Advertisement

Positivity

pp 1–19 | Cite as

Tauberian theorems for the logarithmic summability methods of integrals

  • Muhammet Ali Okur
  • Ümit Totur
Article

Abstract

A real-valued continuous function f on \([1, \infty )\) is said to be summable by the logarithmic summability method of integrals (shortly, \(\ell \) summable) if
$$\begin{aligned} \lim _{x \rightarrow \infty }\frac{1}{\log {x}}\int _1^x\frac{s(t)}{t}dt \end{aligned}$$
exists, where \(s(x)=\int _{1}^{x}f(t)dt\). Our goal in this paper is to obtain some Tauberian theorems for the logarithmic summability method of integrals by some new Tauberian conditions. We also introduce the \(\ell ^{(k)}\) summability method of integrals and give some Tauberian theorems for this method.

Keywords

Tauberian theorem Tauberian condition Logarithmic summability method of integrals General control modulo 

Mathematics Subject Classification

Primary 40E05 Secondary 40A10 40C10 

References

  1. 1.
    Çanak, İ., Totur, Ü.: Tauberian conditions for Cesàro summability of integrals. Appl. Math. Lett. 24, 891–896 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dik, M.: Tauberian theorems for sequences with moderately oscillatory control moduli. Math. Morav. 5, 57–94 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ishiguro, K.: A converse theorem on the summability methods. Proc. Jpn. Acad. 39, 38–41 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ishiguro, K.: Tauberian theorems concerning the summability method of logarithmic type. Proc. Jpn. Acad. 39, 156–159 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kwee, B.: A Tauberian theorem for the logarithmic method of summation. Proc. Camb. Philos. Soc. 63, 401–405 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Móricz, F.: Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences. Studia Math. 219, 109–121 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sezer, S.A., Çanak, İ.: Tauberian theorems for the summability methods of logarithmic type. Bull. Malays. Math. Sci. Soc. (2016).  https://doi.org/10.1007/s40840-016-0437-9 Google Scholar
  8. 8.
    Totur, Ü., Çanak, İ.: One-sided Tauberian conditions for (C,1) summability method of integrals. Math. Comput. Modell. 55(5–6), 1813–1818 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Totur, Ü., Okur, M.A.: On Tauberian conditions for the logarithmic methods of integrability. Bull. Malays. Math. Sci. Soc. 41(2), 879–892 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAdnan Menderes UniversityAydınTurkey
  2. 2.Sena SitesiAydınTurkey

Personalised recommendations