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On the Generalized d’Alembert's and Wilson's Functional Equations on a Compact Group

Published online by Cambridge University Press:  20 November 2018

Belaid Bouikhalene
Belaid Bouikhalene*
Affiliation:
Département de Mathématiques et Informatique, Université Ibn Tofail, Faculté des Sciences, BP 133 Kénitra 14000, Morocco e-mail: bbouikhalene@yahoo.fr
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Abstract

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Let $G$ be a compact group.

Let $\sigma$ be a continuous involution of $G$ . In this paper, we are concerned by the following functional equation

$$\int\limits_{G}{f\left( xty{{t}^{-1}} \right)dt}\,+\int\limits_{G}{f\left( xt\sigma \left( y \right){{t}^{-1}} \right)dt}=2g\left( x \right)h\left( y \right),\,\,\,x,y\in G,$$

where $f,g,h:G\mapsto \mathbb{C}$ , to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d’Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$ .

Keywords

39B3239B4222D1022D1222D15Compact groupsFunctional equationsCentral functionsLie groupsInvariant differential Operators

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005, pp. 505 - 522
Copyright
Copyright © Canadian Mathematical Society 2005

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