Nonstandard Asymptotic Analysis
This research monograph considers the subject of asymptotics from a nonstandard view point. It is intended both for classical asymptoticists - they will discover a new approach to problems very familiar to them - and for nonstandard analysts but includes
- PDF / 11,280,942 Bytes
- 192 Pages / 468 x 684 pts Page_size
- 20 Downloads / 273 Views
1249 Imme van den Berg
Nonstandard Asymptotic Analysis
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Imme van den Berg Institute of Econometrics, Faculty of Economics P.O. Box 800,9700 AV Groningen, Holland
Mathematics Subject Classification (1980): 03E 15, 03H 10, 06F 15, 40A25, 41A60,65B10 ISBN 3-540-17767-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17767-1 Springer-Verlag New York Berlin Heidelberg
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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
ACKNOWLEDGEMENTS.
This book is the fruit of an intensive cooperation with E. Benoit. J.-L. Cal lot and F. and M. Diener. during a stay in Algeria. I was deeply influenced by Professor R. Lutz and Professor G. Reeb. who also gave invaluable advise in the preparation of this manuscript. This text further benefited
from discussions with among others
M. Goze. J. Harthong and J.-P. Reveilles. All the above mathematicians speak the Alsatian dialect of nonstandard analysis. I am greately indebted to Professor E.M. de Jager (Amsterdam) and Professor J.-P. Ramis (Strasbourg) who drew
my attention to asymptotic theory. The latter
insisted on the problem of the "summation to the smallest term" which plays a central role in this book. I 'would like tp express special thanks to the Institute of Econometrics of the University of Groningen. which gave me the opportunity to finish the manuscript. and to Mrs. A. van Oosten. who typed it with such
great savoir-faire.
INTRODUCTION.
1). This book has three main purposes, firstly to present nonstandard methods of
asymptotic reasoning, secondly to present new results obtained by them, and thirdly to present a nonstandard alternative to the classical theory of asymptotic expansions. It is adressed to mathematicians knowing the basic principles of nonstandard analysis, who are willing to
asymptotics and to classical
asymptoticists, who wish to form an opinion about the relevancy of nonstandard analysis to this branch of mathematics; the latter will observe that problems very familiar to them are treated in a manner clearly distinct from the classical one, and may compare the efficiency of both approaches, 2). It is common practice among asympt o t rc.i s t s to speak informally about "fixed" numbers, to be distinguished from numbers depending on a "large" 01' "small" parameter. Nonstandard analysis legalizes this matter of speaking, A formal
Data Loading...
