Tauberian Theorems for Sequences with Moderately Oscillatory Control Modulo

Mehmet Dik

We introduce a general control modulo of the oscillatory behavior of order $m$ of $\{u_n\}$, which leads to new Tauberian conditions and consequently to the new Tauberian theorems. Also the notion of moderately oscillatory and regularly generated sequences is presented and studied. In the first section we give basic definitions, notations and a brief survey of classical results. Next we establish Tauberian theorems by using the general control modulo. The proofs of these theorems are based on the classical and neoclassical Tauberian results, in particular on the corollary to Karamata's Hauptsatz. Finally in the last section we consider the class of moderately oscillatory regularly generated sequences and prove some theorems similar to Tauberian theorems.