A Koksma-Hlawka-Potential Identity on the $d$ Dimensional Sphere and its Applications to Discrepancy

TL;DR

This paper establishes a new identity on the sphere that provides error estimates for harmonic function integrals with respect to signed measures, aiding in understanding quadrature errors on spherical scatterings.

Contribution

It introduces a Koksma-Hlawka-Potential identity on the sphere and derives bounds for quadrature errors involving harmonic functions and signed measures.

Findings

Provides error bounds involving Newtonian potentials

Enables analysis of quadrature errors on the sphere

Applies to scatterings with specified mesh norm

Abstract

Let $d\geq 2$ be an integer, $S^d\subset {\mathbb R}^{d+1}$ the unit sphere and $\sigma$ a finite signed measure whose positive and negative parts are supported on $S^d$ with finite energy. In this paper, we derive an error estimate for the quantity $\left|\int_{S^d}fd\sigma\right|$, for a class of harmonic functions $f:\mathbb R^{d+1}\to \mathbb R$. Our error estimate involves 2 sided bounds for a Newtonian potential with respect to $\sigma$ away from its support. In particular, our main result allows us to study quadrature errors, for scatterings on the sphere with given mesh norm.