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

Zerrin Önder () and İbrahim Çanak ()
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Zerrin Önder: Ege University, Department of Mathematics
İbrahim Çanak: Ege University, Department of Mathematics

A chapter in Advances in Summability and Approximation Theory, 2018, pp 1-22 from Springer

Abstract: 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; 40A05; 40A35; 40E05; 40G05 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/978-981-13-3077-3_1

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