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 be the open unit disc in the complex plane, and let be the unit circle. Given a convex curve in , internally tangent to at one point, then a sequence on , 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 . The Gleason part of a point in the maximal ideal space of which lies in the closure of a convex curve in is proved to be nontrivial. In addition, for each point in the maximal ideal space of which lies in the closure of a compact subset of union a point of , an interpolating Blaschke product is constructed whose extension to the maximal ideal space has modulus less than 1 on , and the relevance of this to the Shilov boundary of is discussed.

**[1]**B. Cole,*One point parts and the peak point conjecture*, Ph.D. Dissertation, Yale University, New Haven, Conn., 1968.**[2]**K. Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Ser. in Modern Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR**24**#A2844. MR**0133008 (24:A2844)****[3]**-,*Bounded analytic functions and Gleason parts*, Ann. of Math. (2)**86**(1967), 74-111. MR**35**#5945. MR**0215102 (35:5945)****[4]**M. L. Weiss,*Note on a theorem of Beurling*, Proc. Amer. Math. Soc.**19**(1968), 793-795. MR**37**#3006. MR**0227422 (37:3006)**

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0361092-7

Keywords:
Interpolating sequence,
convex curve,
pseudohyperbolic metric,
,
maximal ideal space,
Gleason part,
Shilov boundary

Article copyright:
© Copyright 1975
American Mathematical Society