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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Analytic continuation on complex lines
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by Joseph A. Cima and Josip Globevnik PDF
Proc. Amer. Math. Soc. 85 (1982), 411-413 Request permission

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
  • 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 $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 85 (1982), 411-413
  • MSC: Primary 32D15; Secondary 30B40, 32A40
  • DOI: https://doi.org/10.1090/S0002-9939-1982-0656114-3
  • MathSciNet review: 656114