Functional Equations in Several Variables

ENCYCLOPAEDIA OF MATHEMATICS AND ITS APPLICATIONS

EDITED BY G.-C. ROTA

Volume 31

Functional equations in several variables

ENCYCLOPAEDIA OF MATHEMATICS AND ITS APPLICATIONS

Functional equations in several variables with applications to mathematics, information theory and to the natural and social sciences

J. A C Z E L University of Waterloo, Ontario, Canada J. D H O M B R E S Universite de Nantes, France

The right of the University of Cambridge to print and sell all manner of books was granted by Henry VIII in 1534. The University has printed and published continuously since 1584.

CAMBRIDGE UNIVERSITY PRESS Cambridge New York New Rochelle Melbourne Sydney

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521352765 © Cambridge University Press 1989 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 Reprinted 1991 This digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Functional equations in several variables J. Aczel and J. Dhombres. p. cm. - (Encyclopedia of mathematics and its applications) Bibliography: p. Includes indexes. ISBN 0 521 35276 2 1. Functional equations. 2. Functions of several real variables. I. Dhombres, Jean G. II. Title. III. Series. QA431.A334 1989 515.8'4--dcl9 87-38107 CIP ISBN 978-0-521-35276-5 hardback ISBN 978-0-521-06389-0 paperback

CONTENTS

Preface Further information 1

Axiomatic motivation of vector addition Exercises and further results

page

ix xiv 1 8

2 Cauchy's equation. Hamel basis 2.1 General considerations, extensions, and regular solutions 2.2 General solutions Exercises and further results

11 11 18 22

3

Three further Cauchy equations. An application to information theory Exercises and further results

25 31

4

Generalizations of Cauchy's equations to several multiplace vector and matrix functions. An application to geometric objects Multiplace and vector functions A matrix functional equation and a characterization of densities in the theory of geometric objects Pexider equations Cauchy-type equations on semigroups Exercises and further results

38 42 46 48

Cauchy's equations for complex functions. Applications to harmonic analysis and to information measures Cauchy's equation and the exponential equation for complex functions Endomorphisms of the real and complex fields Bohr groups Recursive entropies Exercises and further results

52 52 57 60 66 69

Conditional Cauchy equations. An application to geometry and a characterization of the Heaviside functions Exercises and further results

73 82

4.1 4.2 4.3 4.4 5 5.1 5.2 5.3 5.4 6

34 34

vi

Contents

7

Addundancy, extensions, quasi-extensions and extensions almost everywhere. Applications to harmonic analysis and to rational decision making 7.1 Extensions and quasi-extensions 7.2 Extensions almost everywhere and integral transforms 7.3 Consensus allocations Exercises and further results 8

84 84 92 99 101

D'AIembert's functional equation. An application to noneuclidean mechanics Exercises and further results

103 111

Images of sets and functional equations. Applications to relativity theory and to additive functions bounded on particular sets 9.1 Equations containing images of sets and chronogeometry 9.2 Sets on which bounded additive functions are continuous Exercises and further results

114 114 121 127

9

10 Some applications of functional equations in functional analysis, in the geometry of Banach spaces and in valuation theory 10.1 Functional equations and extreme points 10.2 Totally monotonic functions and extreme rays 10.3 A characterization of strictly convex normed spaces 10.4 Isometries in real normed spaces 10.5 A topology on the set of all solutions of a functional equation: the Bohr group 10.6 Valuations on the fields of rational and of real numbers Exercises and further results

129 129 135 138 142 148 156 161

11 Characterizations of inner product spaces. An application to gas dynamics 11.1 Quadratic functional: a characterization of inner product space 11.2 Triangles in normed spaces: a second characterization of inner product spaces 11.3 Orthogonal additivity 11.4 An application to gas dynamics Exercises and further results

165 165 178 185 191 194

12 Some related equations and systems of equations. Applications to combinatorics and Markov processes Exercises and further results

201 207

13 Equations for trigonometric and similar functions Exercises and further results

209 225

14 A class of equations generalizing d'AIembert and Cauchy Pexider-type equations Exercises and further results

228 237

15 A further generalization of Pexider's equation. A uniqueness theorem. An application to mean values Exercises and further results

