in any space dimension.…”

“…If the kernels are "flat", we get convergence of RBF interpolants to polynomial interpolants. We will report on a recent investigation [38] that reveals a similar connection between RBFs of limited smoothness and piecewise polynomial splines.…”

“…We begin by summarizing the essential insight gained in these papers, and then present some recent results from [38] that deal with radial kernels of finite smoothness in the next subsection. into even powers of r (i.e., κ is infinitely smooth), and that the data X are unisolvent with respect to any set of N linearly independent polynomials of degree at most m. Then lim…”

“…To our knowledge, the flat limit of RBFs with finite smoothness was not studied until the recent paper [38] in which interpolation on R d was investigated.…”

“…Before we explain the results obtained in [38], we look at a few finitely smooth radial kernels as full space Green's functions as discussed in the earlier sections.…”