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

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.