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A theorem of Lohwater and Piranian


Author: Arthur A. Danielyan
Journal: Proc. Amer. Math. Soc. 144 (2016), 3919-3920
MSC (2010): Primary 30H05, 30H10
Published electronically: March 17, 2016
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Abstract: By a well-known theorem of Lohwater and Piranian, for any set $ E$ on $ \vert z\vert=1$ of type $ F_\sigma $ and of measure zero there exists a bounded analytic function in $ \vert z\vert<1$ which fails to have radial limits exactly at the points of $ E$. We show that this theorem is an immediate corollary of Fatou's interpolation theorem of 1906.


References [Enhancements On Off] (What's this?)

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Additional Information

Arthur A. Danielyan
Affiliation: Department of Mathematics and Statistics, University of South Florida, Tampa, Florida 33620
Email: adaniely@usf.edu

DOI: https://doi.org/10.1090/proc/13083
Keywords: Bounded analytic function, Fatou's theorem, radial limit, $F_\sigma$ set
Received by editor(s): October 4, 2015
Received by editor(s) in revised form: October 20, 2015, and November 11, 2015
Published electronically: March 17, 2016
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2016 American Mathematical Society