Radial Function Methods of Approximation Based on Using Harmonic Green's Functions

In this paper we present an explicit method of radial basis function approximation over Rn, using the Green's function for Laplace's equation. We prove convergence of the scheme for all functions that are continuous and of compact support. Interesting variants of formul result, in cases when lower dimensional formul are used to construct higher dimensional ones, and in cases of periodic functions. Various explicit operations are possible on the derived formulae, such as obtaining Fourier and Hilbert transforms.