Highly localized kernels on the sphere induced by Newtonian kernels

Abstract

The purpose of this article is to construct highly localized summability kernels on the unit sphere in ${\mathbb R}^d$ that are restrictions to the sphere of linear combinations of a small number of shifts of the fundamental solution of the Laplace equation (Newtonian kernel) with poles outside the unit ball in ${\mathbb R}^d$. The same problem is also solved for the subspace ${\mathbb R}^{d-1}$ in ${\mathbb R}^d$.