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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Analytic continuation on complex lines

Authors: Joseph A. Cima and Josip Globevnik
Journal: Proc. Amer. Math. Soc. 85 (1982), 411-413
MSC: Primary 32D15; Secondary 30B40, 32A40
MathSciNet review: 656114
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Abstract: The following extension theorem is proved. Let $\Omega \subset {\mathbf {C}}$ be an open set containing $\Delta$, the open unit disc in ${\mathbf {C}}$, and the point 1. Suppose that $f$ is holomorphic on $B$, the open unit ball of ${{\mathbf {C}}^N}$, let $x \in \partial B$ and assume that for all $y \in \partial B$ in a neighborhood of $x$ the function $\varsigma \to f(\varsigma y)$, holomorphic on $\Delta$, continues analytically into $\Omega$. Then $f$ continues analytically into a neighborhood of $x$.

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

  • Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR 0027863
  • Josip Globevnik and Edgar Lee Stout, Highly noncontinuable functions on convex domains, Bull. Sci. Math. (2) 104 (1980), no. 4, 417–434 (English, with French summary). MR 602409
  • Walter Rudin, Function theory in the unit ball of ${\bf C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594

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Article copyright: © Copyright 1982 American Mathematical Society