Multivariate Interpolation by Polynomials and Radial Basis Functions

In many cases, multivariate interpolation by smooth radial basis functions converges towards polynomial interpolants, when the basis functions are scaled to become “wide”. In particular, examples show that interpolation by scaled Gaussians seems to converge towards the de Boor/Ron “least” polynomial interpolant. The paper starts by providing sufficient criteria for the convergence of radial interpolants, and the structure of the polynomial limit interpolation is investigated to some extent. The results lead to general questions about “radial polynomials” ‖x− y‖ 2 and the properties of spaces spanned by linear combinations of their shifts. For their investigation a number of helpful results are collected. In particular, the new notion of a discrete moment basis turns out to be rather useful. With these tools, a variety of well–posed multivariate polynomial interpolation processes can be formulated, leading to interesting questions about their relationships. Part of them can be proven to be “least” in the sense of de Boor and Ron. Finally, the paper generalizes the de Boor/Ron interpolation process and shows 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.