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

abstract

• 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 M, with C metric gij. 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.

authors

• Narcowich, Francis
• Ward, Joseph

published proceedings

• CONSTRUCTIVE APPROXIMATION

author list (cited authors)

• Dyn, N., Narcowich, F. J., & Ward, J. D.

citation count

• 43

complete list of authors

• Dyn, N||Narcowich, FJ||Ward, JD

publication date

• January 1999

publisher

• Springer Nature  Publisher

published in

• Constructive Approximation  Journal

keywords

• Approximation Order
• Positive Definite Kernels
• Riemannian Manifolds
• Sobolev Spaces

Digital Object Identifier (DOI)

• 10.1007/s003659900104

start page

• 175

end page

• 208

volume

• 15

issue

• 2

URL

• http%3A%2F%2Fdx.doi.org%2F10.1007%2Fs003659900104