Some issues on interpolation matrices of locally scaled radial basis functions

Radial basis function interpolation on a set of scattered data is constructed from the corresponding translates of a basis function, which is conditionally positive definite of order m ⩾ 0, with the possible addition of a polynomial term. In many applications, the translates of a basis function are scaled differently, in order to match the local features of the data such as the flat region and the data density. Then, a fundamental question is the non-singularity of the perturbed interpolation (N × N) matrix. In this paper, we provide some counter examples of the matrices which become singular for N ⩾ 3, although the matrix is always non-singular when N = 2. One interesting feature is that a perturbed matrix can be singular with rather small perturbation of the scaling parameter.