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Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. (English) Zbl 1063.40007
Summary: The first named author has recently proved necessary and sufficient Tauberian conditions under which statistical convergence follows from statistical summability $(C,1)$. The aim of the present paper is to generalize these results to a large class of summability methods $(\overline N,p)$ by weighted means. Let $p=(p_k:k=0,1,2,\dots)$ be a sequence of nonnegative numbers such that $p_0>0$ and $$ P_n:=\sum^n_{k=0} p_k\to \infty\quad \text{as}\quad n\to \infty . $$ Let $(x_k)$ be a sequence of real or complex numbers and set $t_n:=P^{-1}_n\sum^n_{k=0} p_kx_k$ for $n=0,1,2,\dots$. We present necessary and sufficient conditions under which the existence of the limit $\text{st-lim}\, x_k=L$ follows that of $\text{st-lim}\, t_n=L$, where $L$ is a finite number. If $(x_k)$ is a sequence of real numbers, then these are one-sided Tauberian conditions. If $(x_k)$ is a sequence of complex numbers, then these are two-sided Tauberian conditions.

40E05Tauberian theorems, general
40G05Cesàro, Euler, Nörlund and Hausdorff methods
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