Invariance Principles for Energy Functionals on Spheres

Abstract.

We study a generalised version of the g-energy functionals defined by Damelin and Grabner. We comment on invariance principles for finite energies and use these principles to obtain expansions of these latter energies in terms of cap discrepancies for a subclass of g. This allows for discrepancy estimates knowing bounds on the energy and vice versa. We are, in particular, able to carefully analyse the case when g gives a Riesz kernel g ^R _s when 0<s≤d or a logarithmic kernel g ^L ₀ in the limits when δ→0⁺.