On the generalized d’Alembert functional equation

Summary.

Let G be a topological locally compact group, and let K be a compact subgroup of G. In this paper we deal with a generalization of d’Alembert’s functional equation on G. We determine those solutions that satisfy a Kannappan type condition. In the case where (G, K) is a central pair, i.e. the algebra of the elements in L₁(G), which are invariant under x → kxk⁻¹, is commutative, then the Kannappan type condition is satisfied, therefore we find all solutions of this equation. Furthermore, if G is a connected Lie group, we prove that the solutions are the joint eigenfunctions of certain operators associated to the left invariant differential operators. As an application, R. Godement’s spherical functions theory is used to give explicit formulas of solutions of this equation.