Approximation orders for interpolation by surface splines to rough functions

TL;DR

This paper investigates the approximation errors of surface splines when interpolating rough functions lacking smoothness, extending existing error estimates beyond smooth functions within the native space.

Contribution

It provides new error estimates for surface spline interpolation of functions with limited smoothness, broadening the understanding of approximation orders for rough functions.

Findings

Error estimates are extended to rough functions with limited smoothness.

Surface splines can approximate less smooth functions with quantifiable accuracy.

The techniques are demonstrated specifically for surface splines within the radial basis function framework.

Abstract

In this paper we consider the approximation of functions by radial basis function interpolants. There is a plethora of results about the asymptotic behaviour of the error between appropriately smooth functions and their interpolants, as the interpolation points fill out a bounded domain in R^d. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function - the native space. In many cases, the native space contains functions possessing a certain amount of smoothness. We address the question of what can be said about these error estimates when the function being interpolated fails to have the required smoothness. These are the rough functions of the title. We limit our discussion to surface splines, as an exemplar of a wider class of radial basis functions, because we feel our techniques are most easily seen and understood in this setting.