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

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