extracta mathematicae Vol. 21, N´ um. 1, 67 – 82 (2006) On Generalized d’Alembert Functional Equation M. Akkouchi 1 , A. Bakali 2 , B. Bouikhalene 2 , E. Elqorachi 3 1 Department of Mathematics, University Cadi Ayyad Faculty of Sciences, Semlalia, Marrakech, Morocco 2 Department of Mathematics, Laboratory L.A.M.A., University Ibn Tofail Faculty of Sciences, Kenitra, Morocco 3 Depatment of Mathematics, University of Ibnou Zohr Faculty of Sciences, Agadir, Morocco e-mail: makkouchi@hotmail.com, bbouikhalene@yahoo.fr, elqorachi@hotmail.com (Presented by Jes´ us M.F. Castillo) AMS Subject Class. (2000): 39B32, 39B42, 22D10 Received May 15, 2004 Let G be a locally compact group. Let σ be a continuous involution of G and let µ be a complex bounded measure. In this paper we study the generalized d’Alembert functional equation D(µ)  G f (xty)dµ(t)+  G f (xtσ(y))dµ(t)=2f (x)f (y) , x, y ∈ G, where f : G → C to be determined is a measurable and essentially bounded function. We give some conditions under which all solutions are of the form ≺ π(x)ξ,ζ ≻ + ≺ π(σ(x))ξ,ζ ≻ 2 , where (π, H) is a continuous unitary representation of G such that π(µ) is of rank one and ξ,ζ ∈H. Furthermore, we also consider the case when f is an integrable solution. In the particular case where G is a connected Lie group, we reduce the solution of D(µ) to a certain problem in operator theory. We prove that the solutions of D(µ) are exactly the common eigenfunctions of some operators associated to a left invariant differential operators on G. Key words: Functional equation, Gelfand measure, μ-spherical function, positive deni- nite function, representation theory, Lie group, invariant differential operator. 67