Resumen de On generalized d'Alembert and Wilson functional equations

Elhoucien Elqorachi, Mohamed Akkouchi

• Let G be a Hausdorff locally compact group and let $\sigma $ be a continuous involution of G. Let be a complex bounded measure on G. We are interested in the functional equations $ \int_{G}f(xty)d\mu(t) + \int_{G}f(xt\sigma(y))d\mu(t) = 2f(x)f(y) $ and $ \int_{G}f(xty)d\mu(t) + \int_{G}f(xt\sigma(y)d\mu(t) = 2f(x)g(y) $. The first one is considered as a generalization of the classical dAlembert functional equation, while the second one is considered as a generalization of Wilsons equation. We treat these equations in two settings: when f satisfies a condition of Kannappan type (see condition K() below) and in the particular case when is a generalized Gelfand measure. In both cases, the solutions will be described by means of -spherical functions and related functions. Also the matrix-valued case is considered.

  The results obtained in this paper are natural extensions of previous works concerning dAlemberts and Wilsons functional equations done in the abelian case.