On Fatou’s theorem

Let E be a set on the unit circle T of C . We prove that there exists an f ∈ H ∞ which has no radial limits on E but has unrestricted limit at each point of T \ E if and only if E is an F σ of measure zero. The necessity of the condition that E is an F σ is almost obvious and the necessity of the co...

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Bibliographic Details
Published in:Analysis and mathematical physics 2020-09, Vol.10 (3), Article 28
Main Author: Danielyan, Arthur A.
Format: Article
Language:English
Subjects:
Analysis
H infinity
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Perspectives in Modern Analysis dedicated to Dov Aharonov
Samuel Krushkal
Simeon Reich and Lawrence Zalcman
Theorems
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Summary:Let E be a set on the unit circle T of C . We prove that there exists an f ∈ H ∞ which has no radial limits on E but has unrestricted limit at each point of T \ E if and only if E is an F σ of measure zero. The necessity of the condition that E is an F σ is almost obvious and the necessity of the condition that E is of measure zero follows from Fatou’s theorem.
ISSN:1664-2368
1664-235X
DOI:10.1007/s13324-020-00368-1