Baxter, Brad J.C. (2008) On spherical averages of radial basis functions. Foundations of Computational Mathematics 8 (3), pp. 395407. ISSN 16153375.

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Abstract
A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(xb_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log ‖x‖ when d is an even positive integer, and univariate splines. Now Bsplines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration.
Item Type:  Article 

Additional Information:  The original publication is available at www.springerlink.com 
Keyword(s) / Subject(s):  Radial basis functions, spherical average, Haar measure  Paley–Wiener 
School or Research Centre:  Birkbeck Schools and Research Centres > School of Business, Economics & Informatics > Economics, Mathematics and Statistics 
Depositing User:  Administrator 
Date Deposited:  01 Feb 2011 11:58 
Last Modified:  17 Apr 2013 12:33 
URI:  http://eprints.bbk.ac.uk/id/eprint/1920 
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