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The Coulomb energy of spherical designs on S 2

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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 S 2 ⊂ ℝ3 that gives an equal weight cubature rule (or equal weight numerical integration rule) on S 2 which is exact for spherical polynomials of degree ⩽ n. (A sequence Ξ of m-point spherical n-designs X on S 2 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(n 2), 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 S 2.

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Authors and Affiliations

  1. School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW, 2052, Australia

    Kerstin Hesse & Paul Leopardi

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  1. Kerstin Hesse
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  2. Paul Leopardi
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Correspondence to Kerstin Hesse.

Additional information

Communicated by: Tomas Sauer.

Dedicated to Edward B. Saff on the occasion of his 60th birthday.

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Hesse, K., Leopardi, P. The Coulomb energy of spherical designs on S 2 . Adv Comput Math 28, 331–354 (2008). https://doi.org/10.1007/s10444-007-9026-7

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