On the integral functional equations: On the integral d'Alembert's and Wilson's functional equations

Abstract

Let G be a locally compact group, and let K be a compact subgroup of G. Let µ : G −→ C\{0} be a character of G. In this paper, we deal with the integral equationsandfor all x, y ∈ G where f, g : G −→ C, to be determined, are complex continuous functions on G. When K ⊂ Z(G), the center of G, Dµ(K) reduces to the new version of d'Almbert's functional equation f (xy) + µ(y)f (xy −1 ) = 2f (x)f (y), recently studied by Davison [18] and Stetkaer [35]. We derive the following link between the solutions of Wµ(K) and Dµ(K) in the following way :. This result is used to establish the superstability problem of Wµ(K). In the case where (G, K) is a central pair, we show that the solutions are expressed by means of K-spherical functions and related functions. Also we give explicit formulas of solutions of Dµ(K) in terms of irreducible representations of G. These formulas generalize Euler's formula cos(x) = e ix +e −ix 2 on G = R.