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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

A century of complex Tauberian theory


Author: J. Korevaar
Journal: Bull. Amer. Math. Soc. 39 (2002), 475-531
MSC (2000): Primary 40E05; Secondary 11M45, 30B50, 44A10, 47A10
Published electronically: July 8, 2002
MathSciNet review: 1920279
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Abstract: Complex-analytic and related boundary properties of transforms give information on the behavior of pre-images. The transforms may be power series, Dirichlet series or Laplace-type integrals; the pre-images are series (of numbers) or functions.

The chief impulse for complex Tauberian theory came from number theory. The first part of the survey emphasizes methods which permit simple derivations of the prime number theorem, associated with the labels Landau-Wiener-Ikehara and Newman. Other important areas in complex Tauberian theory are associated with the names Fatou-Riesz and Ingham. Recent refinements have been motivated by operator theory and include local $H^1$and pseudofunction boundary behavior of transforms. Complex information has also led to better remainder estimates in connection with classical Tauberian theorems. Applications include the distribution of zeros and eigenvalues.


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Additional Information

J. Korevaar
Affiliation: Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands
Email: korevaar@science.uva.nl

DOI: http://dx.doi.org/10.1090/S0273-0979-02-00951-5
PII: S 0273-0979(02)00951-5
Received by editor(s): June 28, 2001
Received by editor(s) in revised form: February 22, 2002
Published electronically: July 8, 2002
Article copyright: © Copyright 2002 American Mathematical Society