A Nonlinear Theory of Generalized Functions
This book provides a simple introduction to a nonlinear theory of generalized functions introduced by J.F. Colombeau, which gives a meaning to any multiplication of distributions. This theory extends from pure mathematics (it presents a faithful generaliz
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Hebe A. Biagioni
A Nonlinear Theory of Generalized Functions
Springer-Verlag Berlin Heidelberg NewYork LondonParis Tokyo Hong Kong
Author
Hebe de Azevedo Biagioni Departamento de Matematica Universidade Estadual de Campinas Caixa Postal 6065 13081 Campinas, SP Brasil
Mathematics Subject Classification (1980): Primary: 46F10 Secondary: 35D05, 35L60, 35L67, 35K55, 65M05, 65M10, 73005, 73J06,76L05 ISBN 3-540-52408-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52408-8 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
PREFACE OF THE SECOND EDITION
This book is the second edition of a text written in 1986, 1987 and reproduced in 1988 in the preprint series Notas de Matematica of the Universidade Estadual de Campinas.
This second edition
has benefitted from a few improvements but it is not substantially different. The set of references has however been enriched by many more recent papers.
INTRODUCTION
In the year
1954 L.Schwartz published a celebrated result
on the "impossibility of the multiplication of distributions",
prov-
ing that there does not exist a differential algebra A containing the space
!ilJ' of
distributions
(on the real line) and having
the
classical properties relative to differentiation and to the algebraic operations of addition and multiplication. At that time
it
had been recognized that physicists had been using "illegal multiplications of distributions", even in classical physical such as Continuum Mechanics. tory results, industry.
theories
In many cases this leads to satisfac-
some of them consisting of numerical codes used
in
This situation suggests to mathematicians that they should
reconsider the problem so as to try to find a solution in form of a suitable underlying mathematical Seven years ago,
theory.
a differential algebra
and having all natural properties
W containing !ilJ',
(of course one of them is in
weakened form relative to Schwartz'
a
impossibility result), has been
constructed. Recently it was recognized that this theory was perfectly welladapted to the solution of problems of physics and engineering involving such multiplications. The aim of this book is to provide a simple introduction
IV
to this nonlinear theory of generalized functions introduced
by
J.F.Colombeau. Now this theory extends from pure mat
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