A note on radial basis function interpolant limits

Radial basis functions (RBFs) are very useful in multivariate interpolation because of their ability to produce highly accurate results for scattered data. Many of them, especially the Gaussian RBF and the multiquadric RBF, contain parameters that need to be adjusted in order to improve the approximations. In fact, it is often of interest to let the parameters tend to certain limits. Here we study if and when the limits of RBF interpolants with parameters exist. Mainly, the dependence of the limit on the properties of the radial functions and on the geometries of the data points is investigated, and some examples are provided.