# 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

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## Authors and Affiliations

1. 1.Department of MathematicsEge UniversityİzmirTurkey