The Theory of Radial Basis Function Approximation in 1990

Abstract

Throughout this work {f(x) : x ∈ ℝ} is a real valued function of d variables that is to be approximated by {s(x): x ∈ ℝ}. The radial basis function method gives many useful choices of s that have several advantages over piecewise polynomials. This approach to multivariable approximation is relatively new, however, so many of the most exciting properties have been proved to hold only in special cases. A coherent account of the known theory is presented. It includes some stunning results on orders of convergence and guaranteed nonsingularity of interpolation matrices. Therefore the author believes that radial basis function methods are becoming as important as piecewise polynomials although good software is not yet available for general computer calculations. We shall enjoy the theoretical discoveries that have been made so far and we will note some interesting conjectures. Earlier reviews of theory are presented by Dyn [12,13]. Many successful applications of radial basis functions are mentioned in ·a recent paper that provides an excellent survey of the practical side of the subject (Hardy [22]).