Tauberian Theorems for Convergence and Subsequential Convergence with Moderately Oscillatory Behavior

Filiz Dik

In the classical Tauberian theory, the main objective is to obtain convergence of a sequence $\{u_n\}$ by imposing conditions about the oscillatory behavior of $\{u_n\}$ in addition to the existence of certain continuous limits. However, there are some conditions of considerable interest from which it is not possible to obtain convergence of $\{u_n\}$. This situation motivates a different kind of Tauberian theory where we do not look for convergence recovery of $\{u_n\}$, rather we are concerned with the subsequential behavior of the sequence $\{u_n\}$. The first section includes definitions, notations and an overview of classical results. Succinct proofs of the Hardy-Littlewood theorem and the generalized Littlewood theorem are given using the corollary to Karamata's Hauptsatz. In the second section subsequential Tauberian theory is introduced and some related Tauberian theorems are proved. Finally, in the last section we study convergence and subsequential convergence of regularly generated sequences.