Functional equations on groups - Stetkar, Henrik (World Scientific Publishing Company, 2013)
This volume provides an accessible and coherent introduction to some of the scientific progress on functional equations on groups in the last two decades. It presents the latest methods of treating the topic and contains new and transparent proofs. Its scope extends from the classical functional equations on the real line to those on groups, in particular, non-abelian groups. This volume presents, in careful detail, a number of illustrative examples like the cosine equation on the Heisenberg group and on the group SL(2, R). Some of the examples are not even seen in existing monographs. Thus, i
1. Introduction. 1.1. A first glimpse at functional equations. 1.2. Our basic philosophy. 1.3. Exercises. 1.4. Notes and remarks -- 2. Around the additive Cauchy equation. 2.1. The additive Cauchy equation. 2.2. Pexiderization. 2.3. Bi-additive maps. 2.4. The symmetrized additive Cauchy equation. 2.5. Exercises. 2.6. Notes and remarks -- 3. The multiplicative Cauchy equation. 3.1. Group characters. 3.2. Continuous characters on selected groups. 3.3. Linear independence of multiplicative functions. 3.4. The symmetrized multiplicative Cauchy equation. 3.5. Exercises. 3.6. Notes and remarks -- 4. Addition and subtraction formulas. 4.1. Introduction. 4.2. The sine addition formula. 4.3. A connection to function algebras. 4.4. The sine subtraction formula. 4.5. The cosine addition and subtraction formulas. 4.6. Exercises. 4.7. Notes and remarks -- 5. Levi-Civita's functional equation. 5.1. Introduction. 5.2. Structure of the solutions. 5.3. Regularity of the solutions. 5.4. Two special cases. 5.5. Exercises. 5.6. Notes and remarks -- 6. The symmetrized sine addition formula. 6.1. Introduction. 6.2. Key formulas and results. 6.3. The case of w being central. 6.4. The case of g being abelian. 6.5. The functional equation on a semigroup with an involution. 6.6. The equation on compact groups. 6.7. Notes and remarks -- 7. Equations with symmetric right hand side. 7.1. Discussion and results. 7.2. Exercises. 7.3. Notes and remarks -- 8. The Pre-d'Alembert functional equation. 8.1. Introduction. 8.2. Definitions and examples. 8.3. Key properties of solutions. 8.4. Abelian pre-d'Alembert functions. 8.5. When is a pre-d'Alembert function on a group abelian? 8.6. Translates of pre-d'Alembert functions. 8.7. Non-abelian pre-d'Alembert functions. 8.8. Davison's structure theorem. 8.9. Exercises. 8.10 Notes and remarks
9. D'Alembert's functional equation. 9.1. Introduction. 9.2. Examples of d'Alembert functions. 9.3. [symbol]d'Alembert functions. 9.4. Abelian d'Alembert functions. 9.5. Non-abelian d'Alembert functions. 9.6. Compact groups. 9.7. Exercises. 9.8. Notes and remarks -- 10. D'Alembert's long functional equation. 10.1. Introduction. 10.2. The structure of the solutions. 10.3. Relations to d'Alembert's equation. 10.4. Exercises. 10.5. Notes and remarks -- 11. Wilson's functional equation. 11.1. Introduction. 11.2. General properties of the solutions. 11.3. The abelian case. 11.4. Wilson functions when g is a d'Alembert function. 11.5. The case of a compact group. 11.6. Examples. 11.7. Generalizations of Wilson's functional equations. 11.8. A variant of Wilson's equation. 11.9. Exercises. 11.10. Notes and remarks --
- 12. Jensen's functional equation. 12.1. Introduction, definitions and set up. 12.2. Key formulas and relations. 12.3. On central solutions. 12.4. The solutions modulo the homomorphisms. 12.5. Examples. 12.6. Other ways of formulating Jensen's equation. 12.7. A Pexider-Jensen's functional equation. 12.8. A variant of Jensen's equation. 12.9. Exercises. 12.10. Notes and remarks -- 13. The quadratic functional equation. 13.1. Introduction. 13.2. The set up and definitions. 13.3. The case of [symbol]. 13.4. General considerations and the abelian case. 13.5. The Cauchy differences of solutions. 13.6. The classical quadratic functional equation. 13.7. Examples. 13.8. Exercises. 13.9. Notes and remarks -- 14. K-spherical functions. 14.1. Introduction and notation. 14.2. From where do K-spherical functions originate? 14.3. The Moroccan school. 14.4. Exercises. 14.5. Notes and remarks --
- 15. The sine functional equation. 15.1. Introduction. 15.2. General results about the solutions. 15.3. The sine equation on cyclic groups. 15.4. A more general functional equation. 15.5. Exercises. 15.6. Notes and remarks -- 16. The cocycle equation. 16.1. Introduction. 16.2. Compact groups. 16.3. Abelian groups. 16.4. Examples. 16.5. The cocycle equation on semidirect products. 16.6. Exercises. 16.7. Notes and remarks