240 250

Contents

vii

16 More about conditional Cauchy equations. Applications to additive number theoretical functions and to coding theory 16.1 Expansions of the Cauchy equation from curves 16.2 Cylindrical conditions 16.3 Additive number theoretical functions and related equations 16.4 An application to mean codeword lengths 16.5 Totally additive number theoretical functions and their generalizations 16.6 Further equations for number theoretical functions Exercises and further results

274 281 283

17 Mean values, mediality and self-distributivity Exercises and further results

287 296

18

Generalized mediality. Connection to webs and nomograms Exercises and further results

298 307

19 Further composite equations. An application to averaging theory 19.1 One-parameter subgroups of affine groups 19.2 Another example of determining one-parameter subgroups 19.3 Two more composite equations 19.4 Reynolds and averaging operators 19.5 Interpolating and extension operators 19.6 Derivation operators Exercises and further results

309 311 319 326 330 334 337 340

20 Homogeneity and some generalizations. Applications to economics Exercises and further results

345 352

21 Historical notes 21.1 Definition of linear and quadratic functions by functional equations in the Middle Ages and application of an implied characterization by Galileo 21.2 The functional equations of the logarithm and of the exponential function 21.3 Some functional equations in the works of Euler 21.4 Functional equations arising from physics 21.5 The binomial theorem and Cauchy's equations 21.6 Cauchy equations after Cauchy 21.7 Further equations 21.8 Recent developments

355

360 362 363 365 371 373 377

Notations and symbols

379

Hints to selected 'exercises and further results'

382

Bibliography

388

Author index

449

Subject index

458

254 254 261 265 267

355

Dedicated to Beckie, Cathy, Julie, Pascale, Robbie, Sylvestre and Thomas

PREFACE

Functional equations are equations in which the unknown (or unknowns) are functions (but we shall not cover such a large domain). As the title shows, we deal in this book with 'functional equations in several variables'. This does not mean that we consider only equations in which the unknown functions are of several variables (in other words, multiplace functions) but, rather, that there are several variables in the equation. Cauchy's equation,

containing two variables x, y and one unknown function / of one variable, is a classical example. On the other hand, iterative equations, differential, difference-differential, difference, integral and similar equations contain (in most cases) just one variable, if the unknown function is of one variable (a one-place function). We will deal here with functional equations in which the number of variables is greater than the number of places in the unknown function (or, if there is more than one unknown function, greater than the number of places in the unknown function with the smallest number of places). There are several thousand works on functional equations, even in this restricted sense, and it is impossible to summarize their contents in a book of this size. So we have tried to make the bibliography encyclopaedic (although not complete) - it is organized by years. The reader is encouraged to refer to as many of the works listed in the bibliography as possible. (Some are quite elementary.) An amazing number of rediscoveries can be spotted in this way. We felt it might be useful to the reader for us to include a historical chapter at the end of the book (Chapter 21). This chapter can be read first in order to get a bird's eye view (which makes a longer introduction unnecessary), ignoring or checking the few references to other sections in it. But, also, we think that this last chapter can be read again with profit after the first twenty chapters have been studied.

