Advances in Computational Mathematics

, Volume 28, Issue 4, pp 331–354

The Coulomb energy of spherical designs on S2

Article

DOI: 10.1007/s10444-007-9026-7

Cite this article as:
Hesse, K. & Leopardi, P. Adv Comput Math (2008) 28: 331. doi:10.1007/s10444-007-9026-7

Abstract

In this work we give upper bounds for the Coulomb energy of a sequence of well separated spherical n-designs, where a spherical n-design is a set of m points on the unit sphere S2 ⊂ ℝ3 that gives an equal weight cubature rule (or equal weight numerical integration rule) on S2 which is exact for spherical polynomials of degree ⩽ n. (A sequence Ξ of m-point spherical n-designs X on S2 is said to be well separated if there exists a constant λ > 0 such that for each m-point spherical n-design X ∈ Ξ the minimum spherical distance between points is bounded from below by \(\frac{\lambda }{{{\sqrt m }}}\).) In particular, if the sequence of well separated spherical designs is such that m and n are related by m = O(n2), then the Coulomb energy of each m-point spherical n-design has an upper bound with the same first term and a second term of the same order as the bounds for the minimum energy of point sets on S2.

Keywords

acceleration of convergence Coulomb energy Coulomb potential equal weight cubature equal weight numerical integration orthogonal polynomials sphere spherical designs well separated point sets on sphere 

Mathematics Subject Classifications (2000)

Primary: 31C20 42C10 Secondary: 41A55 42C20 65B10 65D32 

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia

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