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

 

Recoverability of some classes of analytic functions from their boundary values


Author: Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 87 (1983), 493-498
MSC: Primary 30E25; Secondary 30B30, 46F20
MathSciNet review: 684645
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Abstract: The technique devised by D. J. Patil to recover the functions of the Hardy space $ {H^p}(1 \leqslant p \leqslant \infty )$ from the restrictions of their boundary values to a set of positive measure on the unit circle was modified by S. E. Zarantonello in order to extend the result to $ {H^p}(0 < p < 1)$.

In this paper, we show that Zarantonello's technique can be slightly modified to extend the result to a larger class of analytic functions in the unit disc. In particular, if $ f(z)$ is analytic in the unit disc and satisfies

$\displaystyle \mathop {\lim }\limits_{r \to 1} {(1 - r)^\beta }\log M(r,f) = 0\quad {\text{for}}\;{\text{some}}\;\beta \geqslant 1,$

then $ f(z)$ can be recovered from the restriction of its boundary value to an open arc.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0684645-X
PII: S 0002-9939(1983)0684645-X
Keywords: Recoverability theorem, Beurling distributions, Toeplitz operators
Article copyright: © Copyright 1983 American Mathematical Society