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A sufficient condition for Nevanlinna parametrization and an extension of Heins theorem

Published online by Cambridge University Press:  22 January 2016

Sechiko Takahashi
Sechiko Takahashi*
Affiliation:
Department of Mathematics, Nara Women’s University, Nara 630-8506, Japan, [email protected]
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Abstract

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An extended interpolation problem on a Riemann surface is formulated in terms of local rings and ideals. A sufficient condition for Nevanlinna parametrization is obtained. By means of this, Heins theorem on Pick-Nevanlinna interpolation in doubly connected domains is generalized to extended interpolation.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 153 , 1999 , pp. 87 - 100
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

References

[1] Abrahamse, M. B., The Pick interpolation theorem for finitely connected domains, Michigan Math. J., 26 (1979), 195–203.Google Scholar
[2] Ahlfors, L. V., Conformal Invariants. Topics in Geometric Function Theory, McGraw-Hill, New York, 1973.Google Scholar
[3] Fisher, S. D., Function Theory on Planer Domains, Wiley, New York, 1983.Google Scholar
[4] Garnett, J. B., Bounded Analytic Functions, Academic Press, New York, 1981.Google Scholar
[5] Heins, M. H., Extremal problems for functions analytic and single-valued in a dou bly-connected region, Amer. J. Math., 62 (1940), 91–106.CrossRefGoogle Scholar
[6] Heins, M. H., Nonpersistence of the grenzkreis phenomenon for Pick-Nevanlinna interpo lation on annuli, Ann. Acad. Sci. Fenn. A. I., 596 (1975).Google Scholar
[7] Marshall, D. E., An elementary proof of Pick-Nevanlinna interpolation theorem, Michigan Math. J., 21 (1974), 219–223.Google Scholar
[8] Nevanlinna, R., Über beschränkte Funktionen die in gegebenen Punkten vor-geschriebene Werte annehmen, Ann. Acad. Sci. Fenn., 13, No 1 (1919).Google Scholar
[9] Nevanlinna, R., Über beschränkte analytische Funktionen, Ann. Acad. Sci. Fenn. Ser A, 32, No 7 (1929).Google Scholar
[10] Pick, G., Über die Beschränkungen analytischer Funktionen, welche durch vor-gegebene Funktionswerte bewirkt werden, Math. Ann., 77 (1916), 7–23.Google Scholar
[11] Schur, I., Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J.Reine Angew. Math., 147 (1917), 205–232.Google Scholar
[12] Stray, A., Minimal interpolation by Blaschke products II, Bull. London Math. Soc, 20 (1988), 329–332.Google Scholar
[13] Takahashi, S., Extension of the theorems of Carathéodory-Toeplitz-Schur and Pick, Pacific J. Math., 138 (1989), 391–399.Google Scholar
[14] Takahashi, S., Nevanlinna parametrizations for the extended interpolation problem, Pacific J. Math., 146 (1990), 115–129.CrossRefGoogle Scholar
[15] Takahashi, S., Extended interpolation problem in finitely connected domains, Oper. Theory Adv. Appl., 59 (1992), 305–327.Google Scholar