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

  • Zerrin Önder
  • İbrahim Çanak


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.


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 


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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsEge UniversityİzmirTurkey

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