The s-energy of spherical designs on S ²

Abstract

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 $X=\{\mathbf{x}_1,\ldots,\mathbf{x}_m\}\subset S^2$ of m distinct points is the sum of the potential $\|\mathbf{x}_i-\mathbf{x}_j\|^{-s}$ for all pairs of distinct points $\mathbf{x}_i,\mathbf{x}_j\in X$. A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if $\arccos(\mathbf{x}_i\cdot\mathbf{x}_j)\geq\lambda/\sqrt{m}$ for each pair of distinct points $\mathbf{x}_i,\mathbf{x}_j\in X_m$, where the constant λ 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 E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).