On generalized d’Alembert functional equation

Abstract

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 ƒ (𝓍𝓉𝓎)d μ (𝓉) + ∫ G ƒ (𝓍𝓉𝜎 𝜎 (𝓎)) 𝒹 μ (𝓉) = 2 ƒ (𝓍) ƒ (𝓎) 𝓍, 𝓎 ∈ G, where ƒ: 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.