Tauberian Theory A Century of Developments
Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary
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Series editors
A. Chenciner S.S. Chern B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin 1. Hormander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt
Editor-in-Chief
M. Berger
J. Coates S.R.S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Jacob Korevaar
Tauberian Theory A Century of Developments
Springer
Jacob Korevaar KdV Mathematical Institute University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands
e-mail: [email protected]
Library of Congress Control Number:
2004104246
Mathematics Subject Classification (2000): Primary 40E05 Secondary nM45, N05, P82; 26A12; 30B, C, D; 40-XX; 41Axx; 42Axx; 44Axx; 46A45, F05, Ixx; 47AlO, 30; 60E15
ISSN 0072-7830 ISBN 978-3-642-05919-3 ISBN 978-3-662-10225-1 (eBook) DOI 10.1007/978-3-662-10225-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable to prosecution under the German Copyright Law. springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 1st edition 2004
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To Pia
Preface
Summability methods have been used at least since the days of Euler to assign a reasonable sum to an infinite series, whether it is convergent or not. In its simplest form, Tauberian theory deals with the problem of finding conditions under which a summable series is actually convergent. A first condition of this kind, which applies to Abel summability (the power series method), was given by Alfred Tauber in 1897. However, Tauberian theory began in earnest only around 1910 with the work of Hardy and Littlewood. Over a period of thirty years they obtained a large number of refined 'Tauberian theorems', and they gave the subject its euphonious name. A summability method for a series typically involves an averaging process of the partial sums. The step from summability to convergence requires a reversal of the averaging. For this one generally needs an additional condition on the series, known as a Tauberian condition. There is an endless variety of summability methods, and a corresponding variety of possi
