Approximation orders for interpolation by surface splines to rough functions

Abstract

In this paper we consider the approximation of functions by radial basic 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 IR d . In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basic 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 basic functions, because we feel our techniques are most easily seen and understood in this setting.

Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

“…In the special case of interpolation by means of an integer order thin-plate spline, Brownlee and Light [2] have obtained L p error estimates in terms of |f | W k 2 (Ω) . We will treat the general RBF case here, but we will need to work in the space C k (Ω), rather than W …”

Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants

“…Native spaces are usually comprised of functions which are very smooth, so such error estimates are limited. Our main goal here is to show that when the Fourier transform of φ has algebraic decay, i.e., (2) c 1 …”

Extending the Range of Error Estimates for Radial Approximation in Euclidean Space and on Spheres

“…In recent years, contributions in that direction has been provided by several authors, e.g., [16,17,12] for the sphere and [23,3,18] for the Euclidean case.…”