Approximation orders for interpolation by surface splines to rough functions

Authors: Rob Brownlee; Will Light

Source: IMA Journal of Numerical Analysis, Volume 24, Number 2, April 2004 , pp. 179-192(14)

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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 R^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 example of a wider class of radial basic functions, because we feel our techniques are most easily seen and understood in this setting.

Keywords: surface splines; interpolation; error estimates; Sobolev space; extension theorems

Document Type: Research article

Affiliations: 1: Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, UK

Publication date: 2004-04-01

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The IMA Journal of Numerical Analysis (IMAJNA) publishes original contributions to all fields of numerical analysis; articles will be accepted which treat the theory, development or use of practical algorithms and interactions between these aspects. Occasional survey articles are also published.