Functional Equations: History, Applications and Theory

242 Pages·1984·8.826 MB·English

by  J. Aczél (auth.)

Functional Equations: History, Applications and Theory
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
R. W. BROCKETT, Harvard University, Cambridge, Mass., U.S.A.
J. CORONES, Iowa State University, U.S.A. and Ames Laboratory, U.S. Department of
Energy, Iowa, U.S.A.
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G .. .c. ROTA, M.I. T., Cambridge, Mass., U.S.A.
Functional Equations:
History, Applications
and Theory
edited by
J. Aczel
Centre for Information Theory, University of Waterloo, Ontario, Canada
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster
Library of Congress Cataloging in Publication Data
Main entry under title:
Functional equations: History, Applications and Theory
(Mathematics and its applications)
Includes index.
1. Functional equations-Addresses, essays, lectures.
I. Aczel, J. II. Series: Mathematics and its applications (D. Reidel
Publishing Company)
QA431.F79 1984 515.7 83-24732
ISBN-13: 978-1-4020-0329-5 e-ISBN-13: 978-94-009-6320-7
001: 10.1007/978-94-009-6320-7
Transferred to Digital Print 2001
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
190 Old Derby Street, Hingham, MA 02043, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland
All Rights Reserved
© 1984 by D. Reidel Publishing Company, Dordrecht, Holland
No part of the material protected by this copyright notice may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photocopying,
recording or by any information storage and retrieval system,
without written permission from the copyright owner.
Table of Contents
Editor's Preface vii
Essays
1. J. Aczel: 3
On history, applications and theory of
functional equations (Introduction)
2. Cl. Alsina: 13
The esthetics and usefulness of functional
equations
3. J.G. Dhombres: 17
On the historical role of functional equations
4. W. Gehrig: 33
Functional equation methods applied to economic
problems: some examples
5. L. Reich: 53
On multiple uses of some classical equations
6. A. Sklar: 55
On some unexpected theoretical and practical
applications of functional equations
7. A. Krapez: 57
Functional equations on groupoids and related
structures
Papers
1. J.G. Dhombres: 67
Some recent applications of functional equations
2. A. Krapez: 93
Groupoids with ~-kernels
vi TABLE OF CONTENTS
3. A. Tsutsumi and Sh. Haruki: 99
The regularity of solutions of functional
equations and hypoellipticity
4. H. Haruki: 113
An improvement of the Nevanlinna-P01ya theorem
5. L. Reich and J. Schwaiger: 127
On polynomials in additive and mUltiplicative
functions
6. Cl. Alsina: 161
Truncations of distribution functions
7. B.R. Ebanks: 167
Kurepa's functional equation on Gaussian semigroups
8. J. Aczel: 175
Some recent results on information measures, a new
generalization and some 'real life' interpretations
of the 'old' and new measures
9. W. Gehrig: 191
On a characterization of the Shannon
concentration measure
10. R. Thibault: 207
Closed invariant curves of a noncontinuously
differentiable recurrence
11. R.L. Clerc and C. Hartmann: 217
Invariant curves as solutions of functional
equations
12. A. Sklar: 227
The cycle theorem for flows
Index 231
EDITOR'S PREFACE
Approach your problems from It isn't that they can't see
the right end and begin with the solution. It is that they
the answers. Then one day, can't see the problem.
perhaps you will find the
G.K. Chesterton. The Scandal of
final question.
Father Brown 'The Point of a
'The Hermit Clad ~n Crane Pin' .
Feathers' in R. van Gulik's
The Chinese Haze Murders.
Growing specialization and diversification have brought a host
of monographs and textbooks on increasingly specialized topics.
However, the "tree" of knowledge of mathematics and related
fields does not grow only by putting forth new branches.
It also happens, quite often in fact, that branches which were
thought to be completely disparate are suddenly seen to be
related.
Further, the kind and level of sophistication of mathe
matics applied in various sciences has changed drastically in
recent years: measure theory is used (non-trivially) ~n re
gional and theoretical economics; algebraic geometry interacts
with physics; the Minkowsky lemma, coding theory and the
structure of water meet one another in packing and covering
theory; quantum fields, crystal defects and mathematical pro
gramming profit from homotopy theory; Lie algebras are rele
vant to filtering; and prediction and electrical en~ineering
