Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant

TL;DR

This paper proves that on compact Riemannian manifolds, certain kernels can be used for interpolation with uniformly bounded Lebesgue constants, extending Euclidean results to curved surfaces and establishing decay properties of Lagrange functions.

Contribution

It introduces a family of kernels suitable for interpolation on manifolds, demonstrating bounded Lebesgue constants and decay of Lagrange functions, generalizing Euclidean spline theory to curved spaces.

Findings

Lebesgue constants are uniformly bounded on manifolds with quasi-uniform data sites.

Lagrange functions decay exponentially away from their centers.

Bounded Lebesgue constants extend to surface and Sobolev splines on manifolds.

Abstract

The purpose of this paper is to establish that for any compact, connected C^{\infty} Riemannian manifold there exists a robust family of kernels of increasing smoothness that are well suited for interpolation. They generate Lagrange functions that are uniformly bounded and decay away from their center at an exponential rate. An immediate corollary is that the corresponding Lebesgue constant will be 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 analysis needed for these results was inspired by some fundamental work of Matveev where the Sobolev decay of Lagrange functions associated with certain kernels on \Omega \subset R^d was obtained. With a bit more work, one establishes the following: Lebesgue constants associated with surface splines and Sobolev splines are uniformly bounded on R^d provided the data sites \Xi are quasi-uniformly distributed. The non-Euclidean case is more involved as the geometry of the underlying surface comes into play. In addition to establishing bounded Lebesgue constants in this setting, a "zeros lemma" for compact Riemannian manifolds is established.