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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
DOI: https://doi.org/10.1090/S0002-9947-1993-1106192-4
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 $T = \{ z \in \mathbb {C}:|z| = 1\}$ 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 $\lim \inf |{a_n}|\; > 0$, 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