On Cauchy-type functional equations

Abstract

Let G be a Hausdorff topological locally compact group. Let M(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 ∫Gf(xty)dμ(t)=∑i=1ngi(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.