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Complex Tauberian Theorems

  • Jacob Korevaar
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 329)

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.

Keywords

Zeta Function Dirichlet Series Boundary Behavior Tauberian Theorem Prime Number Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jacob Korevaar
    • 1
  1. 1.KdV Mathematical InstituteUniversity of AmsterdamAmsterdamThe Netherlands

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