Trigonometric formulas and μ-spherical functions

Our main goal is to determine the continuous and bounded solutions of the functional equations ∫Gf(xtσ(y))dμ (t) = f(x)g(y) + f(y)g(x),x,y ∈ G, ∫Gf(xtσ(y))dμ (t) = f(x)g(y) - f(y)g(x) x,y ∈ G ∫Gg(xtσ (y))dμ(t) = g(y)g(x) + f(x)f(y)x,y ∈ G, where G is a locally compact group σ is a continuous homomorphism such that σ ̂ σ=I and μ is a σ-invariant complex bounded measure on G. The solutions are expressed by means of μ-spherical functions and solutions of the functional equation ∫Gf(xty)dμ(t) = f(x)φ(y) + f(y)φ(x)x,y ∈ G, in which φ is a μ-spherical function. The results obtained in the present paper are a natural extension of previous work by Poulsen and Stetkær in which μ is the Dirac measure concentrated at the identity element of G. © Birkhäuser Verlag, Basel, 2006.