The Penalized Lebesgue Constant for Surface Spline Interpolation

TL;DR

This paper investigates the stability and convergence of surface spline interpolation for scattered data with nonuniform density, demonstrating local stability, rapid decay of Lagrange functions, and comparable convergence rates to local approximation schemes.

Contribution

It provides new stability and decay estimates for surface spline interpolation on nonuniform data, extending understanding beyond quasi-uniform cases.

Findings

Surface spline interpolation is locally stable for nonuniform data.

Lagrange functions decay rapidly at a rate depending on local data spacing.

Interpolation achieves convergence rates similar to local approximation methods.

Abstract

Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity, and is far less well understood. In this article we consider the stability of surface spline interpolation (a popular type of RBF interpolation) for data with nonuniform arrangements. Using techniques similar to those recently employed by Hangelbroek, Narcowich and Ward to demonstrate the stability of interpolation from quasi-uniform data on manifolds, we show that surface spline interpolation on R^d is stable, but in a stronger, local sense. We also obtain pointwise estimates showing that the Lagrange function decays very rapidly, and at a rate determined by the local spacing of datasites. These results, in conjunction with a Lebesgue lemma, show that surface spline interpolation enjoys the same rates of convergence as those of the local approximation schemes recently developed by DeVore and Ron.