Results in Mathematics

, Volume 47, Issue 3–4, pp 340–354 | Cite as

Almost Sure Convergence and Square Functions of Averages Of Riemann Sums

Article
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Abstract

Under a very moderate assumption on the Fourier coefficients of a periodic function, we show the convergence almost everywhere of the sequence of averages of its associated Riemann sums. The structure of the set of averages is analyzed by proving a spectral regularization type inequality, which allows to control the corresponding Littlewood-Paley square function.

AMS Subject Classification 2000

42A24 40G99 Secondary 26A42 

En]Keywords

Riemann sums convergence almost everywhere Marcinkiewicz-Salem conjecture spectral regularization Littlewood-Paley square function 
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References

  1. [Bo]
    Bourgain J. [1990] Problems of almost everywhere convergence related to harmonic analysis and number theory, Israël J. Math. 71, 97–127.MathSciNetMATHCrossRefGoogle Scholar
  2. [Br]
    Breiman L. [1992]. Probability, SIAM Ed.Google Scholar
  3. [G]
    IS. Gál [1949] A theorem concerning diophantine approximation, Nieuw. Arch. Wiskunde 23 no2, 13–38.MathSciNetMATHGoogle Scholar
  4. [GK]
    Gál I.S., Koksma J.F. [1950] Sur l’ordre de grandeur des fonctions sommables, Indag. Math. 12, 192–207.Google Scholar
  5. [LW1]
    M. Lifshits and M. Weber [2000] Spectral regularization inequalities, Math. Scand. 86 no1, 75–99. au[LW2]_M. Lifshits and M. Weber [2003] Régularisation Spectrale et Propriétés Métriques des Moyennes Mobiles, J. d’Analyse Math. 89, 1–14.MathSciNetMATHGoogle Scholar
  6. [MS]
    Marczinkiewicz J., Salem R. [1940] Sur les sommes riemanniennes, Comp. Math. 7, 376–389.Google Scholar
  7. [R]
    Rudin W. [1964] An arithmetic property of Riemann sums, Proc. Amer. Math. Soc. 15, 321–324.MathSciNetMATHCrossRefGoogle Scholar
  8. [S]
    Spitzer F.L. [1976] Principles of random walks, Second Edition, Springer, New-York.CrossRefGoogle Scholar
  9. [W1]
    M. Weber [2004] A Theorem related to the Marcinkiewicz-Salem conjecture, Results Math. 45 no. 1–2, 169–184.MathSciNetMATHCrossRefGoogle Scholar
  10. [W2]
    M. Weber [2003] On the order of magnitude of the divisor function, to appear in Acta Math. Sinica.Google Scholar
  11. [W3]
    M. Weber [2004] An arithmetical property of Rademacher sums. Indagationes Math. 15 No.1, 133–150.MATHCrossRefGoogle Scholar
  12. [W4]
    M. Weber [2003] Divisors, spin sums and the functional equation of the Zeta-Riemann function, to appear in Periodica Math. Hungar.Google Scholar
  13. [W5]
    M. Weber [dy2003] Uniform bounds under increments conditions, to appear in T.A.M.S.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Mathématique (IRMA), Université Louis-Pasteur et C.N.R.S.Strasbourg CedexFrance

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