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 So bolev-type estimates for the approximation order resulting from using stric tly 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 approxim ation can be more fully analyzed with rates of approximation given in terms of Sobolev norms. Estimates on the rate of convergence for generalized Her mite and other distributional interpolants can he obtained in certain circu mstances and, finally, the constants appearing in the approximation order i nequalities are explicit. Our focus-in this paper will be on approximation rates in the rases of the circle, other tori, and the 2-sphere.