can use Stein spaces. And in addition to this there are such
new emerging subdisciplines as "completely integrable systems",
"chaos, synergetics and large-scale order", which are almost
impossible to fit into the existinf, classifi~ation schemes.
They draw upon widely different sections of mathematics.
This program, Mathematics and Its Applications, is
devoted to such (new) interrelations as exempla gratia:
- a central concept which plays an important role in several
different mathematical and/or scientific specialized areas;
- new applications of the results and ideas from one area of
scientific endeavor into another;
- influences which the results, problems and concepts of one
field of enquiry have and have had on the development of
another.
vii
J. Aczel (ed.), Functional Equations: History, Applications and Theory, vii-ix.
<0 1984 by D. Reidel Publishing Company.
viii EDITOR'S PREFACE
The Hathematics and Its Applications programme tries to
make available a careful selection of books which fit the
philosophy outlined above. Hith such books, which are stimu
lating rather than definitive, intriguing rather than encyclo
paedic, we hope to contribute something towards better commu
nication among the practitioners in diversified fields.
The topic of the present volume in the HIA Series is
functional equations, that is the search for functions which
satisfy certain functional relationships such as, for example,
f(x + y) = f(x) + fey) - one of the very simplest examples.
They are remarkable in that they arise in most parts of mathe
matics. A number of relatively well known examples are
Cauchy's equations (the one just written down), the functional
equations for the Riemann zeta function (are there more solu
tions, a question studied by A. Heil), the equation for entropy,
numerous equations in combinatorics, and quite recently for
example the important Yang-Baxter equations in lattice sta
tistical dynamics and quantum field theory which asks for
n2 x n2 matrix valued functions satisfying
(I <8> R(u-v)) (R(u) <8> I) (I ® R(v)) = (R(v) <8> I) (I <8> R(u)) (R(u-v)<8>I)
(both sides are n3 x n3 matrices). Still other examples arise
in probability theory (f(x2 + y2)1/2) = f(x) fey)) in operator
theory and in geometry (find all transformations Rn + Rn
which take straight lines into straight lines).
In addition, there are functional equations which govern
the behaviour of iterations of maps such as the Poincare
equation F(az) = aF(z)(I-F(z)) where a is a parameter and the
equation which is at the heart of various universality results
(Feigenbaum theory, chaos theory), G(n) = _A-1 G(G(An)),
A = -G(I). And for that matter a formal group is an n-tuple
of power series in 2n variables FI (XI' ... , ~; YI, ... , Yn),
... , Fn (XI' ... , Xn; Y I' ... , Yn) satisfying the functional
equations F(O; Y) = Y, F(X, 0) = X, F(F(X, Y), Z) = X,
F(F(X, Y), z) = F(X, F(Y, Z)).
In turn, formal groups have, for example, application in
algebraic geometry, algebraic number theory and algebraic
topology. Thus functional equations seem to occur in virtually
all parts of mathematics.
The subject does not fit well into any of the established
mathematical specialisms, neither in terms of its problems,
nor in terms of its results and techniques. Though there are
several spectacular results such as Heierstrass' theorem in
which kinds of functions admit some sort of polynomial addition
formula, a general, universal powerful technique for dealing
with functional equations does not yet exist and thus the
EDITOR'S PREFACE ix
material in this book poses a considerable challenge to the
mathematical community. This fits in well with the philo
sophy behind the Mathematics and Its Applications book series.
Also a glance at the Table of Contents will convince the
reader that the subject impinges on even more specialisms
in mathematics then are mentioned above, making this a truly
multispecialistic volume, another aim of the MIA programme.
The unreasonable effective As long as algebra and geometry
ness of mathematics ln proceeded along separate paths,
SClence .... their advance was slow and
their applications limited.
Eugene Higner
But when these sciences
joined company they drew from
Well, if you knows of a
each other fresh vitality and
better 'ole, go to it.
thenceforward marched on at a
Bruce Bairnsfather rapid pace towards perfection.
Joseph Louis Lagrange
What is now proved was once
only imagined.
William Blake
Amsterdam, October 1983 Michiel Hazewinkel
Essays