Approximation on the sphere using radial basis functions plus polynomials
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Approximation on the sphere using radial basis functions plus polynomials Ian H. Sloan · Alvise Sommariva
Received: 20 January 2006 / Accepted: 20 January 2007 / Published online: 11 May 2007 © Springer Science + Business Media B.V. 2007
Abstract In this paper we analyse a hybrid approximation of functions on the sphere S2 ⊂ R3 by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear system. The analysis is first carried out in the native space associated with the kernel (with no explicit polynomial component, and no side conditions). A more refined error estimate is obtained for functions in a still smaller space. Numerical calculations support the utility of this hybrid approximation. Keywords Scattered data · Radial basis functions · Spherical harmonics · Error estimate Mathematics Subject Classifications (2000) 41A30 · 65D30
1 Introduction In recent decades many authors have investigated the approximation of functions on the sphere S2 ⊂ R3 by means of polynomials or radial basis functions (for example [3, 5, 9, 12, 17, 18]). Often the underlying motivation has been the need to
Communicated by J. Ward. I. H. Sloan (B) · A. Sommariva School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia e-mail: [email protected] A. Sommariva e-mail: [email protected]
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approximate geophysical quantities. It is well understood that polynomial approximants may have a place in the approximation of data that is slowly varying and global in nature. (Indeed, the NASA model EGM 96 [2] takes this to the extreme, by representing the earth’s gravitational potential as a 360-degree spherical polynomial.) On the other hand, for scattered data, and for data that varies rapidly over short distances, radial basis functions come into their own. Our purpose in this paper is to study combined approximation by radial basis functions and polynomials – the radial basis functions to handle scattered data and rapid changes, and the polynomial component to handle the slowly varying large-scale features. Combined polynomial and radial basis function approximations have often been studied in the context of radial basis functions constructed from conditionally positive definite kernels (in which case a polynomial part is needed to make the theory work). Here, however, our point of view is different: we restrict attention to the case of (strictly) positive definite kernels. Thus our inclusion of a polynomial component, like that in [7], is voluntary rather than forced, and is motivated by the view that hybrid approximations of this kind offer real advantages. We first consider the approximation of functions that belong to the native space associated with the positive definite kernel, using the native space introduced by Schaback [13], with no poly
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