The Coulomb energy of spherical designs on S 2

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 ⊂ R 3 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 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 ∈ the minimum spherical distance between points is bounded from below by λ √ 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 .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.

The s-energy of spherical designs on S 2

“…Theorem 1, Corollary 1, and Theorem 2 extend results from Hesse and Leopardi [8], and contain the results from Hesse and Leopardi [8] as the special case s = 1 (Coulomb energy).…”

“…The construction of a suitable split (3.2) is rather delicate and generalizes ideas from [8, Lemmas 1 to 3, and Section 4]. Once the correct split is chosen, we proceed analogously to Hesse and Leopardi [8] applying techniques which were developed in Hesse and Sloan [9,10].…”

“…Next we write Q ± i, as a Riemann-Stieltjes integral, a trick that was also used in Hesse and Leopardi [8], Hesse and Sloan [9,10] and that is originally due to Reimer [14]. For each i = 1, 2, .…”

“…Spherical n-designs of a fixed strength n can have points arbitrary close together (see [8,Section 5]). Hence, the s-energy of an infinite sequence = {X m(n) } of m(n)-point spherical n-designs X m(n) of increasing strength n can grow arbitrarily quickly and can thus have no asymptotic upper bounds in terms of the strength n and the cardinality m(n), without further assumptions.…”

Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

“…First the tensor product construction of Korevaar and Meyers [17], as well as Bajnok [2], has O t 3 points (compared to (t + 1) 2 points for the current construction) and very bad geometrical properties. Second Hesse and Leopardi [14] have pointed out that any nonoverlapping union of spherical t-designs is also a spherical t-design. This makes it possible to construct a spherical t-design with an arbitrarily small minimum distance between points.…”