Tauberian theorems for sequences whose oscillations are controlled

One of the main objectives of the classical Tauberian theory is to recoverconvergence of sequences, whose divergence is manageable, out of the existenceof certain limits, which is known as Abel?s necessary conditions or its general-izations, and certain additional conditions that control the oscillatory behavior.These conditions are called Tauberian conditions and this type of theorems arecalled Tauberian theorems. In terms of the control modulo of oscillatory behaviorof integer order, introduced by Stanojevi´ (1998), we now summarize the classicalcresults. Tauber (1897) proved that if the classical control modulo of a sequence isa null sequence, then one obtains convergence of sequence out of its Abel?s neces-sary condition. Littlewood (1910) showed that Tauber?s condition can be replacedby the boundedness of the classical control modulo of a sequence. Later, Schmidt(1925) introduced the slowly oscillating sequences and proved the more generaltheorem, which is known as the generalized Littlewood theorem. The theoremgiven by Karamata (1930) simplified the proofs of the early classical Tauberiantheorems and provided viewpoint to obtain some Tauberian theorems that wasgiven in terms of control modulo of oscillatory behavior of a sequence. Using theKaramata?s Hauptsatz and employing the definition of moderately oscillating se-quences defined by Stanojevi´ (1999a), Dik (2002b) proved several theorems thatcadmitted the higher order control modulo of a sequence as a Tauberian condition.Dik (2002a) pointed out that there were some conditions that we could not inferconvergence of a sequence out of the existence of Abel?s necessary condition. Thismotivated a different kind of Tauberian theory, so called subsequential Tauberiantheory.