Order of approximation by electrostatic fields due to electrons

Abstract

LetD be a Jordan domain in the complex plane andA q (D) the Bers space with norm ∥ ∥ q . IfD is the unit disk, it is known that ∥S n 0∥2≥π/18, whereS n =∑ k=1 n l/(z−z nk ) withz nk ∈∂D, so that approximation in ∥ ∥ q ,q<-2, is not possible. In this paper, we give an order of estimate of ∥f−S n ∥ q for 2<q<∞ when ∂D is a sufficiently smooth Jordan curve, and prove that this order of approximation is in general best possible.