Paley-Wiener type theorems by transmutations

The classical Paley-Wiener theorem for functions in L-dx(2) relates the gro wth of the Fourier transform over the complex plane to the support of the f unction. In this work we obtain Paley-Wiener type theorems where the Fourie r transform is replaced by transforms associated with self-adjoint operator s on L-d mu(2), with simple spectrum, where d mu is a Lebesgue-Stieltjes me asure. This is achieved via the use of support preserving transmutations.