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Subsequential Convergence Conditions


Let be a sequence of real numbers and let be any regular limitable method. We prove that, under some assumptions, if a sequence or its generator sequence generated regularly by a sequence in a class of sequences is a subsequential convergence condition for, then for any integer, the repeated arithmetic means of,, generated regularly by a sequence in the class, is also a subsequential convergence condition for.



  1. Stanojević ČV: Analysis of divergence: applications to the Tauberian theory, Graduate Research Seminar. University of Missouri-Rolla, Rolla, Mo, USA, 1999.

    Google Scholar 

  2. Stanojević ČV: Analysis of Divergence: Control and Management of Divergent Process, edited by İ. Çanak, Graduate Research Seminar Lecture Notes. University of Missouri-Rolla, Rolla, Mo, USA; 1998.

    Google Scholar 

  3. Hardy GH: Divergent Series. The Clarendon Press, Oxford University Press, New York, NY, USA; 1949:xvi+396.

    MATH  Google Scholar 

  4. Boos J: Classical and Modern Methods in Summability, Oxford Mathematical Monographs. Oxford University Press, Oxford, UK; 2000:xiv+586.

    Google Scholar 

  5. Dik M: Tauberian theorems for sequences with moderately oscillatory control modulo. Mathematica Moravica 2001, 5: 57–94.

    MATH  Google Scholar 

  6. Dik F: Tauberian theorems for convergence and subsequential convergence with moderately oscillatory behavior. Mathematica Moravica 2001, 5: 19–56.

    MATH  Google Scholar 

  7. Tauber A: Ein Satz aus der Theorie der unendlichen Reihen. Monatshefte für Mathematik und Physik 1897,8(1):273–277. 10.1007/BF01696278

    Article  MathSciNet  MATH  Google Scholar 

  8. Littlewood JE: The converse of Abel's theorem on power series. Proceedings of the London Mathematical Society 1911,9(2):434–448.

    Article  MATH  Google Scholar 

  9. Rényi A: On a Tauberian theorem of O. Szász. Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum 1946, 11: 119–123.

    MATH  Google Scholar 

  10. Çanak İ, Totur Ü: Tauberian theorems for Abel limitability method. submitted for publication submitted for publication

  11. Çanak İ: Tauberian theorems for a generalized Abelian summability methods. Mathematica Moravica 1998, 2: 21–66.

    MATH  Google Scholar 

  12. Çanak İ, Totur Ü: A note on Tauberian theorems for regularly generated sequences. submitted for publication submitted for publication

  13. Çanak İ, Totur Ü: A Tauberian theorem with a generalized one-sided condition. Abstract and Applied Analysis 2007, 2007: 12 pages.

    Article  Google Scholar 

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Correspondence to İbrahim Çanak.

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Çanak, İ., Totur, Ü. & Dik, M. Subsequential Convergence Conditions. J Inequal Appl 2007, 087414 (2007).

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