Spectral Analysis for Radial Basis Function Collocation Matrices

The aim of this paper is to provide tools and results for the analysis of the linear systems arising from radial basis function (RBF) approximations of partial differential equations (PDEs), see e.g., [1,9]. Informally, a radial function \(\phi (x) : \mathbb{R}^n \rightarrow \mathbb{R} \) is a function of the Euclidean norm \(\|x\|\) of x, i.e., \(\phi (x) = \eta (\| x\|) \), for \( \eta (t) : \mathbb{R}^n \rightarrow \mathbb{R}\)