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

Let $d\ge 2$ 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 $\mid {\int }_{S^d}fd\sigma \mid$, 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.