Energy functionals, numerical integration and asymptotic equidistribution on the sphere
- Damelin, Steven B.
- Grabner, Peter J.
DOIAbstract
AbstractIn 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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