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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A theorem of Lohwater and Piranian
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by Arthur A. Danielyan PDF
Proc. Amer. Math. Soc. 144 (2016), 3919-3920 Request permission

Abstract:

By a well-known theorem of Lohwater and Piranian, for any set $E$ on $|z|=1$ of type $F_\sigma$ and of measure zero there exists a bounded analytic function in $|z|<1$ 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
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  • Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
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  • Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
  • A. J. Lohwater and G. Piranian, The boundary behavior of functions analytic in a disk, Ann. Acad. Sci. Fenn. Ser. A. I. 1957 (1957), no. 239, 17. MR 91342
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Additional Information
  • Arthur A. Danielyan
  • Affiliation: Department of Mathematics and Statistics, University of South Florida, Tampa, Florida 33620
  • MR Author ID: 241417
  • Email: adaniely@usf.edu
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 3919-3920
  • MSC (2010): Primary 30H05, 30H10
  • DOI: https://doi.org/10.1090/proc/13083
  • MathSciNet review: 3513548