x

Preface

From this historical chapter, the reader will see that a somewhat general theory of functional equations in several variables is a relatively recent development. It can also be seen that, from their very beginnings, functional equations arose from applications, were developed mostly for the sake of applications and, indeed, were applied quite intensively as soon as they were developed. Such a course of development is not typical in all theoretical aspects of mathematics. A characteristic example of this process can be found in the first, introductory, chapter which is devoted to the parallelogram law for vector addition. This brings us to the natural question of motivation: To what kind of problems can functional equations be applied (and not only within mathematics). G.H. Hardy said that mathematicians are 'makers of patterns'. They often construct new notions, partly based on older ones, which may come from mathematics or from the natural, behavioural and social sciences, for instance from physics or economics. Thus the notion of affine vectors evolved in order to represent forces, the (homogeneous) linear function to describe proportionality, the logarithmic function to transform geometric sequences into algebraic ones, and the trigonometric functions to determine unknown parts of triangles from known ones. The next task of the mathematician is to describe the properties of these new objects, that is, to establish their relation to other objects in order to include them in a systematic and orderly way into an existing or new theory. (This is important because it is convenient to organize the notions in a pattern ruled by the same structure, with unity in the style of proofs.) The vectors, for instance, form the foundation of linear algebra, the trigonometric functions that of trigonometry. After stating the properties of the new objects (or at least some of them), what next? If we are lucky, the properties deduced from the definitions of mathematical objects can be quite numerous, even too rich. According to the principle of economy of reasoning and to a legitimate desire for elegance, it is desirable to check whether the newly introduced objects are the only ones which have some of the most important properties that we have just established. It is here where functional equations may enter. One takes these properties as points of departure and tries to determine all objects satisfying them and, in particular, to find conditions under which there is unicity. This is done in the framework of the classical structures of mathematics, be they algebraic, order-theoretic, topological, or simply the real axis which carries all these structures. This is the fundamental procedure which motivates functional equations. We see that it is axiomatic in nature. But we see also that the axioms are not arbitrarily chosen, we do not generalize just for the sake of generalization. In fact, one prefers to take as points of departure those which are considered the most useful for applications. (Also practical concerns determine the procedure since we want to attain uniqueness.) Thus there is a feedback from the applications to the theory.

Preface

xi

This is not all. After such a uniqueness result is established, for instance that concerning the equation of the cosine (d'Alembert's equation), one also tries to deduce (using the equation, not the solutions) the classical properties of the cosine directly. One thus arrives at an elegant way of developing trigonometry which, with some changes, can also be used to introduce and develop elliptic and, especially, hyperbolic trigonometry. This procedure was followed (with varying degrees of clarity) by mathematicians in different ages, such as Oresme, Euler, d'Alembert, Abel, Cauchy, Lobachevskii, Darboux, Picard, etc. Another famous example, outside the framework of our book, is the introduction of the gamma function through its functional equation, as developed by Artin. As in all parts of mathematics, different authors' viewpoints and levels of discussion of functional equations are different. Also, sometimes the methods of reduction of one functional equation to another or the proof of their equivalence or discussion and/or reduction of regularity conditions, extension of domains, etc. give rise to further intrinsic developments. This is a further sign of the increasing maturity of this field of mathematics. On the other hand, the characterization of functions by their equations, as described above, involves many branches of pure and applied mathematics in the development of the theory of functional equations. Our general intention in this book is to give the reader at least a general impression of what this subject is about by focusing on a relatively small number of examples, chosen with particular regard to applications but without neglecting theory. This explains why most technical connections among the different chapters are minimal, so that every chapter may constitute a unit of study in itself. Indeed, the reader can start with any chapter (and even many sections) with only occasional backward references. (Here the subject index at the end of the book will prove helpful.) An obvious exception is Chapter 2 which deals with Cauchy's equation and is intended to provide the reader with a certain number of definitions, techniques and results which will be used throughout the book (such as domain, extension of solutions, general solutions, regular solutions, etc.). However, from a more general mathematical point of view, connections between chapters can be found, and such connections can be used, for instance in both undergraduate and graduate courses or seminars. So can the several chapters and sections (or parts of them) devoted to connections with other parts of mathematics and to applications in and outside of mathematics. We give some examples below: Combinatorics, probability and information theory 3; 5.4; 12; 16.4 Economics, decision making and mean values 7.3; 15; 17; 20 Almost periodic functions and harmonic analysis 5.3; 7.2; 8; 10.5

xii

Preface

Functional analysis, operator theory and Heaviside functions Geometry, nomography and physics

