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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Two counterexamples involving inner functions


Author: Kenneth Stephenson
Journal: Proc. Amer. Math. Soc. 73 (1979), 22-24
MSC: Primary 30D50
MathSciNet review: 512050
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Abstract: Two questions about the existence of bounded analytic functions with prescribed behavior are answered using a construction technique of McLaughlin and Piranian. If A is a relatively closed, countable subset of the unit disc, the construction gives

(I) Function $ f \in {H^\infty }$ so that the inner factor of $ f - \alpha $ is a finite Blaschke product if and only if $ \alpha \in A$.

(II) Inner function $ \phi $ so that $ \phi - \alpha $ has a discrete singular inner factor if and only if $ \alpha \in A$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0512050-4
PII: S 0002-9939(1979)0512050-4
Keywords: Bounded analytic function, inner function, Riemann surface
Article copyright: © Copyright 1979 American Mathematical Society