*(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 $p=({p}_{k}:k=0,1,2,\cdots )$ be a sequence of nonnegative numbers such that ${p}_{0}>0$ and

Let $\left({x}_{k}\right)$ be a sequence of real or complex numbers and set ${t}_{n}:={P}_{n}^{-1}{\sum}_{k=0}^{n}{p}_{k}{x}_{k}$ for $n=0,1,2,\cdots $. We present necessary and sufficient conditions under which the existence of the limit $\text{st-lim}\phantom{\rule{0.166667em}{0ex}}{x}_{k}=L$ follows that of $\text{st-lim}\phantom{\rule{0.166667em}{0ex}}{t}_{n}=L$, where $L$ is a finite number. If $\left({x}_{k}\right)$ is a sequence of real numbers, then these are one-sided Tauberian conditions. If $\left({x}_{k}\right)$ is a sequence of complex numbers, then these are two-sided Tauberian conditions.