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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Taylor series with limit-points on a finite number of circles

Author: Emmanuel S. Katsoprinakis
Journal: Trans. Amer. Math. Soc. 337 (1993), 437-450
MSC: Primary 30B10; Secondary 42A99
MathSciNet review: 1106192
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Abstract: Let $ S(z):\sum\nolimits_{n = 0}^\infty {{a_n}{z_n}} $ be a power series with complex coefficients. For each $ z$ in the unit circle $ T = \{ z \in \mathbb{C}:\vert z\vert = 1\} $ we denote by $ L(z)$ the set of limit-points of the sequence $ \{ {s_n}(z)\} $ of the partial sums of $ S(z)$. In this paper we examine Taylor series for which the set $ L(z)$, for $ z$ in an infinite subset of $ T$, is the union of a finite number, uniformly bounded in $ z$, of concentric circles. We show that, if in addition $ \lim \inf \vert{a_n}\vert\; > 0$, a complete characterization of these series in terms of their coefficients is possible (see Theorem 1).

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Keywords: Partial sums, limit-points, Taylor series
Article copyright: © Copyright 1993 American Mathematical Society