Variational splines on Riemannian manifolds with applications to integral geometry

TL;DR

This paper generalizes variational spline theory to compact Riemannian manifolds, providing explicit formulas, proving existence and uniqueness, and demonstrating convergence with applications to integral geometry problems like Radon transforms.

Contribution

It extends classical spline theory to Riemannian manifolds, deriving explicit formulas and establishing convergence and optimality for applications in integral geometry.

Findings

Explicit formulas for variational splines using Laplace-Beltrami eigenfunctions

Proof of existence and uniqueness of splines on manifolds

Convergence of splines to functions in $C^{k}$ norms

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.