Functional Equations in Several Variables

Cover......Page 1
ENCYCLOPAEDIA OF MATHEMATICS AND ITS APPLICATIONS......Page 2
Title......Page 4
Copyright......Page 5
CONTENTS......Page 6
Dedication......Page 9
PREFACE......Page 10
FURTHER INFORMATION......Page 15
1 Axiomatic motivation of vector addition......Page 16
Exercises and further results......Page 23
2.1 General considerations, extensions, and regular solutions......Page 26
2.2 General solutions......Page 33
Exercises and further results......Page 37
3 Three further Cauchy equations. An application to information theory......Page 40
Exercises and further results......Page 46
4.1 Multiplace and vector functions......Page 49
4.2 A matrix functional equation and a characterization of densities in the theory of geometric objects......Page 53
4.3 Pexider equations......Page 57
4.4 Cauchy-type equations on semigroups......Page 61
Exercises and further results......Page 63
5.1 Cauchy\'s equation and the exponential equation for complex functions......Page 67
5.2 Endomorphisms of the real and complex fields......Page 72
5.3 Bohr groups......Page 75
5.4 Recursive entropies......Page 81
Exercises and further results......Page 84
6 Conditional Cauchy equations. An application to geometry and a characterization of the Heaviside function......Page 88
Exercises and further results......Page 97
7.1 Extensions and quasi-extensions......Page 99
7.2 Extensions almost everywhere and integral transforms......Page 107
7.3 Consensus allocations......Page 114
Exercises and further results......Page 116
8 D\'Alembert\'s functional equation. An application to noneuclidean mechanics......Page 118
Exercises and further results......Page 126
9.1 Equations containing images of sets and chronogeometry......Page 129
9.2 Sets on which bounded additive functions are continuous......Page 136
Exercises and further results......Page 142
10.1 Functional equations and extreme points......Page 144
10.2 Totally monotonic functions and extreme rays......Page 150
103 A characterization of strictly convex normed spaces......Page 153
10.4 Isometries in real normed spaces......Page 157
10.5 A topology on the set of all solutions of a functional equation: the Bohr group......Page 163
10.6 Valuations on the fields of rational and of real numbers......Page 171
Exercises and further results......Page 176
11.1 Quadratic functional: a characterization of inner product spaces......Page 180
11.2 Triangles in normed spaces: a second characterization of inner product spaces......Page 193
11.3 Orthogonal additivity......Page 200
11.4 An application to gas dynamics......Page 206
Exercises and further results......Page 209
12 Some related equations and systems of equations. Applications to combinatorics and Markov processes......Page 216
Exercises and further results......Page 222
13 Equations for trigonometric and similar functions......Page 224
Exercises and further results......Page 240
14 A class of equations generalizing d\'Alembert and Cauchy Pexider-type equations......Page 243
Exercises and further results......Page 252
15 A further generalization of Pexider\'s equation. A uniqueness theorem. An application to mean values.......Page 255
Exercises and further results......Page 265
16.1 Expansions of the Cauchy equation from curves......Page 269
16.2 Cylindrical conditions......Page 276
16.3 Additive number theoretical functions and related equations......Page 280
16.4 An application to mean codeword lengths......Page 282
16.5 Totally additive number theoretical functions and their generalizations......Page 289
16.6 Further equations for number theoretical functions......Page 296
Exercises and further results......Page 298
17 Mean values, mediality and self-distributivity......Page 302
Exercises and further results......Page 311
18 Generalized mediality. Connection to webs and nomograms......Page 313
Exercises and further results......Page 322
19 Further composite equations. An application to averaging theory......Page 324
19.1 One-parameter subgroups of affine groups......Page 326
19.2 Another example of determining one-parameter subgroups......Page 334
19.3 Two more composite equations......Page 341
19.4 Reynolds and averaging operators......Page 345
19.5 Interpolating and extension operators......Page 349
19.6 Derivation operators......Page 352
Exercises and further results......Page 355
20 Homogeneity and some generalizations. Application to  economics......Page 360
Exercises and further results......Page 367
21.1 Definition of linear and quadratic functions by functional equations in the Middle Ages and application of an implied characterization by Galileo......Page 370
21.2 The functional equations of the logarithm and of the exponential function......Page 375
21.3 Some functional equations in the works of Euler......Page 377
21.4 Functional equations arising from physics......Page 378
21.5 The binomial theorem and Cauchy\'s equations......Page 380
21.6 Cauchy equations after Cauchy......Page 386
21.7 Further equations......Page 388
21.8 Recent developments......Page 392
Notation and symbols......Page 394
Hints to selected exercises and further results......Page 397
BIBLIOGRAPHY......Page 403
AUTHOR INDEX......Page 464
SUBJECT INDEX......Page 473