D'Alembert's functional equation on groups
Volume 99 / 2013
Abstract
Given a (not necessarily unitary) character $\mu:G \to (\mathbb C \setminus \{ 0\},\cdot)$ of a group $G$ we find the solutions $g:G \to \mathbb C$ of the following version of d'Alembert's functional equation \begin{equation} g(xy) + \mu(y)g(xy^{-1}) = 2g(x)g(y), x,y \in G. \tag{$*$} \label{*} \end{equation} The classical equation is the case of $\mu = 1$ and $G = \mathbb R$. The non-zero solutions of $(*)$ are the normalized traces of certain representations of $G$ on $\mathbb C^2$. 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 directly with d'Alembert's functional equation. In the process we find for any non-abelian solution $g$ of $(*)$ the corresponding solutions $w:G \to \mathbb C$ of \begin{equation} w(xy) + w(yx) = 2w(x)g(y) + w(y)g(x), x,y \in G. \tag{$**$} \label{**} \end{equation} A novel feature is our use of the theory of group representations and their matrix-coefficients which simplifies some arguments and relates the results to harmonic analysis on groups.
