Invariance Principles for Energy Functionals on Spheres

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<sd or a logarithmic kernel g^L₀ in the limits when 0⁺.The author is supported by the START project Y96-MAT of the Austrian Science Fund.