Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

TL;DR

This paper develops a unified theory for kernel interpolation and approximation on compact Riemannian manifolds, demonstrating bounded Lebesgue constants and decay properties for kernels derived from differential operators, applicable to spheres and SO(3).

Contribution

It introduces a general framework for kernel-based interpolation on manifolds, including explicit kernels like surface splines, with proven boundedness and decay properties, extending Euclidean results to curved spaces.

Findings

Lebesgue constants are uniformly bounded depending only on mesh ratio.

Kernels exhibit algebraic or exponential decay away from centers.

The theory applies to important manifolds like spheres and SO(3).

Abstract

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and SO(3), we establish, using techniques involving differential geometry and Lie groups, that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for $L_2$ minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a "zeros lemma" for domains on compact Riemannian manifolds -- one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.