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 ${\mathbb{R}}^d$.