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 ^∞ Riemannian manifold , with C ^∞ 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 be 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 cases of the circle, other tori, and the 2 -sphere. April 10, 1996. Dates revised: March 26, 1997; August 26, 1997. Date accepted: September 12, 1997. Communicated by Ronald A. DeVore.