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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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A century of complex Tauberian theory
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by J. Korevaar PDF
Bull. Amer. Math. Soc. 39 (2002), 475-531 Request permission

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
  • Received by editor(s): June 28, 2001
  • Received by editor(s) in revised form: February 22, 2002
  • Published electronically: July 8, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 39 (2002), 475-531
  • MSC (2000): Primary 40E05; Secondary 11M45, 30B50, 44A10, 47A10
  • DOI: https://doi.org/10.1090/S0273-0979-02-00951-5
  • MathSciNet review: 1920279