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A converse to the Lusin-Privalov radial uniqueness theorem

Author: Robert D. Berman
Journal: Proc. Amer. Math. Soc. 87 (1983), 103-106
MSC: Primary 30D40
MathSciNet review: 677242
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Abstract: Let $ E$ be a subset of the unit circumference $ C$. If for every nonempty open arc $ A$ of $ C$, the set $ E$ is not both metrically dense and of second category in $ A$, then there exists a nonconstant analytic function $ f$ on the open unit disk $ \Delta $, such that $ {f^ * }(\eta ) = 0$, $ \eta \in E$, where $ {f^ * }$ is the radial limit function of $ f$.

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  • [1] F. Bagemihl and W. Seidel, Some boundary properties of analytic functions, Math. Z. 61 (1954), 186-199. MR 0065643 (16:460d)
  • [2] P. Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335-400. MR 1555035
  • [3] N. N. Lusin and I. I. Privalov, Sur l'unicité et la multiplicité des fonctions analytiques, Ann. Sci. École Norm. Sup. (3) 42 (1925), 143-191. MR 1509265
  • [4] S. N. Mergelyan, Amer. Math. Soc. Transl. 3 (1962), 287-293.
  • [5] I. I. Privalov, Randeigenschaften analytischer Funktionen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956. MR 0083565 (18:727f)
  • [6] F. Riesz and M. Riesz, Über die Randwerte einer analytischen Funktion, Quatrième Congrès des Math. Scand. Stockholm, 1916, pp. 27-44.
  • [7] W. J. Schneider, Approximation and harmonic measure, Aspects of Contemporary Complex Analysis (D. H. Brannan and J. G. Clunie, eds.), Academic Press, New York, 1980, pp. 334-340. MR 623476 (82g:30058)

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Keywords: Lusin-Privalov, radial uniqueness
Article copyright: © Copyright 1983 American Mathematical Society

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