Extending the range of error estimates for radial approximation in Euclidean space and on spheres

TL;DR

This paper extends error estimates for radial basis function interpolation to functions with intermediate smoothness levels in Euclidean space and on spheres, using an adapted error doubling technique.

Contribution

It adapts Schaback's error doubling trick to provide error bounds for functions with intermediate smoothness, applicable to both Euclidean spaces and spheres.

Findings

Error estimates for functions with intermediate smoothness.

Convergence of pseudoderivatives of the interpolation error.

Extension of error bounds to spherical domains.

Abstract

We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68(225):201--216, 1999.] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of R^d and spheres. As a step on the way to our ultimate goal we also show convergence of pseudoderivatives of the interpolation error.