6; 10; 11; 19.4; 19.5; 19.6 1; 4.2; 5.2; 6; 8; 9.1; 11.4; 18; 19.4 Groups, groupoids and semigroups 4.4; 5.3; 16.2; 16.3; 19.1; 19.2; 19.3 Trigonometric functions and number theory 1; 8; 13; 14; 16.3; 16.4; 16.5; 16.6. (There are several more items related to these and other applications in the 'exercises and further results' at the end of different chapters.) We also indicate some longer sequences which can be used for individual courses. The fundamental Chapter 2 should be added to each of them, and, clearly, some paragraphs of the following chapters and sections can be omitted if they do not fit into the subject of the course: - 3; 4.1; 4.2; 4.4; 9.1; 11.2; 16.3; 16.5; 16.6; 19.1; 19.2; 20 - 5.1; 5.2; 5.3; 8; 10.5; 12; 13; 14 - 5.2; 6; 9.2; 11.3; 11.4; 14; 16.3; 16.4; 16.5; 16.6 - 10.4; 15; 16.4; 17; 18; 19.3 - 3 ; 6; 7; 10.2; 10.3; 11.2; 16.1; 16.2 -5.1; 5.4; 6; 7.2; 10.1; 10.2; 19.4; 19.5; 19.6 - 1; 4.3; 4.4; 5.2; 7.2; 8; 11.1; 12; 13; 15; 16.2; 17; 18; 19.1; 19.2. The pedagogical purpose of the book and the wish to at least indicate some further results and directions of research has led us to include more than 400 items as 'exercises and further results'. Some can be solved by a direct application of the material explained in the chapter in which the exercise appears; these exercises serve the usual aims of practice and of further applications of the results and proofs. Others are 'further results': extensions of the results described in the text to more general (algebraic, topological) settings and (more or less) related theorems. We sometimes give details of such generalizations and further results. Even where we do not generalize, the reader may want to try to prove such generalizations. Some 'exercises and further results' go deeper in the theory (and the most difficult ones are marked with a * ). For some selected exercises and results we provide the reader with hints (which we hope won't become hindrances!), but we do not give hints to exercises which we consider to be easy or to results which are preceded by easily accessible references. In a large number of cases, these references are to the original contributions to the solution of the problem; in some other cases we preferred to give more modern or more easily accessible literature. On the other hand, in the main text, we have tried to be accurate concerning priorities, except for general background references for which we again wanted to quote easily accessible sources.

Preface

xiii

Now we have to say a few words about the prerequisites which are necessary for a fruitful use of the book. We do try to avoid all unnecessary complications by limiting ourselves to simpler cases and we think that most chapters in the book can be read at the sophomore level, after mastering calculus, general and linear algebra, and perhaps basic Lebesgue theory of integration in 1R or Un. Incidentally, a previous monograph by one of the authors (Aczel 1966c) provides a more elementary approach to some parts of this book and also some further topics. Some applications require more specialized knowledge which we indicate there (some Banach algebra techniques for example, duality in functional analysis, and some measure theory). We try to explain at least the results we use. But, generally, no previous knowledge of the field to which functional equations are applied is required. This is true, for instance, in information theory, number theory, mean values, consensus allocations, geometric objects, almost periodic functions, relativity theory, etc. While we have tried to 'homogenize' the presentation to some degree, the attentive reader will notice which parts were written by which author, due to their different styles (and mother tongues). Hopefully, this is not too terrible. Acknowledgements Che Tat Ng has suggested several essential improvements and shortcuts. Gord Sinnamon helped checking the galley proofs. Richard Atkins and Gord Sinnamon helped finishing the subject index. Susan Aczel has compiled the author index and worked long and hard to put the bibliography together. Linda Gregory and Brenda Law have typed (in several versions), Dong Nguyen, Richard Atkins, Nicole Brillouet-Belluot and, last but not least, Bert Schweizer have read and corrected the manuscript and suggested improvements. Our warmest thanks are due to all of them and to colleagues, including anonymous referees, who have helped us with their comments. Support by the Natural Sciences and Engineering Research Council of Canada and the France-Canada Scientific Cooperation Agreement is gratefully acknowledged. We express also our thanks to the publishing and printing houses for their conscientious work. Let us add that we welcome comments on the book or on any aspect of the subject. Janos Aczel Jean Dhombres

FURTHER INFORMATION

References, numbering. The references in the bibliography are listed by years, within the same year alphabetically according to the authors' names, and works of the same author(s) in the same year are distinguished by letters. References are quoted in the text with the authors' names, the years, and, where necessary, with distinguishing letters. Theorems, propositions, lemmata, and corollaries are numbered consecutively in each chapter (so that Lemma 1 may be followed by Theorem 2 and that by Corollary 3) and formulas are separately numbered, also consecutively, within the chapters. In the same chapter they are referred to by these numbers, while in the references to another chapter the number of that chapter is attached, for example Theorem 2.3 or formula (3.5). (Some longer chapters are subdivided into sections but numbering of formulas, etc., is within chapters, not sections.) As usual, easier theorems are called 'Propositions'. Exercises and further results at the end of the chapters are numbered (separately) and quoted in a similar way. Those with stars (*) are thought (by the authors) to be more difficult. An Author Index and a Subject Index are, as usual, at the end. Notations are mostly standard, but we also summarize less standard notations at the end of the book.

