An integral functional equation on groups under two measures.

Let G be a locally compact Hausdorff group, let σ be a continuous involutive automorphism on G, and let μ, ν be regular, compactly supported, complex-valued Borel measures on G. We find the continuous solutions f : G → C of the functional equation ∫G f (σ(y)xt)dμ(t) + ∫G f (xyt)dν(t) = f (x)f (y), x,y ∈ G, in terms of continuous characters of G. This equation provides a common generalization of many functional equations (d'Alembert's, Cauchy's, Gajda's, Kannappan's, Stetkær's, Van Vleck's equations...). So, a large class of functional equations will be solved. [ABSTRACT FROM AUTHOR]