Real Paley-Wiener theorems for Fourier series

Consider an integrable function f on the torus Tm, with Fourier coefficients f˜(n), n∈Zm. For functions with support in a neighbourhood of the origin, we describe the support by means of lp-growth properties of difference operators acting on the Fourier coefficients f˜(n). We then generalize the result to compact Lie groups, using a certain Weyl group invariant difference operator. We also consider l2-Bernstein inequalities for the difference operators.