RADIAL FUNCTION METHODS OF APPROXIMATION BASED ON USING HARMONIC GREEN’S FUNCTIONS

Frank Stenger; Elaine Cohen; Richard Riesenfeld

Title	RADIAL FUNCTION METHODS OF APPROXIMATION BASED ON USING HARMONIC GREEN’S FUNCTIONS
Publication Type	Journal Article
Year of Publication	2002
Authors	Stenger, F, Cohen, E, Riesenfeld, R
Volume	6
Issue	1
Start Page	1
Pagination	15
Date Published	2002
ISSN	1083-2564
AMS	41A30
Abstract	In this paper we present an explicit method of radial basis function approximation over ${ \mathbb{R}^n }$, using the Green’s function for Laplace’s equation. We prove convergence of the scheme for all functions that are continuous and of compact support. Interesting variants of formulae result, in cases when lower dimensional formulae are used to construct higher dimensional ones, and in cases of periodic functions. Various explicit operations are possible on the derived formulae, such as obtaining Fourier and Hilbert transforms.
URL	http://www.acadsol.eu/en/articles/6/1/1.pdf
Refereed Designation	Refereed
Full Text	REFERENCES [1] P.J. Davis, Interpolation and Approximation, Blaisdell, Waltham, MA, 1963. [2] N. Dyn, Interpolation and approximation by radial basis functions in approximation theory, VI: Volume 1, Eds. C.K. Chui, L.L. Shumaker, and J.D. Ward, Academic Press (New York), (1989) 211–234. [3] R. Franke, Scattered data interpolation: tests of some methods, Math. Comp., 38 (1982), 181–200. [4] C.A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), 11–22. [5] M.J.D. Powell, The theory of radial basis function approximation in 1990, Advances in Numerical Analysis, Vol II”, edited by Will Wright, Oxford (1992), 105-210. [6] F. Stenger, Approximate methods in control theory, Computational Control, eds. K. Bowers and J. Lund, Birkh¨auser (1989), 229-316. [7] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer–Verlag, 1993. [8] F. Stenger, Kronecker product extension of linear operators, SIAM J. Numer. Anal., 5 (1968), 422-435. [9] O. Szasz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. NBS, 45 (1950), 239–245.