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An Alternative Proof of a Tauberian Theorem for the Weighted Mean Method of Summability

  • Çağla Kambak
  • İbrahim ÇanakEmail author
Short Communication
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Abstract

Let \((p_n)\) be a sequence of nonnegative numbers such that \(p_0>0\) and
$$\begin{aligned} P_n:=\sum _{k=0}^{n}p_k\rightarrow \infty, n\rightarrow \infty . \end{aligned}$$
Let \((u_n)\) be a sequence of real or complex numbers. The nth weighted mean of \((u_n)\) is defined by
$$\begin{aligned} \sigma _n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k u_k \quad (n =0,1,2,\ldots ) \end{aligned}$$
We give an alternative proof of a Tauberian theorem stating that the existence of the limit \(\lim _{n \rightarrow \infty } u_n=s\) follows from that of \(\lim _{n \rightarrow \infty } \sigma _n=s\) and a Tauberian condition.

If \((u_n)\) is a sequence of real numbers, then these Tauberian conditions are one-sided. If \((u_n)\) is a sequence of complex numbers, these Tauberian conditions are two-sided.

Significance Statement: If a sequence converges, then its weighted means converge to the same number. But, the converse of this implication is not true in general and its partial converse might be valid. This manuscript presents an alternative proof of a well-known Tauberian theorem stating that convergence of a slowly decreasing sequence (in case of sequences of real numbers) or a slowly oscillating sequence (in case of sequences of complex numbers) follows from its weighted mean summability. Corollaries of the main results in this manuscript consist of well-known Tauberian theorems for Cesàro and logarithmic summability methods.

Keywords

Weighted mean method of summability Tauberian conditions and theorems Slowly decreasing sequences Slowly oscillating sequences 

Mathematics Subject Classification

40E05 40G05 

Notes

References

  1. 1.
    Hardy GH (1991) Divergent series. Chelsea Publishing Company, New YorkzbMATHGoogle Scholar
  2. 2.
    Móricz F, Rhoades BE (2004) Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability II. Acta Math Hung 102(4):279–285MathSciNetCrossRefGoogle Scholar
  3. 3.
    Çanak İ, Totur Ü (2011) Some Tauberian theorems for the weighted mean methods of summability. Comput Math Appl 62(6):2609–2615MathSciNetCrossRefGoogle Scholar
  4. 4.
    Çanak İ, Totur Ü (2013) Extended Tauberian theorem for the weighted mean method of summability. Ukr Math J 65(7):1032–1041MathSciNetCrossRefGoogle Scholar
  5. 5.
    Totur Ü, Çanak İ (2012) Some general Tauberian conditions for the weighted mean summability method. Comput Math Appl 63(5):999–1006MathSciNetCrossRefGoogle Scholar
  6. 6.
    Tietz H, Zeller K (1997) Tauber-Sätze für bewichtete Mittel. Arch Math (Basel) 68(3):214–220MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borwein D, Kratz W (1989) On relations between weighted mean and power series methods of summability. J Math Anal Appl 139(1):178–186MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sezer SA, Çanak İ (2015) On a Tauberian theorem for the weighted mean method of summability. Kuwait J Sci 42(3):1–9MathSciNetzbMATHGoogle Scholar
  9. 9.
    Schmidt R (1925) Über divergente Folgen und lineare Mittelbildungen. Math Z 22:89–152MathSciNetCrossRefGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  1. 1.Department of MathematicsEge UniversityİzmirTurkey

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