Some observations regarding interpolants in the limit of flat radial basis functions

Radial basis functions (RBFs) form a primary tool for multivariate interpolation. Some of the most commonly used radial functions feature a shape parameter, allowing them to vary from being nearly flat ( small) to sharply peaked ( large). The former limit can be particularly accurate when interpolating a smooth function based on scattered data. This study discusses theoretical and computational aspects of the → 0 limit, and includes the conjecture that Gaussian RBF interpolants will never diverge in this limit.