Energy functionals, numerical integration and asymptotic equidistribution on the sphere

In this paper, we study the numerical integration of continuous functions on d-dimensional spheres S ⊆ R by equally weighted quadrature rules based at N ≥ 1 points on S which minimize a generalized energy functional. Examples of such points are configurations, which minimize energies for the Riesz kernel ‖x− y‖−s 0 < s ≤ d and logarithmic kernel − log ‖x− y‖. We deduce that extremal point configurations are asymptotically equidistributed on S as N → ∞ and we present explicit rates of convergence for the special case s = d.