Locality properties of radial basis function expansion coefficients for equispaced interpolation

摘要

Many types of radial basis functions (RBFs) are global in terms of having large magnitude across the entire domain. Yet, in contrast, e.g. with expansions in orthogonal polynomials, RBF expansions exhibit a strong property of locality with regard to their coefficients. That is, changing a single data value mainly affects the coefficients of the RBFs which are centred in the immediate vicinity of that data location. This locality feature can be advantageous in the development of fast and well-conditioned iterative RBF algorithms. With this motivation, we employ here both analytical and numerical techniques to derive the decay rates of the expansion coefficients for cardinal data, in both 1D and 2D. Furthermore, we explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs.