Recoverability of some classes of analytic functions from their boundary values
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- by Ahmed I. Zayed PDF
- Proc. Amer. Math. Soc. 87 (1983), 493-498 Request permission
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 Zarantonelloโs 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 \[ \lim \limits _{r \to 1} {(1 - r)^\beta }\log M(r,f) = 0\quad {\text {for}}\;{\text {some}}\;\beta \geqslant 1,\] then $f(z)$ can be recovered from the restriction of its boundary value to an open arc.References
- D. J. Patil, Representation of $H^{p}$-functions, Bull. Amer. Math. Soc. 78 (1972), 617โ620. MR 298017, DOI 10.1090/S0002-9904-1972-13031-3
- Walter Rudin, Division in algebras of infinitely differentiable functions, J. Math. Mech. 11 (1962), 797โ809. MR 0153796 G. Walker, Analytic representations, values, and recoverability of distributions, Ph.D. thesis, University of Wisconsin, Milwaukee, 1974.
- Sergio E. Zarantonello, A representation of $H^{p}$-functions with $0<p<\infty$, Pacific J. Math. 79 (1978), no.ย 1, 271โ282. MR 526683 A. Zayed, Topological vector spaces of analytic functions, submitted. โ, On Beurling distributions and holomorphic functions in the unit disc, submitted.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 493-498
- MSC: Primary 30E25; Secondary 30B30, 46F20
- DOI: https://doi.org/10.1090/S0002-9939-1983-0684645-X
- MathSciNet review: 684645