The s-energy of spherical designs on S2

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Abstract

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S2. A spherical n-design is a point set on S2 that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy Es(X) of a point set X = {x1,…,xm} ⊂ S2 of m distinct points is the sum of the potential |xi−xj|−s for all pairs of distinct points xi, xj ∈ X. A sequence Ξ = {Xm} of point sets Xm ⊂ S2, where Xm has the cardinality card(Xm) = m, is well separated if arccos(xi.xj) ≥ λ/√m for each pair of distinct points xi, xj ∈ Xm, where the constant λ is independent of m and Xm. For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy Es(Xm(n)) for well separated sequences Ξ = {Xm(n)} of spherical n-designs Xm(n) with card(Xm(n))=m(n).

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