On the superstability of generalized d’Alembert harmonic functions

Iz-iddine EL-Fassi

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Given a (not necessarily unitary) character μ:G → (ℂ∖0,·) of a group G we find the solutions g: G → ℂ of the following version of d’Alembert’s functional equation $g (x y) + μ (y) g (x y^{- 1}) = 2 g (x) g (y)$ , x,y ∈ G. (*) The classical equation is the case of μ = 1 and G = ℝ. The non-zero solutions of (*) are the normalized traces of certain representations of G on ℂ². Davison proved this via his work [20] on the pre-d’Alembert functional equation on monoids. The present paper presents a detailed exposition of these results working...

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We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] in 1962. Our solution is based on a generalization of Radó’s theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.