On Cauchy-type functional equations

LetG be a Hausdorff topological locally compact group. LetM(G) denote the Banach algebra of all complex and bounded measures on G. For all integers n ≥ 1 and all μ ∈ M(G), we consider the functional equations ∫ G f(xty)dμ(t) = ∑n i=1gi(x)hi(y), x,y ∈ G, where the functions f , {gi}, {hi}: G → C to be determined are bounded and continuous functions on G. We show how the solutions of these equations are closely related to the solutions of the μ-spherical matrix functions. When G is a compact group and μ is a Gelfand measure, we give the set of continuous solutions of these equations.