The s-energy of spherical designs on S2

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S-2. A spherical n-design is a point set on S-2 that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree <= n. The s-energy E-s(X) of a point set X = {x(1),..., x(m)} subset of S-2 of m distinct points is the sum of the potential parallel to x(i) - x(j) parallel to(-s) for all pairs of distinct points x(i), x(j) epsilon X. A sequence Xi = {X-m} of point sets X-m subset of S-2, where Xm has the cardinality card (X-m) = m, is well separated if arccos(x(i) center dot x(j)) >= lambda/root m for each pair of distinct points x(i), x(j) epsilon X-m, where the constant lambda is independent of m and X-m. For all s > 0, we derive upper bounds in terms of orders of n and m(n) of the s-energy Es(X-m(n)) for well separated sequences Xi = {X-m(n)} of spherical n-designs X-m(n) with card (X-m(n)) = m(n).