Localization Properties of Rbf Expansion Coefficients for Cardinal Interpolation. I. Equispaced Nodes

Many types of radial basis functions (RBFs) are global, in terms of having large magnitude across the entire domain. Yet, in contrast, for example, 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 centered 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 1-D and 2-D. Furthermore, we explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs.