Regularity properties of functional equations in several variables

366 Pages·2005·14.07 MB·English

by  Antal Járai

REGULARITY PROPERTIES OF
FUNCTIONAL EQUATIONS IN
SEVERAL VARIABLES
Advances in Mathematics
VOLUME 8
Series Editor:
J. Szep, Budapest University of Economics, Hungary
Advisory Board:
S.-N. Chow, Georgia Institute of Technology, U.S.A.
G. Erjaee, Shiraz University, Iran
W. Fouche, University of South Africa, South Africa
P. Grillet, Tulane University, U.S.A.
H.J. Hoehnke, Institute of Pure Mathematics of the Academy of
Sciences, Germany
F. Szidarovszky, University of Airzona, U.S.A.
P.G. Trotter, University of Tasmania, Australia
P. Zecca, Universitb di Firenze, Italy
REGULARITY PROPERTIES OF
FUNCTIONAL EQUATIONS IN
SEVERAL VARIABLES
ANTAL JARAI
Eotvos Lorand University, Budapest, Hungary
- Springer
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9 8 7 6 5 4 3 2 1 SPIN 113 77986
This book is dedicated to JBnos Aczkl, my "mathematical grandfather", the
teacher of several of us in the field of functional equations, and to my teacher
ZoltAn Dar6czy who introduced me to functional equations.
TABLE OF CONTENTS
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter I . PRELIMINARIE. S. . . . . . . . . . . . . . . . . 1
$1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
$ 2. Notation and terminology . . . . . . . . . . . . . . . . . . 4 0
Chapter I1. STEINHAUTYSP E THEOREMS . . . . . . . . . . . .53
$ 3. Generalizations of a theorem of Steinhaus . . . . . . . . . . . 5 3
5 4 . Generalizations of a theorem of Piccard . . . . . . . . . . . . 6 6
Chapter I11 . BOUNDEDNEASNSD CONTINUITY OF SOLUTIONS . . . . 7 3
$ 5. Measurability and boundedness . . . . . . . . . . . . . . . . 7 3
$ 6 . Continuity of bounded measurable solutions . . . . . . . . . . 7 6
5 7. On a problem of Mazur . . . . . . . . . . . . . . . . . . . 8 1
5 8. Continuity of measurable solutions . . . . . . . . . . . . . . 8 6
$ 9 . Continuity of solutions having Baire property . . . . . . . . . 9 4
$10. Almost solutions . . . . . . . . . . . . . . . . . . . . . 100
Chapter IV . DIFFERENTIABILITAYN D ANALYTICITY . . . . . . . 109
5 11 . Local Lipschitz property of continuous solutions . . . . . . . 109
5 12. Holder continuity of solutions . . . . . . . . . . . . . . . 128
$ 13. Solutions of bounded variation . . . . . . . . . . . . . . . 132
$ 14. Differentiability . . . . . . . . . . . . . . . . . . . . . 137
5 15. Higher order differentiability . . . . . . . . . . . . . . . . 141
5 16 . Analyticity . . . . . . . . . . . . . . . . . . . . . . . 144
Chapter V . REGULARITTYH EOREMS ON MANIFOLDS . . . . . . 157
5 17. Local and global results on manifolds . . . . . . . . . . . . 157
Chapter VI . REGULARITRYES ULTS WITH FEWER VARIABLES . . . 169
5 18. ~wiatak'sm ethod . . . . . . . . . . . . . . . . . . . . 170
§ 19. Between measurability and continuity . . . . . . . . . . . . 174
5 20 . Between Baire property and continuity . . . . . . . . . . . 204
$ 21 . Between continuity and differentiability . . . . . . . . . . . 218
Chapter VII . APPLICATIONS . . . . . . . . . . . . . . . . 2 31
5 22 . Simple applications . . . . . . . . . . . . . . . . . . . . 231
$23. Characterization of the Dirichlet distribution . . . . . . . . . 275
$24. Characterization of Weierstrass's sigma function . . . . . . . 285
viii Table of contents
References . . . . . . . . . . . . . . . . . . . . . . . . . . 335
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
PREFACE
This book is about regularity properties of functional equations. It con-
tains, in a unified fashion, most of the modern results about regularity of
non-composite functional equations with several variables. It also contains
several applications including very recent ones. I hope that this book makes
these results more accessible and easier to use for everyone working with
functional equations.
This book could not have been written without the stimulating atmo-
sphere of the International Symposium on Functional Equations conference
series and thus I am grateful to all colleagues working in this field. This
series of conferences was created by JBnos Aczitl. I am especially grateful to
him for inviting me to the University of Waterloo, Canada, which provided a
peaceful working environment. I started this book in September 1998 during
my stay in Waterloo.
I thank Mikl6s Laczkovich, the referee of my C.Sc. dissertation very
much for his remarks and suggestions.
Between 1974 and 1997 I worked at Kossuth Lajos University, Debrecen,
Hungary. Naturally, I am grateful to all members of the Debrecen School of
functional equations.
Finally, I would like to thank those of my colleagues, joint pieces of work
with whom in one way or another are contained in this book: JAnos Aczkl,
ZoltBn Darbczy, Roman Ger, Gyula Maksa, Zsolt PAles, Wolfgang Sander,
and LAsz16 Szitkelyhidi.
I thank my sons, Antal JArai Jr. and ZoltAn JBrai for correcting several
errors of my "Hunglish"; all the remaining errors are mine.
Finally, I wish to express my gratitude to Kluwer Academic Publishers,
in particular, Professor Jen6 Sz6p and John C. Martindale for the first rate
and patient technical help.
Financial support for this book was provided mainly by Szitchenyi Schol-
arship for Professors, Hungary and partly by OTKA TO31995 grant.
Budapest, May 25, 2003.
Antal JBrai
Department of Computer Algebra
Eotvos LorAnd University
PBzmAny Pitter sittAny 1/C
H-1117 Budapest, Hungary
e-mail: [email protected]
http://compalg.inf.elte. hu/-ajarai
Chapter I.
PRELIMINARIES
In this chapter, starting with simple examples, we describe the problems
with which we will deal in this book. We also present simple examples of
our methods. First we formulate the fundamental problem, then analyse its
conditions and explore its applicability. We then formulate theorems that
follow from our results as corollaries to that fundamental problem. Then we
survey possibilities for generalization. We close this chapter by summarizing
our notation and terminology, including the formulation of theorems not
readily available in the literature or usually formulated in a different way.
1. INTRODUCTION
1.1. General considerations and simple examples. As a first, illus-
trative example let us consider the best-known functional equation, Cauchy's
equation
with unknown function f. In a wider sense differential equations, integral
equations, variational problems, etc. are also functional equations, but here
we will use this expression in a more restrictive sense for functional equations
2 Chapter I. Preliminaries
without infinitesimal operations such as integration and differentiation. For a
more formal definition, see Aczd [3], 0.1. To formulate a functional equation
exactly we have to give the set of functions in which we look for solutions.
We also have to give the domain of the functional equation. In the above
example this is the set of the pairs (x,y ) of the variables x and y for which
equality has to be satisfied. For example, we may look for all measurable
functions f : R -+ R such that (1) is satisfied for all (x, y) E [0, m[ x R.
Conditions such as measurability, Baire property, continuity everywhere or
in a point, boundedness, differentiability, analyticity, etc. are called regularity
conditions. If this kind of conditions are imposed on the solution, then we
say that we look for regular solutions. Otherwise, if we look for solutions
among all maps from a given set into another given set, then we say that we
look for the general solution of the functional equation.
Usually, the domain of the functional equation is the set of all tuples of
the variables for which both sides are defined. For example, if we say that
f : R -+ R is a solution of Cauchy's functional equation, then it is implicitly
understood that (1) is satisfied for all (x, y) E R x R. If the domain of the
equation is not the largest possible for which both sides are defined, then
we speak about an equation with restricted domain; the term conditional
equation is also used, especially if the domain of the equation also depends
on the solution or solutions.
Cauchy's equation is a functional equation with two variables; the vari-
ables denoted by x and y in (1). Equations like f (x) = f (-x), f (x) =
-f (-x), f (22) = f (x)~o,r difference equations are called functional equa-
tions in a single variable. The "single variable" may also be a vector variable;
it is understood that there are no more variables in the equation than the
number of places in the unknown function or the minimal number of places
in the unknown functions if there is more than one. Otherwise we speak
-
about a functional equation in several variables. This distinction is very use-
ful in practice. There is a large difference between functional equations with
a single variable and several variables: the methods used in the two cases
are quite different. In this book we deal with functional equations in several
variables. About equations in a single variable see the books Kuczma [I261
and Kuczma, Choczewski, Ger [128].
The distinction between functional equations in a single variable and in
several variables and what we have said about variables, domain, regular and
general solutions also apply to systems of functional equations.
Further simple examples of functional equations are Cauchy's exponen-