This treatise deals with modern theory of functional equations in several variables and their applications to mathematics, information theory, and the natural, behavioural and social sciences. The authors have chosen to emphasize applications, though not at the expense of theory, so they have kept the prerequisites to a minimum; the reader need be familiar only with calculus and elementary algebra, and have a basic knowledge of Lebesgue integration. Where, for certain applications, more advanced topics are needed, the authors have included references and explained the results used. Moreover, the book has been designed so that the chapters can be read almost independently of each other, enabling a selection of material to be chosen for introductory and advanced courses. At the end of each chapter are included exercises and further results, some 400 in all, which extend the material presented in the text and also test it. The history of functional equations is well documented in a final chapter which is complemented by an encyclopedic bibliography running to over 1600 items.

### Contents

Preface; Further information; 1. Axiomatic motivation of vector addition; 2. Cauchy's equation: Hamel basis; 3. Three further Cauchy equations: an application to information theory; 4. Generalizations of Cauchy's equations to several multiplace vector and matrix functions: an application to geometric objects; 5. Cauchy's equations for complex functions: applications to harmonic analysis and to information measures; 6. Conditional Cauchy equations: an application to geometry and a characterization of the Heaviside functions; 7. Addundancy, extensions, quasi-extensions and extensions almost everywhere: applications to harmonic analysis and to rational decision making; 8. D'Alembert's functional equation: an application to noneuclidean mechanics; 9. Images of sets and functional equations: applications to relativity theory and to additive functions bounded on particular sets; 10. Some applications of functional equations in functional analysis, in the geometry of Banach spaces and in valauation theory; 11. Characterizations of inner product spaces: an application to gas dynamics; 12. Some related equations and systems of equations: applications to combinatorics and Markov processes; 13. Equations for trigonometric and similar functions; 14. A class of equations generalizing d'Alembert and Cauchy Pexider-type equations; 15. A further generalization of Pexider's equation: a uniqueness theorem: an application to mean values; 16. More about conditional Cauchy equations: applications to additive number theoretical functions and to coding theory; 17. Mean values, mediality and self-distributivity; 18. Generalized mediality: connection to webs and nomograms; 19. Further composite equations: an application to averaging theory; 20. Homogeneity and some generalizations: applications to economics; 21. Historical notes; Notations and symbols; Hints to selected 'exercises and further results'; Bibliography; Author index; Subject index.