A radial basis function approximation is typically a linear combination of shifts of a radially symmetric function, possibly augmented by a polynomial of suitable degree, that is, it takes the form s(~) = ~ ckΒ’(ll~ -~kll) +p(~), ~ e R d. k=l In the mid 1980s, Micchelli, building on pioneering work of Schoenberg in the 1930s and 1940s, provided simple sufficient conditions on Β’ that imply radial basis functions can interpolate scattered data. However, when the data density varies locally, several authors, such as Hon and Kansa [1], have suggested scaling the translates. In other words, it can be advantageous to replace the Euclidean norm by stone more general distance functional A(..), that is s(~) = 9 ekΒ’(~(x,~k)) +p(~), x e a a.
k=l This distance functional A need not be a metric, but we shall require that A be symmetric and satisfy A(x, x) = 0, for all x C ]~d. Unfortunately, the Micchelll-Schoenberg theory does not obviously apply in this more general setting, but some papers have observed that interpolation is well defined if the distance functional is a sufficiently small perturbation of the Euclidean norm. However, in this study we follow a different approach which returns to the roots of Schoenberg's work. Specifically, we use Schoenberg's classification of Euclidean distance matrices to provide a simple technique which, given a suggested distance functional A, calculates a perturbed distance functional /k for which the underlying interpolation matrix is invertible, when the function Β’ is strictly positive definite (i.e., a Mercer kernel) or strictly conditionally positive (or negative) definite of order one. As a simple by-product of this method, we can also apply the Narcowich-Ward [2] norm estimate results easily, since the minimum distance between points is now under our control via /k. @