Norming Sets and Scattered Data Approximation on Spheres

This short note deals with approximation order of spaces spanned by (x;), x 2 X, with a positive deenite kernel on a sphere and X a given set of nodes. We estimate both the L 2-and the L 1-error, if the function to be approximated is assumed to be rather smooth, thus deviating from the usual assumption that the function stems from thèna-tive space' of the kernel. It is of interest that, based on our former results on norming sets, we can bound p-norms of right inverses of collocation matrices which originate from evaluating the orthonormal basis of spherical harmonics up to a given degree L at X. The notion of norming sets proved to be an essential tool for our investigation of scattered data interpolation on spheres S n?1 IR n ; see 2]. There, our study was based on the so-called native spaces; these are certain Hilbert spaces of continuous functions on S n?1 which possess a positive deenite reproducing kernel (x; y), x; y 2 S n?1. In such a setting, the same kernel serves two diierent purposes: rstly, it deenes the Hilbert space H of functions to be interpolated (see Section 3), and secondly, it also generates the space of interpolants, (1) where X S n?1 is the given ((nite) set of interpolation nodes. Then the Golomb-Weinberger theory of optimal interpolation of linear functionals can be employed in order to nd error estimates for the interpolation. Norming sets have entered our estimation technique, and they can be favourably used when the knots are scattered. In the present paper we wish to establish further results of a similar character, but in a more general setting. In fact, we use the same space V X as before, generated by a positive deenite kernel. But we investigate upper bounds for the best approximation (rather than interpolation) of functions All rights of reproduction in any form reserved.