Interpolating sequences on convex curves in the open unit disc
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- by Dennis H. Wortman PDF
- Proc. Amer. Math. Soc. 48 (1975), 157-164 Request permission
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.References
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B. Cole, One point parts and the peak point conjecture, Ph.D. Dissertation, Yale University, New Haven, Conn., 1968.
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111. MR 215102, DOI 10.2307/1970361
- Max L. Weiss, Note on a theorem of Beuring, Proc. Amer. Math. Soc. 19 (1968), 793–795. MR 227422, DOI 10.1090/S0002-9939-1968-0227422-8
Additional Information
- © Copyright 1975 American Mathematical Society
- 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