Interpolation by Basis Functions of Different Scales and Shapes

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, 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 be distorted into functions with ellipsoids as level sets. In such cases, the symmetry and the definiteness of the matrices are lost. However, this paper provides sufficient conditions for the unique solvability of such interpolation processes. The basic technique is a matrix perturbation argument combined with the Ball–Narcowich–Ward stability results.