The story we wish to tell is about the progress in the last 10–20 years of the theory of a number of trigonometric functional equations. We shall present recent results about these functional equations on groups, in particular nonabelian groups, reveal how they contain earlier results as special cases and apply them to important and interesting groups. Many of the earlier results deal with functions defined on R or more generally R n. But not all groups look like R n with its algebraic, topological and differentiable structures. Just think of the groups Z and SL(2, C). New, interesting and non-obvious phenomena make their appearance in the transition from R~n to other groups. The theory throws light on the properties of solutions of the functional equations that are valid on all groups or at least on all groups of a special type (like connected, solvable Lie groups). The table of contents gives a schematic survey of which functional equations we study. We illustrate the theory by thoroughly calculating explicit formulas for the solutions of the functional equations on concrete examples of such groups, like the Heisenberg group, the (ax + b)-group, SL(2, R) etc., which have not been mentioned, let alone discussed, in earlier monographs about functional equations.
We wish to tell our story to graduate students and professional mathematicians seeking an accessible and self-contained introduction to those functional equations that generalize classical relations between the trigonometric functions, like the sine addition formula. The present book is meant as
(1) a place where some of the new results of the last 10–20 years are validated and archived in an accessible way.
(2) a source where the reader can find references for further study and get an account of recent and current research in this area of mathematics.
(3) a springboard for research in the area.
(4) a textbook that can be used in a graduate course to give an easily read introduction to important functional equations on groups, because we want to induct a new generation into the recent developments.
Much of the material of the book has appeared in research journals, written for experts, but not in book form. Our exposition is detailed and requires little background material, so that the non-expert reader can follow the arguments. Special technical results are collected in appendices. Combining this with notes and remarks we hope that the book may serve the purposes mentioned above.
I collected the results of the present book, because they are interesting and beautiful extensions of classical results about functional equations and because they interconnect with parts of harmonic analysis. They combine concepts and results from algebra, analysis and topology. I confess that I have had fun playing around with various identities that extend classical ones, and that I appreciate the clever tricks, intricate computations and ingenious methods invented by the mathematicians of the field. Hopefully the reader will enjoy my choice of topics.

