Some Tauberian theorems for the statistical limit and statistical summability of $\ell^{(k)}$ mean of functions

Abstract

If a function $s$ has a limit $\xi$ at $\infty$, then the statistical limit of function $s$ exists and equals $\xi$. However, the converse implication is not always true.  In this paper, we present Tauberian conditions on general logarithmic control modulo that allow us to obtain the existence of ordinary limit of a function that has statistical limit $\xi$.  Additionally, we introduce the statistical summability of $\ell^{(k)}$ mean of functions for each nonnegative integer $k$ and provide conditions for Tauberian theorems. Our theorems generalize some well-known Tauberian theorems in the literature.