On generalized Gajda's functional equation of d'Alembert type (Q2911043)

\textit{M. Akkouchi}, \textit{A. Bakali} and \textit{I. Khalil} [J. Lond. Math. Soc., II. Ser. 57, No. 3, 694--705 (1998; Zbl 0932.43001)] investigated the properties of solutions of the functional equation NEWLINE\[NEWLINE\int_Kf(xk.y)dk=\sum_{i=1}^nh_i(x)g_i(y)\qquad(x,y\in G),NEWLINE\]NEWLINE where \(f, g_i, h_i\) \((i=1, \dots, n)\) are functions from a locally compact Hausdorff group \(G\) to \(\mathbb{C}\) and \(K\) is a compact group of automorphisms of \(G\). Applying harmonic analysis and representation theory for groups the authors use the positive definite kernel theory and operator-valued spherical functions to derive properties of the solutions of NEWLINE\[NEWLINE\int_Kf(y^{-1}k.x)dk=\langle a(x),a(y)\rangle \qquad(x,y\in G),\tag{\(*\)} NEWLINE\]NEWLINE where \(f: G\to \mathbb{C}\) is a continuous function, \(a: G \to H\) is a weakly continuous function into a Hilbert space \(H\). This equation is a generalization of Gajda's functional equation of d'Alembert's type. In the case where \(H=\mathbb{C}^n\) they find the matrix-valued solutions of the equation (\(*\)) without imposing Stetkær's assumption. As an application, they give explicit formulas for solutions of the equation (\(*\)) in terms of strongly continuous unitary representation of \(G\) in the case where \(G\) is abelian.