ENERGY ESTIMATES AND THE WEYL CRITERION ON COMPACT HOMOGENEOUS MANIFOLDS By S.B. Damelin

The purpose of this paper is to demonstrate how many results concerning approximation, integration, and density on the sphere can be generalised to a much wider range of manifolds M , namely the compact homogeneous mani-folds. The essential ingredient is that invariant kernels (the generalisation of zonal or radial kernels) have a spectral decomposition in terms of projection kernels onto invariant polynomial subspaces. In particular, we establish a Weyl criterion for such manifolds M and announce an energy result which generalizes work of Damelin and Grabner.