Variational splines on Riemannian manifolds with applications to integral geometry

Abstract

We extend the classical theory of variational interpolating splines to the case of compact Riemannian manifolds. Our consideration includes in particular such problems as interpolation of a function by its values on a discrete set of points and interpolation by values of integrals over a family of submanifolds. The existence and uniqueness of interpolating variational spline on a Riemannian manifold is proven. Optimal properties of such splines are shown. The explicit formulas of variational splines in terms of the eigen functions of Laplace-Beltrami operator are found. It is also shown that in the case of interpolation on discrete sets of points variational splines converge to a function in $C^k$ norms on manifolds. Applications of these results to the hemispherical and Radon transforms on the unit sphere are given.

Adv. in Appl. Math. 33 (2004), no. 3, 548-572