Tauberian conditions with controlled oscillatory behavior for statistical convergence

Abstract

We present new Tauberian conditions in terms of the general logarithmic control modulo of the oscillatory behavior of a real sequence $(s_n)$ to obtain
$$\lim_{n\to\infty}s_n=\xi \ \ \text{from} \ st-\lim_{n\to\infty}s_n=\xi,$$
where $\xi$ is a finite number. We also introduce the statistical $(\ell,m)$ summability method and extend some Tauberian theorems to this method. The main results improve some well-known Tauberian theorems obtained for the statistical convergence.