Solutions and stability of a generalization of wilson's equation

Abstract

Abstract. In this paper we study the solutions and stability of the generalized Wilson's functional equation G f (xty)dµ(t)+ G f (xtσ(y))dµ(t) = 2f (x)g(y), x, y ∈ G, where G is a locally compact group, σ is a continuous involution of G and µ is an idempotent complex measure with compact support and which is σ-invariant. We show that G g(xty)dµ(t) + G g(xtσ(y))dµ(t) = 2g(x)g(y), x, y ∈ G if f = 0 and G f (t.)dµ(t) = 0. We also study some stability theorems of that equation and we establish the stability on noncommutaive groups of the classical Wilson's functional equation f (xy) + χ(y)f (xσ(y)) = 2f (x)g(y) x, y ∈ G , where χ is a unitary character of G.