The Coulomb energy of spherical designs on S 2
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The Coulomb energy of spherical designs on S2 Kerstin Hesse · Paul Leopardi
Received: 12 July 2005 / Accepted: 24 July 2006 / Published online: 17 February 2007 © Springer Science + Business Media B.V. 2007
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 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 ∈ the minimum spherical distance between points is bounded from below by √λm .) 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
Communicated by: Tomas Sauer. Dedicated to Edward B. Saff on the occasion of his 60th birthday. K. Hesse (B) · P. Leopardi School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia e-mail: [email protected] P. Leopardi e-mail: [email protected]
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K. Hesse, P. Leopardi
1 Introduction This paper examines the (discrete) Coulomb (or r−1 ) energy of spherical designs on the unit sphere S2 ⊂ R3 . The Coulomb energy of a set of m distinct points (or an m-point set) X := {x1 , . . . , xm } on S2 is defined as E(X) :=
m m −1 1 xi − x j , 2 i=1 j=1, j=i
where |x| denotes the Euclidean norm (in R3 ) of the vector x. There have been many equivalent definitions of spherical designs since the original one by Delsarte, Goethals, and Seidel [7]. The following is the most convenient one for our purposes. Definition 1 For n ∈ N0 , a spherical n-design is a finite set of points X ⊂ S2 , such that the corresponding equal weight cubature rule Q X is exact for Pn (S2 ), the space of all spherical polynomials of degree up to and including n; that is, X ⊂ S2 is a spherical n-design if it satisfies Q X ( p) =
p(x) dω(x) S2
for all p ∈ Pn (S2 ),
where Q X ( f ) :=
4π f (x), |X|
f ∈ C(S2 ),
(1)
x∈X
and where ω is the Lebesgue surface measure on S2 , and where |X| denotes the cardinality of X. Noting that a spherical n-design is also a spherical k-design for all k ∈ N0 with k < n, we say that n is the strength of X if X is a spherical n-design but not a spherical (n + 1)-design. By a spherical design we mean a spherical n-design without specify
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