GENERALIZED HERMITE INTERPOLATION AND POSITIVE-DEFINITE KERNELS ON A RIEMANNIAN MANIFOLD Academic Article uri icon

abstract

  • In this paper we introduce a class of positive definite kernels defined on a closed, compact, Riemannian manifold. These kernels provide a grid-free method for solving uniquely a generalized version of the Hermite interpolation problem, a version in which one fits a smooth surface to multi-dimensional scattered data-including data generated by derivatives, fluxes, or any other quantity one can obtain by integrating a function against a compactly supported distribution. The positive definite kernels introduced here include the C members of the class of spherical positive definite functions introduced by Schoenberg [Duke Math, J. 9 (1942), 96-108] and shown to be strictly positive definite by Xu and Cheney [Proc. Amer. Math, Sec.116 (1992), 977-981]. Thus, as a consequence of our results, the generalized Hermite interpolation problem is well-poised-i.e., a solution exists and is unique-for the class of interpolants constructed out of C spherical positive definite functions. We also provide similar results when the underlying manifold is the m-dimensional torus. 1995 Academic Press. All rights reserved.

published proceedings

  • JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

author list (cited authors)

  • NARCOWICH, F. J.

citation count

  • 45

complete list of authors

  • NARCOWICH, FJ

publication date

  • January 1995