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Tauberian Conditions Under Which Convergence Follows from Statistical Summability by Weighted Means

  • Zerrin Önder
  • İbrahim Çanak
Chapter

Abstract

Let \((p_n)\) be a sequence of nonnegative numbers such that \(p_0>0\) and
$$ P_n:=\sum _{k=0}^{n}p_k\rightarrow \infty \,\,\,\,\text {as}\,\,\,\,n\rightarrow \infty . $$
Let \((s_n)\) be a sequence of real and complex numbers. The weighted mean of \((s_n)\) is defined by
$$ t_n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k s_k\,\,\,\,\text {for}\,\,\,\,n =0,1,2,\ldots $$
We obtain some sufficient conditions, under which the existence of the limit \(\lim s_n=\mu \) follows from that of st-\(\lim t_n=\mu \), where \(\mu \) is a finite number. If \((s_n)\) is a sequence of real numbers, then these Tauberian conditions are one-sided. If \((s_n)\) is a sequence of complex numbers, these Tauberian conditions are two-sided. These Tauberian conditions are satisfied if \((s_n)\) satisfies the one-sided condition of Landau type relative to \((P_n)\) in the case of real sequences or if \((s_n)\) satisfies the two-sided condition of Hardy type relative to \((P_n)\) in the case of complex numbers.

Keywords

Statistical convergence Slow decreasing Slow decreasing relative to \((P_n)\) Slow oscillation Slow oscillation relative to \((P_n)\) The one-sided conditions of Landau type The two-sided conditions of Hardy type Tauberian theorems Weighted mean summability method 

2010 Mathematics Subject Classification

40A05 40A35 40E05 40G05 

References

  1. 1.
    E. Kolk, Matrix summability of statistically convergent sequences. Analysis 13(1–2), 77–83 (1993)Google Scholar
  2. 2.
    R. Schmidt, Über divergente folgen und lineare mittelbildungen. Mathematische Zeitschrift 22(1), 89–152 (1925)Google Scholar
  3. 3.
    J. Boos, Classical and Modern Methods in Summability, Oxford Mathematical Monographs Series (Oxford University Press, Oxford, 2000), p. xiv+586Google Scholar
  4. 4.
    G.H. Hardy, Divergent Series (Clarendon, Oxford, 1949), p. xvi+396Google Scholar
  5. 5.
    H. Tietz, Tauber-Bedingungen für potenzreihenverfahren und bewichtete mittel. Hokkaido Math. J. 20(3), 425–440 (1991)Google Scholar
  6. 6.
    F. Móricz, B.E. Rhoades, Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability. Acta Math. Hung. 66(1–2), 105–111 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    H. Tietz, K. Zeller, Tauber-Sätze für bewichtete mittel. Arch. Math. (Basel) 68(3), 214–220 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    F. Móricz, U. Stadtmüller, Necessary and sufficient conditions under which convergence follows from summability by weighted means. Int. J. Math. Math. Sci. 27(7), 399–406 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    F. Móricz, B.E. Rhoades, Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability. II. Acta Math. Hung. 102(4), 279–285 (2004)Google Scholar
  10. 10.
    S.A. Sezer, İ. Çanak, On a Tauberian theorem for the weighted mean method of summability. Kuwait J. Sci. 42(3), 1–9 (2015)Google Scholar
  11. 11.
    H. Fast, Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)MathSciNetCrossRefGoogle Scholar
  12. 12.
    I.J. Schoenberg, The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361–375 (1959)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Zygmund, Trigonometrical Series, 1st edn. (Z Subwencji Funduszu Kultury Narodowej, Warszawa-Lwow, 1935), p. xii+320Google Scholar
  14. 14.
    T. Šalát, On statistically convergent sequences of real numbers. Math. Slovaca 30(2), 139–150 (1980)Google Scholar
  15. 15.
    J.A. Fridy, On statistical convergence. Analysis 5(4), 301–313 (1985)Google Scholar
  16. 16.
    J.A. Fridy, M.K. Khan, Statistical extensions of some classical Tauberian theorems. Proc. Am. Math. Soc. 128(8), 2347–2355 (2000)Google Scholar
  17. 17.
    F. Móricz, Ordinary convergence follows from statistical summability \((C,1)\) in the case of slowly decreasing or oscillating sequences. Colloq. Math. 99(2), 207–219 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ü. Totur, İ. Çanak, Some Tauberian conditions for statistical convergence. C. R. Acad. Bulg. Sci. 67(7), 889–896 (2014)Google Scholar
  19. 19.
    H. Çakallı, A study on statistical convergence. Funct. Anal. Approx. Comput. 1(2), 19–24 (2009)Google Scholar
  20. 20.
    I.J. Maddox, A Tauberian theorem for statistical convergence. Math. Proc. Camb. Philos. Soc. 106(2), 277–280 (1989)MathSciNetCrossRefGoogle Scholar
  21. 21.
    F. Móricz, C. Orhan, Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. Stud. Sci. Math. Hung. 41(4), 391–403 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ü. Totur, İ. Çanak, On Tauberian theorems for statistical weighted mean method of summability. Filomat 30(6), 1541–1548 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    F. Móricz, Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences. Analysis (Munich) 24(2), 127–145 (2004)Google Scholar
  24. 24.
    G.A. Mikhalin, Theorems of Tauberian type for \((J,\,p_{n})\) summation methods. Ukr. Mat. Žh. 29(6), 763–770 (1977). English translation: Ukr. Math. J. 29(6), 564–569 (1977)Google Scholar
  25. 25.
    F. Móricz, U. Stadtmüller, Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim’s sense. Int. J. Math. Math. Sci. 65–68, 3499–3511 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsEge UniversityİzmirTurkey

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