Dirichlet integral point-source harmonic interpolation over ${\mathbb{R}}^3$ spherical interiors: DIDACKS II

Abstract

This article addresses the interpolation of harmonic functions over the interior of a $\mathbb{R}^3$ unit sphere by linear combinations of fundamental-solution point-source basis functions, where all the sources are assumed to be outside the sphere. Since the source and field points are in different domains, the fundamental-solution basis functions are bounded and can be regarded as defining a new type of kernel space that is related to reproducing kernel Hilbert space (RKHS), but is different from it and which is labeled a Dirichlet integral dual-access collocation-kernel space (DIDACKS). DIDACKS theory has direct implications for the method of fundamental solutions (MFS) and some for the fast multipole method (FMM) and boundary element method (BEM).