Closed form representations for a class of compactly supported radial basis functions

Authors:
Simon Hubbert
Affiliations:
School of Economics, Mathematics and Statistics, Birkbeck College, London, England WC1E 7HX
Venue:
Advances in Computational Mathematics
Year:
2012

Citing 2
Cited 2

Operators on radial functions

Journal of Computational and Applied Mathematics - Special issue on scattered data
The missing Wendland functions

Advances in Computational Mathematics

Full length article: Closed form representations and properties of the generalised Wendland functions

Journal of Approximation Theory
Wendland functions with increasing smoothness converge to a Gaussian

Advances in Computational Mathematics

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Abstract

In this paper we re-examine Wendland's strategy for the construction of compactly supported positive definite radial basis functions. We acknowledge that this strategy can be modified to capture a much larger range of functions, including the so-called missing Wendland functions which have been the subject of a recent paper by Schaback (Adv Comput Math 34:67---81, 2011). Our approach is to focus on a general integral representation of such functions and we will show how a careful evaluation of this integral leads to new closed form expressions for both Wendland's original functions and the missing ones. The resulting expressions are easy to code and so provide the potential user with a quick way of accessing a desired example for a given application.