Tauberian conditions for a general limitable method

Abstract

Let (un) be a sequence of real numbers, L an additive limitable method with some property, and and different spaces of sequences related to each other. We prove that if the classical control modulo of the oscillatory behavior of (un) in is a Tauberian condition for L, then the general control modulo of the oscillatory behavior of integer order m of (un) in or is also a Tauberian condition for L.