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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Diophantine approximation and convergence of alternating series


Author: N. V. Rao
Journal: Proc. Amer. Math. Soc. 93 (1985), 420-422
MSC: Primary 11K60
MathSciNet review: 773994
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a continuous periodic function of period $ 2\pi $ and

$\displaystyle \sum {(\vert{a_n}\vert + \vert{b_n}\vert)\log n < \infty } ,$

where $ {a_n},{b_n}$ are the Fourier coefficients of $ f$. Let $ E$ be the set of all points $ \theta $ in $ [0,2\pi )$ for which the series

$\displaystyle \sum {\frac{{{{( - 1)}^n}}}{n}f(n\theta )} $

does not converge. It is established here that hausdorff outer measure $ {h_\alpha }(E) = 0$ for every $ \alpha > 0$.

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

  • [1] A. S. Besicovitch, Sets of fractional dimensions. IV, J. London Math. Soc. (2) 9 (1934), 126-131.
  • [2] V. Jarnik, Zur metrischen Theorie der Diophantischen Approximationen, Prace Mat.-Fiz. 36 (1928-29), 91-106.
  • [3] K. Mahler, On the approximation of 𝜋, Nederl. Akad. Wetensch. Proc. Ser. A. 56=Indagationes Math. 15 (1953), 30–42. MR 0054660 (14,957a)
  • [4] Norbert Wiener, The Fourier integral and certain of its applications, dover Publications, Inc., New York, 1959. MR 0100201 (20 #6634)
  • [5] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776 (21 #6498)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0773994-4
PII: S 0002-9939(1985)0773994-4
Article copyright: © Copyright 1985 American Mathematical Society