Interpolation by basis functions of different scales and shapes

Abstract

Under very mild additional assumptions, translates of conditionally positive definite radial basis functions allow unique interpolation to scattered multivariate data, because the interpolation matrices have a symmetric and positive definite dominant part. In many applications the data density varies locally according to the signal behaviour, and then the translates should get different scalings that match the local data density. Furthermore, if there is a local anisotropy in the data, the radial basis functions should possibly be distorted into functions with ellipsoids as level sets. In such cases, the symmetry and the definiteness of the matrices are no longer guaranteed. However, this brief note is the first paper to provide sufficient conditions for the unique solvability of such interpolation processes. The basic technique is a simple matrix perturbation argument combined with the Ball-Narcowich-Ward stability results.