Functional Equations On Hypergroups

by Laszlo Szekelyhidi

Overview

The theory of hypergroups is a rapidly developing area of mathematics due to its diverse applications in different areas like probability, harmonic analysis, etc. This book exhibits the use of functional equations and spectral synthesis in the theory of hypergroups. It also presents the fruitful consequences of this delicate “marriage” where the methods of spectral analysis and synthesis can provide an efficient tool in characterization problems of function classes on hypergroups.This book is written for the interested reader who has open eyes for both functional equations and hypergroups, and who dares to enter a new world of ideas, a new world of methods — and, sometimes, a new world of unexpected difficulties.

Product Details

ISBN-13:	9789814407007
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/28/2012
Pages:	212
Product dimensions:	6.20(w) x 9.10(h) x 0.70(d)

Table of Contents

Preface vii

1 Introduction 1

1.1 Basic concepts and facts 1

1.2 Convolution of subsets 6

1.3 Invariant means on hypergroups 7

1.4 Haar measure on hypergroups 11

1.5 Exponential functions on hypergroups 20

1.6 Exponential families on hypergroups 22

1.7 Additive and multi-additive functions on hypergroups 23

1.8 Moment functions on hypergroups 26

1.9 Exponentials and additive functions on a special hypergroup 33

2 Polynomial hypergroups in one variable 37

2.1 Polynomial hypergroups in one variable 37

2.2 Exponential and additive functions on polynomial hypergroups 39

2.3 Moment functions on polynomial hypergroups 43

2.4 Moment functions on the SU(2)-hypergroup 48

3 Polynomial hypergroups in several variables 51

3.1 Polynomial hypergroups in several variables 51

3.2 Exponential and additive functions on multivariate polynomial hypergroups 52

3.3 Moment function sequences on multivariate polynomial hypergroups 55

4 Sturm-Liouville hypergroups 59

4.1 Sturm-Liouville functions 59

4.2 Exponentials and additive functions on Sturm-Liouville hypergroups 62

4.3 Moment functions on Sturm-Liouville hypergroups 65

5 Two-point support hypergroups 73

5.1 Conditional functional equations 73

5.2 Two-point support hypergroups of noncompact type 77

5.3 Moment functions on two-point support hypergroups of noncompact type 78

5.4 Two-point support hypergroups of compact type 86

5.5 The cosh hypergroup 87

5.6 Associated pairs of moment functions 88

6 Spectral analysis and synthesis on polynomial hypergroups 95

6.1 Spectral analysis and spectral synthesis on hypergroups 95

6.2 Basic concepts and facts 96

6.3 Spectral analysis on polynomial hypergroups in a single variable 99

6.4 Exponential polynomials on polynomial hypergroups in a single variable 100

6.5 Spectral synthesis on polynomial hypergroups in a single variable 103

6.6 Spectral analysis and spectral synthesis on multivariate polynomial hypergroups 103

6.7 Spectral analysis and moment functions 106

7 Spectral analysis and synthesis on Sturm-Liouville hypergroups 109

7.1 Exponential monomials on Sturm-Liouville hypergroups 109

7.2 Linear independence of special exponential monomials 110

7.3 Spectral analysis on Sturm-Liouville hypergroups 119

8 Moment problems on hypergroups 123

8.1 The moment problem in general 123

8.2 Uniqueness on polynomial hypergroups 124

8.3 The case of Sturm-Liouville hypergroups 128

8.4 An approximation result 131

9 Special functional equations on hypergroups 133

9.1 The sine functional equation on polynomial hypergroups 133

9.2 The cosine functional equation on polynomial hypergroups 138

9.3 The Lévi-Civitá functional equation 142

10 Difference equations on polynomial hypergroups 149

10.1 Introduction 149

10.2 Difference equations with 1-translation 150

10.3 Difference equations with general translation 153

11 Stability problems on hypergroups 161

11.1 Stability of exponential functions on hypergroups 161

11.2 Stability of additive functions on hypergroups 163

11.3 Superstability of a mixed-type functional equation 165

11.4 Superstability of generalized moment functions on hypergroups 168

Bibliography 173

Index 187