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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
DOI: https://doi.org/10.1090/S0002-9939-1982-0663875-6
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.

References [Enhancements On Off] (What's this?)

  • [1] D. Patil, Representation of $ {H^p}$-functions, Bull. Amer. Math. Soc. 4 (1972), 617-620. MR 0298017 (45:7069)
  • [2] W. Rudin, Division in algebras of infinitely differentiable functions, J. Math. Mech. 11 (1962), 797-810. MR 0153796 (27:3757)
  • [3] G. Walker, Analytic representations, values, and recoverability of distributions, Ph.D. thesis, University of Wisconsin, Milwaukee, 1974.
  • [4] S. Zarantonello, A representation of $ {H^p}$ functions with $ 0 < p < \infty $, Pacific J. Math. 79 ( 1978), 271-282. MR 526683 (80g:30023)
  • [5] A. Zayed, Topological vector spaces of analytic functions, submitted.
  • [6] -, On Beurling distributions and holomorphic functions in the unit disc, submitted.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0663875-6
Keywords: Recoverability theorem, Beurling distributions, Toeplitz operators
Article copyright: © Copyright 1982 American Mathematical Society

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