Scaling radial basis functions via euclidean distance matrices

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 forms(x)=∑k=1nckφ(∥x−xk∥)+p(x),x∈ℝdIn the mid 1980s, Micchelli, building on pioneering work of Schoenberg in the 1930s and 1940s, provided simple sufficient conditions on ƒ 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 some more general distance functional Δ(·,·), ), that iss(x)=∑k=1nckφ(Δ(x,xk))+p(x),x∈ℝdThis distance functional A need not be a metric, but we shall require that Δ be symmetric and satisfy Δ (χ, χ) = 0, for all χ ∈ ℝd. Unfortunately, the Micchelli-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 Δ, calculates a perturbed distance functional Δ for which the underlying interpolation matrix is invertible, when the function θ 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 Δ.