Axiomatic motivation of vector addition

In this book we try to emphasize applications as motivation for functional equations. Of course we cannot present them in all exact details (as we try to do in the treatment of functional equations themselves). This first chapter is meant to give such a motivation for the Cauchy and d'Alembert equations. The reader who needs no convincing of their importance (or has any difficulty with Chapter 1) may proceed directly to Chapter 2. One of the first problems to which functional equations were ever applied, and for whose sake functional equations were first solved, was the so called 'parallelogram of forces' or, in more modern language, the axiomatic motivation of the customary rule for addition of vectors (d'Alembert 1769; Poisson 1804, 1811; Picard 1928, pp. 4-17; Aczel 1966c, pp. 120-4, 1969a, pp. 7-11,1976a are a few relevant references). Here we give a somewhat more satisfactory treatment, based on slightly weaker assumptions. We start with the space Fof all 3-tuples p = (xl9x29x3)9 xteR. This space is equipped with the euclidean distance dip, q) - ( f o - y, )2 + (x2 - y2)2 + (x3 -

y,)2)1'2

between p and q = (yl9 y2, y3). Our purpose is to characterize the additive law of such 3-tuples by geometric conditions. Our fundamental tools will then be rotations on V, which are bijective transformations from Konto itself preserving 0 = (0,0,0) and distances in V. We recall that a rotation is completely determined by an axis through 0 and an angle, which is a real number modulo 2n. Given any two 3-tuples, p and q, having the same positive distance from the origin, there exists one and only one rotation mapping p into q. The angle of this rotation, say ©, is the angle of the two 3-tuples, which can be determined by the analytic formula cos © = —

2

Axiomatic motivation of vector addition

where 0e[O,7r] or 0e[7c,27i] according to the orientation chosen on V (for example by choosing (1,0,0), (0,1,0) and (0,0,1) to be a directed basis). The first will be called the smaller, the second the larger angle of the two vectors. For the sake of simplicity, we call the 3-tuple P = (*i,x 2 ,x 3 ) the vector p. We say that two vectors p = (x1?X2,X3) and q = (yl9y 2^3) have the same (or opposite) direction if there exist real numbers k, \i of the same (respectively, opposite) sign such that ^i = fiyi9

i = 1,2,3.

We allow that one (but not both) of k and fi be 0. For convenience we call p and q of the same direction in this case too. Naturally, the length of a vector p is its euclidean norm With this in mind, we can state our theorem: Theorem 1. //, and only if, a binary operation ° on the set V of all vectors in the oriented three dimensional euclidean space (i) is invariant under (really, covariant with) the rotations of the space, that is, the result of the operation (the 'resultant vector) undergoes the same rotation as the two factors ('components'), (ii) is commutative and associative, (iii) for two vectors of the same direction, ° reduces to arithmetic addition (the resultant vector also points in that direction and its length is the sum of the lengths of the components), and (iv) the resultant of two vectors of equal length depends continuously upon their angle, then the operation ° is the usual vector addition (the resultant is obtained by the 'parallelogram rule). Remark. Strictly speaking, suppositions like (i) and (iii) are themselves functional equations. However, for this introductory chapter we prefer the above verbal descriptions. Proof The most important consequence of (i) and of the commutativity (ii) is that the resultant of two vectors of equal lengths lies in the plane spanned by them, more exactly, in the direction of the bisector of their angle (using this direction as an axis for a rotation). Drawn from their intersection, the resultant could still lie either inside the smaller or the larger angle formed by the two vectors. Condition (iii) assures that the zero-vector is a unit under °(p°0=0°p = p),

