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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Some results concerning the boundary zero sets of general analytic functions


Author: Robert D. Berman
Journal: Trans. Amer. Math. Soc. 293 (1986), 827-836
MSC: Primary 30D40
MathSciNet review: 816329
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Abstract: Two results concerning the boundary zero sets of analytic functions on the unit disk $ \Delta $ are proved. First we consider nonconstant analytic functions $ f$ on $ \Delta $ for which the radial limit function $ {f^{\ast}}$ is defined at each point of the unit circumference $ C$. We show that a subset $ E$ of $ C$ is the zero set of $ {f^{\ast}}$ for some such function $ f$ if and only if it is a $ {\mathcal{G}_\delta }$ that is not metrically dense in any open arc of $ C$. We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0816329-6
Article copyright: © Copyright 1986 American Mathematical Society