Summary.
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlfors, L.V.: Conformal Invariants. McGraw-Hill, 1973
Andrievskii, V.V., Blatt, H.-P.: Discrepancy of Signed Measures and Polynomial Approximation. Springer-Verlag, 2002
Bronstein, I.N., Semendjajew, K.A., Musiol, G., Mühlig, H.: Taschenbuch der Mathematik. Verlag Harri Deutsch, 4th edition, 1999
Frank, P., von Mises, R.: Die Differential- und Integralgleichungen der Mechanik und Physik. Volume I. Vieweg, 2nd edition, 1961
Gaier, D.: Konstruktive Methoden der konformen Abbildung. Springer Verlag, 1964
Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Academic Press, 1981
Günter, N.M.: Die Potentialtheorie und ihre Anwendung auf Grundaufgaben der mathematischen Physik. B.G. Teubner Verlagsgesellschaft, 1957
Koch, T., Liesen, J.: The conformal ‘bratwurst’ maps and associated Faber polynomials. Numerische Mathematik 86, 173–191 (2000)
Kühnau, R.: Geometrie der konformen Abbildung auf der hyperbolischen und der elliptischen Ebene. Deutscher Verlag der Wissenschaften, 1972
Kühnau, R.: Koeffizientenbedingungen für schlicht abbildende Laurentsche Reihen. Bull. Acad. Polon. Sciences, Sér. math., astr., phys. 20, 7–10 (1972)
Menke, K.: On the distribution of Tsuji points. Mathematische Zeitschrift 190, 439–446 (1985)
Menke, K.: Tsuji points and conformal mapping. Annales Polonici Mathematici XLVI, 183–187 (1985)
Menke, K.: Distortion theorems for functions schlicht in an annulus. Journal für die reine und angewandte Mathematik 375/376, 346–361 (1987)
Menke, K.: Point systems with extremal properties and conformal mapping. Numerische Mathematik 54, 125–143 (1988)
Pólya, G., Szegő, G.: Aufgaben und Lehrsätze aus der Analysis I. Springer Verlag, 4th edition, 1971
Pommerenke, C.: Univalent Functions. Vandenhoek & Ruprecht, 1975
Saff, E.B., Totik, V.: Logarithmic Potentials with external Fields. Springer-Verlag, 1997
Stiemer, M.: Effective discretization of the energy integral and Grunsky coefficients in annuli. To appear in Constructive Approximation, 2004
Tsuji, M.: Potential Theory in Modern Function Theory. Maruzen, 1959
Author information
Authors and Affiliations
Additional information
Mathematics Subject Classification (1991): 30C85, 30E10, 31C20
The notation Menke points has been introduced by D. Gaier.
Rights and permissions
About this article
Cite this article
Stiemer, M. On the approximation order of extremal point methods for hyperbolic minimal energy problems. Numer. Math. 99, 533–555 (2005). https://doi.org/10.1007/s00211-004-0565-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-004-0565-2