On the approximation order of extremal point methods for hyperbolic minimal energy problems

Abstract

Let Γ be an analytic Jordan curve in the unit disk **. We regard the hyperbolic minimal energy problem ** where ** (Γ) denotes the set of all probability measures on Γ. There exist several extremal point discretizations of μ*, among others introduced by M. Tsuji (Tsuji points) or by K. Menke (hyperbolic Menke points). In the present article, it is proven that hyperbolic Menke points approach the images of roots of unity under a conformal map from ** onto Ω geometrically fast if the number of points tends to infinity. This establishes a conjecture of K. Menke. In particular, explicit bounds for the approximation error are given. Finally, an effective method for the numerical determination of μ* providing a geometrically shrinking error bound is presented.