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An asymptotic property of gap series

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Abstract

Takahashi [15] gave a concrete upper bound estimate of the law of the iterated logarithm for ∑ f(n k x). We extend this result and prove the best possibility of this bound.

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References

  1. I. Berkes, On the asymptotic behavior of ∑ f(n k x), Applications, Z. Wahr. verw. Geb., 34 (1976), 347-365.

    Article  MATH  MathSciNet  Google Scholar 

  2. I. Berkes, An optimal condition for the LIL for trigonometric series, Trans. Amer. Math. Soc., 347 (1995), 515-530.

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Berkes and W. Philipp, The size of trigonometric and Walsh series and uniform distribution mod 1, J. London Math. Soc., 50 (1994), 454-464.

    MATH  MathSciNet  Google Scholar 

  4. I. Berkes, On the convergence of ∑ c n f(nx) and the Lip 1/2 class, Trans. Amer. Math. Soc., 349 (1997), 4143-4158.

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Berkes and W. Philipp, A limit theorem for lacunary series ∑ f(n k x), Studia Sci. Math. Hungar., 34 (1998), 1-13.

    MATH  MathSciNet  Google Scholar 

  6. S. Dhompongsa, Uniform laws of the iterated logarithm for Lipschitz classes of functions, Acta Sci. Math. (Szeged), 50 (1986), 105-124.

    MATH  MathSciNet  Google Scholar 

  7. P. Erdős, On trigonometric sums with gaps, Magyar Tud. Akad. Mat. Kut. Int. Közl., 7 (1962), 37-42.

    Google Scholar 

  8. K. Fukuyama and B. Petit, Le théorème limite central pour les suites de R.C. Baker, Ergod. Theory Dynam. Sys., 21 (2001), 479-492.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Maruyama, On an asymptotic property of a gap sequence, Kôdai Math. Sem. Rep., 2 (1950), 31-32. (Reproduced version is available in: Selection of Gisiro Maruyama, 63–66, Kaigai, Tokyo, 1988.)

    MathSciNet  Google Scholar 

  10. E. Péter, A probability limit theorem for { f(nx)}, Acta Math. Hungar., 87 (2000), 23-31.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Takahashi, An asymptotic property of a gap sequence, Proc. Japan Acad., 38 (1962), 101-104.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Takahashi, The law of the iterated logarithm for a gap sequence with infinite gaps, Tôhoku Math. J., 15 (1963), 281-288.

    MATH  Google Scholar 

  13. S. Takahashi, On the law of the iterated logarithm for lacunary trigonometric series, Tôhoku Math. J., 24 (1972), 319-329.

    MATH  Google Scholar 

  14. S. Takahashi, On the law of the iterated logarithm for lacunary trigonometric series II, Tôhoku Math. J., 27 (1975), 391-403.

    MATH  Google Scholar 

  15. S. Takahashi, An asymptotic behavior of { f(n k t)}, Sci. Rep. Kanazawa Univ., 33 (1988), 27-36.

    MathSciNet  Google Scholar 

  16. W. Philipp, Limit theorem for lacunary series and uniform distribution mod 1, Acta Arith., 26 (1975), 241-251.

    MATH  MathSciNet  Google Scholar 

  17. A. Zygmund, Trigonometric Series, Vol I, Cambridge University Press (1959).

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Authors and Affiliations

  1. Department of Mathematics, Kobe University, Rokko Kobe, 657-8501, Japan

    Katusi Fukuyama

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  1. Katusi Fukuyama
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Fukuyama, K. An asymptotic property of gap series. Acta Mathematica Hungarica 97, 257–264 (2002). https://doi.org/10.1023/A:1020859112807

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