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. 86 (1982), 97-102
MSC: Primary 30D55; Secondary 30B30, 30E25
MathSciNet review: 663875
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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

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

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

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Keywords: Recoverability theorem, Beurling distributions, Toeplitz operators
Article copyright: © Copyright 1982 American Mathematical Society