A Tauberian Theorem with a Generalized One-Sided Condition

Abstract

We prove a Tauberian theorem to recover moderate oscillation of a real sequence $u=\left(u_n\right)$ out of Abel limitability of the sequence $\left(V_n^{\left(1\right)}\left(\Delta u\right)\right)$ and some additional condition on the general control modulo of oscillatory behavior of integer order of $u=\left(u_n\right)$.