A theorem of Lohwater and Piranian
HTML articles powered by AMS MathViewer
- 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
- E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- S. V. Kolesnikov, On the sets of nonexistence of radial limits of bounded analytic functions, Mat. Sb. 185 (1994), no. 4, 91–100 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 81 (1995), no. 2, 477–485. MR 1272188, DOI 10.1070/SM1995v081n02ABEH003547
- 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
- G. Piranian, Review of [3], MR1272188 (95g:30043).
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