Tauberian Remainder Theorems
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232 Tord H. Ganelius University of Goteborq, Goteborg/Sweden
Tauberian Remainder Theorems
Springer-Verlag Berlin· Heidelberg· New York 1971
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich Series: Institute of Mathematical Sciences, Madras Adviser: K. R. Unni
232 Tord H. Ganelius University of Goteborq, Goteborg/Sweden
Tauberian Remainder Theorems
Springer-Verlag Berlin· Heidelberg· New York 1971
AMS Subject Classifications (1970): 40E05, 44A 35, 41A25, 42A 92
ISBN 3-540-05657-2 Springer Verlag Berlin' Heidelberg· New York ISBN 0-387-05657-2 Springer Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 78-179438. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr,
PREFACE The main part of these notes consists of the material in a series of lectures I gave at the Institute of mathematical sciences "Matscience" in Madras in January 1969 and which has been available in a mimeographed report from that institute. In this edition I have added some topics I have given in lectures at other places. Many theorems represent unpublished work of mine and I have also included some references to work of my students. The main theme is the application of Wiener's general method to different kinds of tauberian problems, in particular to remainder problems. As will be seen the proofs of general theorems are often simpler than special proofs for special cases. The first application of Wiener's method to remainders in tauberian theorems was given by Beurling in 1938. His work has been continued by Lyttkens. Among other things she has proven theorems where conditions on the Fourier transform of the kernel are required only in the upper halfplane. Such theorems are certainly important e.g. in number theory. I have chosen to formulate my theorems in a less general way in this respect, but it is easy to see that the method can be used in many cases also under the weaker assumptions. For help in the preparation of the notes for publication thanks are due to the staff of Matscience but also to J.E.Andersson, A.Ganelius, T.O.Ganelius and K.Lidhag.
August 1971
Tord Ganelius
CON TEN T S CHAPTER 1. DIFFERENT APPROACHES TO TAUBERIAN THEOREMS 1.1
Introduction.
1.2
Approximation methods
3
1.3
Wiener's method • • •
6
1.4
The theory of distributions-
• 13
1 .5
Examples of kernels .
• 15
• • •
CHAPTER 2. A TYPICAL GENERAL REMAINDER THEOREM 2.1
A general theorem generalizing HardyLittlewoo
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