Energy functionals, numerical integration and asymptotic equidistribution on the sphere

Authors:
Steven B. Damelin;Peter J. Grabner
Affiliations:
Department of Mathematics and Computer Science, Georgia Southern University, P.O. Box 8093, Statesboro, GA;Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
Venue:
Journal of Complexity
Year:
2003

Citing 3
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Abstract

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, 0s≤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.