Preface v

Our story v

The organization of this book vi

Topics we do not discuss viii

Acknowledgements viii

1. Introduction 1

1.1 A first glimpse at functional equations 1

1.2 Our basic philosophy 3

1.3 Exercises 6

1.4 Notes and remarks 6

2. Around the Additive Cauchy Equation 7

2.1 The additive Cauchy equation 7

2.2 Pexiderization 13

2.3 Bi-additive maps 14

2.4 The symmetrized additive Cauchy equation 15

2.5 Exercises 18

2.6 Notes and remarks 26

3. The Multiplicative Cauchy Equation 29

3.1 Group characters 29

3.2 Continuous characters on selected groups 31

3.3 Linear independence of multiplicative functions 37

3.4 The symmetrized multiplicative Cauchy equation 39

3.5 Exercises 40

3.6 Notes and remarks 49

4. Addition and Subtraction Formulas 51

4.1 Introduction 51

4.2 The sine addition formula 52

4.3 A connection to function algebras 58

4.4 The sine subtraction formula 61

4.5 The cosine addition and subtraction formulas 64

4.6 Exercises 68

4.7 Notes and remarks 72

5. Levi-Civita’s Functional Equation 75

5.1 Introduction 75

5.2 Structure of the solutions 76

5.3 Regularity of the solutions 79

5.4 Two special cases 81

5.5 Exercises 84

5.6 Notes and remarks 90

6. The Symmetrized Sine Addition Formula 93

6.1 Introduction 93

6.2 Key formulas and results 94

6.3 The case of w being central 101

6.4 The case of g being abelian 101

6.5 The functional equation on a semigroup with an involution 105

6.6 The equation on compact groups 106

6.7 Notes and remarks 106

7. Equations with Symmetric Right Hand Side 107

7.1 Discussion and results 107

7.2 Exercises 109

7.3 Notes and remarks 109

8. The Pre-d’Alembert Functional Equation 111

8.1 Introduction 111

8.2 Definitions and examples 112

8.3 Key properties of solutions 113

8.4 Abelian pre-d’Alembert functions 118

8.5 When is a pre-d’Alembert function on a group abelian? 120

8.6 Translates of pre-d’Alembert functions 123

8.7 Non-abelian pre-d’Alembert functions 124

8.8 Davison’s structure theorem 128

8.9 Exercises 131

8.10 Notes and remarks 132

9. D’Alembert’s Functional Equation 135

9.1 Introduction 135

9.2 Examples of d’Alembert functions 137

9.3 µ-d’Alembert functions 144

9.4 Abelian d’Alembert functions 147

9.5 Non-abelian d’Alembert functions 151

9.6 Compact groups 152

9.7 Exercises 153

9.8 Notes and remarks 160

10. D’Alembert’s Long Functional Equation 165

10.1 Introduction 165

10.2 The structure of the solutions 166

10.3 Relations to d’Alembert’s equation 169

10.4 Exercises 175

10.5 Notes and remarks 175

11. Wilson’s Functional Equation 177

11.1 Introduction 177

11.2 General properties of the solutions 178

11.3 The abelian case 180

11.4 Wilson functions when g is a d’Alembert function 183

      11.4.1 The case of g non-abelian 183

      11.4.2 Discussion for g abelian 184

11.5 The case of a compact group 184

11.6 Examples 186

11.7 Generalizations of Wilson’s functional equations 190

11.8 A variant of Wilson’s equation 190

11.9 Exercises 194

11.10 Notes and remarks 197

12. Jensen’s Functional Equation 199

12.1 Introduction, definitions and set up 199

12.2 Key formulas and relations 202

12.3 On central solutions 205

12.4 The solutions modulo the homomorphisms 210

12.5 Examples 215

12.6 Other ways of formulating Jensen’s equation 218

12.7 A Pexider-Jensen’s functional equation 219

12.8 A variant of Jensen’s equation 219

12.9 Exercises 220

12.10 Notes and remarks 225

13. The Quadratic Functional Equation 227

13.1 Introduction 227

13.2 The set up and definitions 229

13.3 The case of R~n 229

13.4 General considerations and the abelian case 231

13.5 The Cauchy differences of solutions 234

13.6 The classical quadratic functional equation 236

13.7 Examples 239

13.8 Exercises 243

13.9 Notes and remarks 249

14. K-Spherical Functions 253

14.1 Introduction and notation 253

14.2 From where do K-spherical functions originate? 258

14.3 The Moroccan school 259

14.4 Exercises 262

14.5 Notes and remarks 262

15. The Sine Functional Equation 263

15.1 Introduction 263

15.2 General results about the solutions 264

15.3 The sine equation on cyclic groups 270

15.4 A more general functional equation 273

15.5 Exercises 274

15.6 Notes and remarks 278

16. The Cocycle Equation 279

16.1 Introduction 279

16.2 Compact groups 281

16.3 Abelian groups 282

16.4 Examples 286

16.5 The cocycle equation on semidirect products 287

16.6 Exercises 288

16.7 Notes and remarks 290

Appendices 291

A. Basic Terminology and Results 293

A.1 Numbers and matrices 293

A.2 Functions 294

A.3 Vector spaces 295

A.4 Algebras 296

A.5 Groups 297

      A.5.1 General theory and notation 297

      A.5.2 Groups generated by their squares 300

      A.5.3 Topological groups 301

      A.5.4 Examples of groups 302

      A.5.5 Semidirect products of groups 305

      A.5.6 Functions on coset spaces 307

A.6 Semigroups and involutions 308

A.7 Exercises 311

A.8 Notes and remarks 313

B. Substitutes for Commutativity 315

B.1 Kannappan’s condition etc 315

B.2 Four derived functions 317

B.3 Exercises 319

B.4 Notes and remarks 321

C. The Casorati Determinant 323

C.1 Around the Casorati determinant 323

C.2 Exercises 327

C.3 Notes and remarks 327

D. Regularity 329

D.1 Introduction 329

D.2 Continuity of homomorphisms 330

D.3 Smoothness of solutions 332

D.4 Gajda’s result 335

D.5 Exercises 336

D.6 Notes and remarks 336

E. Matrix-Coefficients of Representations 337

E.1 Matrix-coefficients 337

E.2 On representations 340

E.3 On the regular representations 344

E.4 On compact groups 345

F. The Small Dimension Lemma 347

G. Group Cohomology 351

Bibliography 355

Glossary 371

Index 375

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