Multivariate Interpolation by Polynomials and Radial Basis Functions

Abstract

In many cases, multivariate interpolation by smooth radial basis functions converges toward polynomial interpolants, when the basis functions are scaled to become flat. In particular, examples show and this paper proves that interpolation by scaled Gaussians converges toward the de Boor/Ron “least” polynomial interpolant. To arrive at this result, a few new tools are necessary. The link between radial basis functions and multivariate polynomials is provided by ”radial polynomials” $\|x-y\|_2^{2\ell}$ that already occur in the seminal paper by C.A. Micchelli of 1986. We study the polynomial spaces spanned by linear combinations of shifts of radial polynomials and introduce the notion of a discrete moment basis to define a new well-posed multivariate polynomial interpolation process which is of minimal degree and also “least” and “degree-reducing” in the sense of de Boor and Ron. With these tools at hand, we generalize the de Boor/Ron interpolation process and show that it occurs as the limit of interpolation by Gaussian radial basis functions. As a byproduct, we get a stable method for preconditioning the matrices arising with interpolation by smooth radial basis functions.