Variational splines on Riemannian manifolds with applications to\n integral geometry

2011/04/09 by Isaac Z. Pesenson, Pesenson, Isaac
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1104.1709

openalex publication_date 2011/04/09 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Computing Fourier Transforms and Convolutions on the 2-Sphere

Related

• Variational time discretization of Riemannian splines
• Spline Interpolation on Compact Riemannian Manifolds
• Higher-order Riemannian spline interpolation problems: a unified approach by gradient flows
• Multiresolution analysis on compact Riemannian manifolds
• A note on the regularity and the existence of Riemannian splines