Axiomatic motivation of vector addition

3

and then again (i) shows that in (K, °) a vector p and a vector — p of the same length but opposite direction are inverses: ( —p)°p = p°( —p) = 0. Thus, with (ii), we have (v) (V, °) is an abelian group with 0 as unit and — p as the inverse of p. By (iii), the resultant of two vectors of the same length and of the same direction (angle 0) lies in the same direction, that is, in the direction of the bisector of their smaller angle (0). If there existed an angle of two vectors of the same length for which the resultant would lie in the bisector of their larger angle (Figure 1) then, by the continuity (iv), there would exist an angle, smaller than n9 under which two vectors of the same length would have the zero-vector as resultant. But these would then be inverses to each other in (F,°), while we have just seen that the inverse vectors always form angles n. (By (v) there exists just one inverse.) This contradiction shows that the resultant of two vectors of equal length always lies inside their smaller angle (and is the zero-vector if and only if the two vectors of equal and non-zero length have the angle n). So we have determined the direction (in the bisector of their smaller angle) of the resultant of two vectors of equal length x. Keeping their angle fixed,

Fig. 1

Axiomatic motivation of vector addition

we now determine its length, which we will denote by g(x). This function g is defined on the set of all nonnegative numbers and is, in view of the above, nonnegative. Also (see Figure 2), if the vectors p l5 p2 have the same direction, and also the vectors q l5 q 2 , and (|p| denoting the length of the vector p) then, by (iii), IPi°P2l = lqioq2l = * + }; and, in view of the definition of g, l*iI = IPI°qiI = g(x)> \*i\ = lP2°q21 = Q(y\

l(Pi°p2)°(qi°q2)l = o(x + y). o

But also, by (ii) and (iii), since (Pi°qi) and (p2 q2) a r e collinear, l(PioP2)o(qi°q2)l = l(Pioqi)o(P2oq2)l = ^W + ^ X

(i)

g(x + y) = g(x) + g(y) for all nonnegative x, y.

(2)

and so As mentioned in the introduction, this is Cauchy's equation (Cauchy 1821). Furthermore, as pointed out above, g(x) ^ 0

for all nonnegative x.

(3)

We will prove in Chapter 2 that (2) and (3) can hold if, and only if, there exists a nonnegative constant c such that g(x) = ex (x ^ 0). The constant c has to be nonnegative, but, by (v), it cannot be 0 except when the two vectors of equal length have opposite directions. This shows that the length of the resultant of two vectors of equal length and of angle different from n is proportional to the length of the components. Here we have kept the angle of the two vectors constant. If we remove this restriction, then c will depend upon that angle. We now determine the dependence of the length of the resultant of two vectors of equal length on their angle (/TC). Because of what we have just proved, it is enough to take two vectors of unit length ('unit vectors'). Denote their angle, for convenience, by 2(j> and the length of their resultant by 2/(0) (this is really our c, now dependent upon (/>). Take four unit vectors (Figure 3) p 1 ,p 2 ,qi,q 2 (IPi I = |p2l = l«i I = I«2I = *) with angles (Pi,p 2 )* =2ifr = ( q 1 , q 2 ) * , (Pi,qi)* = 2(0 + R of f(x + y) + f(x-y) = 2f(x)cosy (x,yeU) is given by f(x) —tfcosx + bsinx (xeR), where a,b are arbitrary real constants. (As usual, the set of all real numbers is denoted by R.) 2. Let £ be a euclidean space of dimension n>2. Let f:ExE->R be invariant under any linear unitary transformation T of E (Tx = Ax where A is a matrix with determinant 1), that is, f(Tx, Ty) — f(x,y) for x,y in E. Suppose also that / is linear in the first variable [/(ax + fiz, y) = a/(x, y) + /?/(z, y) for all a,/?eR, x,y,zeE]. Show that / is, up to a scalar multiple, the scalar product on E. 3. Let R3 be the usual euclidean space with three coordinates. Let /:R 3 x R3->R3 be such that, for any rotation T, f(Tx, Ty) = T(f(x,y)); x,yeU3. Suppose that / is linear in the first variable. Show that / is, up to a scalar multiple, the vector product on R3. 4. (Aczel 1966c, pp. 223-4.) Find the general solution of the functional equation where x,y and z belong to a nonempty set S and F:S x S~>G, where (G, + ) is a group. 5. Find the general solution F:R 2 -»R of the functional equation F(x, y)F(y, z) = F(x, z), (x, y, ze R). 6. (Moszner 1965.) Find the general solution F: R2 -» R of the equation in Exercise 5 if we suppose only that the equation in the previous exercise holds for ordered triples x < y ^ z. 7. Show that, among / : R -> R, only constant functions satisfy f

\ l )

=

^

x +

^ ^

x

" ^

(x,yeR).

