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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.

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

  • [1] A. F. Beardon, The generalized capacity of Cantor sets, Quart. J. Math. Oxford Ser. (2) 19 (1968), 301-304. MR 0230884 (37:6442)
  • [2] S. J. Taylor, On the connexion between Hausdorff measures and generalized capacity, Math. Proc. Cambridge Philos. Soc. 57 (1961), 524-531. MR 0133420 (24:A3254)
  • [3] G. Csordas, A. J. Lohwater and L. T. Ramsey, Lacunary series and the boundary behavior of Bloch functions, Michigan Math. J. 29 (1982), 281-288. MR 674281 (84b:30035)

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

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