Complex Tauberian Theorems

Abstract

In the course of Chapters I and II, the notion of ‘Tauberian theorem’ has evolved. We would now say that such a theorem involves a class of objects S (functions, series, sequences) and a transformation T. The transformation is an ‘averaging operation’ with attendant continuity property: certain limit behavior of the original S implies related limit behavior of the image ’T S. The aim of a Tauberian theorem is to reverse the averaging, or pass to a different average. One wants to go from a limit property of T S to a limit property of S, or another transform of S. Such theorems typically require an additional condition, a ‘Tauberian condition’, on S and perhaps a condition on the transform T S.