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Proceedings of the American Mathematical Society

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



On divergent lacunary trigonometric series

Author: L. Thomas Ramsey
Journal: Proc. Amer. Math. Soc. 90 (1984), 397-400
MSC: Primary 42A55; Secondary 28A12
MathSciNet review: 728355
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Abstract: Let $ S = \left\{ {{\lambda _n}} \right\}_{n = 1}^\infty $ be a sequence of positive real numbers such that $ {\lambda _{n + 1}}/{\lambda _n} \geqslant q > 1$ for all $ n$. If $ q > 8$ then any divergent series with frequencies in $ S$ has its real part diverging (uniformly) to $ + \infty $ on a set of positive logarithmic capacity. It is necessary that $ q > 2$. A new sufficient condition for the generalized capacity of a set to be positive is developed and then applied in the proof.

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Keywords: Lacunary series, generalized capacity, logarithmic capacity
Article copyright: © Copyright 1984 American Mathematical Society

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