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Hardy-Type Tauberian Conditions on Time Scales

  • Ceylan Turan Yalçın
  • Oktay Duman
Article
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Abstract

Hardy’s well-known Tauberian theorem for number sequences states that if a sequence \(x=\left( x_{k}\right) \) satisfies \(\lim Cx=L\) and \(\Delta x_{k}:=x_{k+1}-x_{k}=O\left( 1/k\right) \), then \(\lim x=L\), where Cx denotes the Cesàro mean (arithmetic mean) of x. In this study, we extend this result to the theory of time scales. We also discuss the theory on some special time scales.

Keywords

Tauberian theorems Cesàro summability Statistical convergence Time scales 

Mathematics Subject Classification

40E05 40G15 26E70 
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References

  1. 1.
    Aktuglu, H., Bekar, S.: \(q\)-Cesáro matrix and \(q\)-statistical convergence. J. Comput. Appl. Math. 235, 4717–4723 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aslim, G., Guseinov, G.S.: Weak semirings \(\omega \)-semirings, and measures. Bull. Allahabad Math. Soc. 14, 1–20 (1999)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  4. 4.
    Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser Boston, Inc., Boston (2001)CrossRefGoogle Scholar
  5. 5.
    Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fekete, Á., Móricz, F.: A characterization of the existence of statistical limit of real-valued measurable functions. Acta Math. Hung. 114, 235–246 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fridy, J.A., Khan, M.K.: Statistical extensions of some classical Tauberian theorems. Proc. Am. Math. Soc. 128, 2347–2355 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guseinov, G.S.: Integration on time scales. J. Math. Anal. Appl. 285, 107–127 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hardy, G.H.: Divergent Series. Clarendon Press, Oxford (1949)zbMATHGoogle Scholar
  11. 11.
    Móricz, F.: Statistical limits of measurable functions. Analysis (Munich) 24, 1–18 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Turan, C., Duman, O.: Statistical convergence on timescales and its characterizations. In: Advances in Applied Mathematics and Approximation Theory. Springer Proceedings in Mathematics and Statistics, vol. 41, pp. 57–71. Springer, New York (2013)CrossRefGoogle Scholar
  13. 13.
    Turan, C., Duman, O.: Convergence methods on time scales. AIP Conf. Proc. 1558, 1120–1123 (2013)CrossRefGoogle Scholar
  14. 14.
    Turan, C., Duman, O.: Fundamental properties of statistical convergence and lacunary statistical convergence on time scales. Filomat 31(14), 4455–4467 (2017)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsTOBB Economics and Technology UniversityAnkaraTurkey

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