Error estimates for scattered data interpolation on spheres

Abstract

We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the $n$-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.