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 Sd ⊂ Rd+1 by equally weighted quadrature rules based at N≥2 points on Sd 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||, s = 0. We deduce that point configurations which are extremal for the Riesz energy are asymptotically equidistributed on Sd for 0 ≤s≤d as N → ∞ and we present explicit rates of convergence for the special case s = d, which had been open.

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