Solutions and stability of a generalization of Wilson's equation

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In this paper we study the solutions and stability of the generalized Wilson's functional equation ∫Gf(xty)dμ(t)+∫Gf(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 ∫Gg(xty)dμ(t)+∫Gg(xtσ(y))dμ(t)=2g(x)g(y), x,y∈ G if f≠ 0 and ∫Gf(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.