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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Interpolating sequences on convex curves in the open unit disc


Author: Dennis H. Wortman
Journal: Proc. Amer. Math. Soc. 48 (1975), 157-164
MSC: Primary 30A80; Secondary 46J15
DOI: https://doi.org/10.1090/S0002-9939-1975-0361092-7
MathSciNet review: 0361092
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Abstract: Let $ D$ be the open unit disc in the complex plane, and let $ C$ be the unit circle. Given a convex curve $ \Gamma $ in $ D \cup C$, internally tangent to $ C$ at one point, then a sequence on $ \Gamma $, successive points of which are equally spaced in the hyperbolic (Poincaré) metric, is shown to be interpolating. This result is then applied to the study of the Banach algebra $ {H^\infty }$. The Gleason part of a point in the maximal ideal space of $ {H^\infty }$ which lies in the closure of a convex curve in $ D$ is proved to be nontrivial. In addition, for each point $ m$ in the maximal ideal space of $ {H^\infty }$ which lies in the closure of a compact subset of $ D$ union a point of $ C$, an interpolating Blaschke product is constructed whose extension to the maximal ideal space has modulus less than 1 on $ m$, and the relevance of this to the Shilov boundary of $ {H^\infty }$ is discussed.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0361092-7
Keywords: Interpolating sequence, convex curve, pseudohyperbolic metric, $ {H^\infty }$, maximal ideal space, Gleason part, Shilov boundary
Article copyright: © Copyright 1975 American Mathematical Society