Geodesic
On the Generalized d’Alembert's and Wilson's Functional Equations on a Compact Group
Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 505-522

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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 $G$ .
DOI : 10.4153/CMB-2005-047-3
Mots-clés : 39B32, 39B42, 22D10, 22D12, 22D15, Compact groups, Functional equations, Central functions, Lie groups, Invariant differential Operators 
Bouikhalene, Belaid. On the Generalized d’Alembert's and Wilson's Functional Equations on a Compact Group. Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 505-522. doi: 10.4153/CMB-2005-047-3
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     title = {On the {Generalized} {d{\textquoteright}Alembert's} and {Wilson's} {Functional} {Equations} on a {Compact} {Group}},
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-047-3/}
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