Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold

The purpose of this paper is to study certain variational principles and Sobolev-type estimates for the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation on a closed (i.e., no boundary), compact, connected, orientable, m-dimensional C-infinity Riemannian manifold-M, with CM metric g(ij). The rate of approximation can be more fully analyzed with rates of approximation given in terms of Sobolev norms. Estimates on the rate of convergence for generalized Hermite and other distributional interpolants can he obtained in certain circumstances and, finally, the constants appearing in the approximation order inequalities are explicit. Our focus-in this paper will be on approximation rates in the rases of the circle, other tori, and the 2-sphere.