Error Bounds for Solving Pseudodiierential Equations on Spheres by Collocation with Zonal Kernels

The problem of solving pseudodiierential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodiierential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation. Data tting and solving diierential and integral equations on the sphere are areas of growing interest with applications to physical geodesy, potential theory , oceanography, and meteorology 6,10]. As more and more satellites are being launched into space, the acquisition of global data is becoming more important and more widespread, and the demand for spherical data processing and solving problems of a global nature is increasing. In this paper we investigate the solution of pseudodiierential equations on spheres by collocation at scattered data points with zonal kernels. Denoting