8. For any XER, one defines the (leibnizian) dilogarithm by ^——X-dt

.i(x)=Jo

f

and, for x T^O, x # 1, Show that the following functional equation holds for x # 1, y ^ 1: M(xy) = M(x) + M(y) + M f - ^ - ( l - j \x-l + M|-^—(1 - x) j - 2A(x + >> + xy)M(2), where A is the characteristic function of [1, oo[ (A(x) = 1 if x > 1, X(x) = 0 if x < 1). 9. (Aczel 1956Z?; Samuelson 1974; Christian 1983.) By forming the difference quotient and using the continuity and convexity of ax, show that the derivative of a* exists

10

Axiomatic motivation of vector addition

everywhere and is proportional to ax: dax ~ = axl(a). dx Give a geometric motivation to /(a') = tl(a) for all real numbers t. Show that there exists an exponential function whose derivative at 0 is 1. Call its base e. Then show that l(s) — loge s = In s for all s > 0. (The above can serve as a definition of the natural logarithm.) 10. (Aczel 1966c, p.26.) Determine all solutions f:R->R of /(*•+ y) + fix - y) = f(x){y + 2) - y(x2 - 2y). 11. Suppose that /:R+-*R + (R + is the set of positive numbers) is continuous, strictly decreasing and satisfies /(x + y) + /(/(x) + /(y)) = / [ / ( x + /(y)) + /(y + /(x))] for all xyyeU + . Prove that /(x) = / - 1 ( x ) for all xeU+ (f~l is the inverse function of/).

Cauchy's equation. Hamel basis

The purpose of this chapter is to introduce the main notions for functional equations: domain of the equation, solution and extension of a solution, regularity conditions, and the general solution of a functional equation. We use the example of the Cauchy equation on the real axis of which we give the solutions under different regularity conditions, and also the general solution.

2.1 General considerations, extensions, and regular solutions We first consider Cauchy's equation (Cauchy 1821): f(x + y) = f(x) + f(y) forall(x,y)eU2.

(1)

(As usual, R is the set of all reals, R+ the set of positive numbers, and iR+ that of all nonnegative numbers, all equipped with the usual topology; and, for any set 5, the cartesian square is 52:= {{x,y)\xeS,yeS}.) If we compare (1) to (1.2) we notice a difference. There the equation was supposed to hold only for nonnegative x, y\ here for all real x, y. The set of all values of the variables, on which the equation is supposed to hold, is called the domain of the equation (not to be confused with the domain(s) of the unknown function(s); the domain of (1) is IR2, while the domain of / is R). A function satisfying a functional equation on a given domain is called a solution of the equation on that domain. It is sometimes (but not always, see equations (3.1), (3.18), and Chapter 7) possible to extend a solution of an equation from a more restricted domain to a wider one. For (1.2), that is, for 2 (2) Q(X + y) = g(X) + g{y) for all (x, y)e R + , this means the following (cf. Aczel-Erdos 1965; Aczel-Baker-DjokovicKannappan-Rado 1971; Grz^slewicz-Sikorski, 1979, for general algebraic structures):

12

Cauchy's equation. Hamel basis

Theorem 1. To every solution g:U+-+Mof(2) there exists a solution of (1), for which f(x) = g(x) forallxeU

f:U-*U

+.

(3)

Proof. We construct this / , that is, we extend g to all of U by the definition f(s-t)

= g(s)-g(t)

forall(s,f)eR 2 + .

(4)

This definition is unambiguous (and so is indeed a definition) because, whenever s — t = u — v, where s,f,w and v are in R+, that is, whenever s + u = M + r, then, by (2), g(s) + g(v) = g(s + v) = #(w + t) =