Tauberian theorems for smooth functions that never vanish

Abstract

It is well known that if a function has slow growth in the ordinary sense, then it has slow growth in the Cesàro sense. The converse holds if special extra conditions are assumed; results of this form are called Tauberian Theorems. In this paper, we use techniques from generalized functions to investigate this phenomenon, with particular emphasis on the space of multipliers of Schwartz functions. We then apply our results to study series of delta functions and the division problem for tempered distributions.