Harmonic analysis and generalized functional equations for the cosine

Let G be a locally compact group and let mu be a regular, compactly supported, complex-valued Borel measure on G. In the present paper, we determine the continuous solutions f, g : G -> C of the functional equation integral(G) {f(xyt) + f(xy(-1) t)} d mu(t) = 2 g(x)f(y), x, y is an element of G, in terms of characters, additive maps and matrix elements of irreducible, two-dimensional representations of G. Many consequences of this result are presented.