The Vijayaraghavan and Hardy theorem

A Tauberian theorem aims at the convergence of a sequence, In a first step one might establish its boundedness. A rather general approach to this boun dedness is described by a theorem due to Vijayaraghavan and Hardy. The assu mptions in this theorem are rather complicated: One demands that three rema inders tend to zero if five parameters tend to infinity. We demand only tha t the remainders are suitably bounded if the parameters are large enough. T his is further simplified in Satz 4.1. In addition we explain several refin ements and extensions. We conclude with some applications and remarks.