WIENERS TAUBERIAN THEOREM FOR SPHERICAL-FUNCTIONS ON THE AUTOMORPHISMGROUP OF THE UNIT DISK

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spheri cal integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of funct ions with integral zero. This is then used to prove a generalization o f Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a ''two circles'' Morera type theorem (earlier announced by Agranovskii).