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

Summary. Let be an analytic Jordan curve in the unit disk We regard the hyperbolic minimal energy problem where () denotes the set of all probability measures on . There exist several extremal point discretizations of *, 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 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 * providing a geometrically shrinking error bound is presented.Mathematics Subject Classification (1991): 30C85, 30E10, 31C20The notation Menke points has been introduced by D. Gaier.