Highly localized kernels on the sphere induced by Newtonian kernels

TL;DR

This paper constructs highly localized kernels on the sphere by using linear combinations of shifted Newtonian kernels, aiming for improved localization properties for applications in spherical analysis.

Contribution

It introduces a method to create localized kernels on the sphere from Newtonian kernels, extending the approach to subspaces within the ambient space.

Findings

Successfully constructed localized kernels on the sphere

Extended the construction to subspaces of the ambient space

Provided theoretical analysis of the kernels' properties

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 ${\mathbb R}^d$.