On generalized d?Alembert and Wilson functional equations

Abstract

Let G be a Hausdorff locally compact group and let $\sigma $ be a continuous involution of G. Let μ 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 d’Alembert functional equation, while the second one is considered as a generalization of Wilson’s equation. We treat these equations in two settings: when f satisfies a condition of Kannappan type (see condition K(μ) below) and in the particular case when μ is a generalized Gelfand measure. In both cases, the solutions will be described by means of μ-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 d’Alembert’s and Wilson’s functional equations done in the abelian case.