Moving Least Squares Approximation on the Sphere

We introduce moving least squares approximation as an approximation scheme on the sphere. We prove error estimates and approximation orders. Finally, we show certain numerical results. x1. Introduction Recently, approximation on the sphere has become important because of its obvious applications to Meteorology, Oceanography and Geoscience and Geo-engineering in general. Over the last years several approaches were made to reconstruct a continuous function on the sphere from a nite number of discrete data (see 4] for a recent overview). One possibility is based on the theory of radial basis functions. Much work in this direction was done by 5,6,9,15] and this list is far from being complete. Even if this approach allows to use scattered data for the reconstruction process, the computation of the approximating function needs the solution of a full N N linear system where N denotes the number of centers, and is intolerably expensive when N is large. Another technique 14] is hyperinterpolation approximation. It uses quadra-ture rules to discretize the L 2-orthogonal projection operator onto the space of spherical harmonics. Here, the advantage is that no linear system has to be solved. On the other hand, this method depends on the chosen quadrature rule, and since it is based on projection onto the space of spherical harmonics, it inherits the problems of projections, i.e., the Lebesgue constants cannot be uniformly bounded. Other local methods are based on spherical splines. These macro-element methods depend on a spherical triangulation (cf. 1]). In this paper we want to introduce and investigate the method of moving least squares approximation on the sphere. It will turn out that the eeort of computing the approximant is bounded by a constant which depends only on the space dimension and the desired approximation order. Furthermore, it can be shown that the involved Lebesgue constants are uniformly bounded. ISBN 0-8265-xxxx-x. All rights of reproduction in any form reserved.