Weighted polynomials, radial basis functions and potentials on locally compact spaces

Abstract

We discuss the classical notions of capacity, transfinite diameter, Chebyshev constant and equilibrium distributions in the context of locally compact spaces. We also demonstrate how the analogues of certain theorems in classical potential theory can be extended to this framework and applied to various problems in approximation theory. In particular, we explore the similarities between the theory of weighted polynomials and the theory of radial basis functions from a potential theoretic perspective.