Generalized Hermite Interpolation and Positive Definite Kernels on a Riemannian Manifold
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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.
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