Tauberian conditions with controlled oscillatory behavior

Abstract

Let $\left(u_n\right)$ be a sequence of real numbers. In this paper, we give new Tauberian conditions imposed on the general control modulo of the oscillatory behavior of integer order $m\ge 1$ of a sequence $\left(u_n\right)$, under which, convergence follows from the Abel summability. These are generalizations of some of the well-known classical Tauberian conditions.