A Wiener Tauberian Theorem for weighted convolution algebras of zonal functions on the automorphism group of the unit disc

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the algebra L-1(G parallel to K, omega), the convolution algebra of zonal functions on the automorphism group on the unit disc which are integrable with respect to the weight; function omega, to be dense in the algebra, or to have as closure an ideal of functions whose set of common zeros of the Fourier transforms is a finite set on the boundary of the maximal ideal space of the algebra. The weights considered behave like Legendre functions of the first kind.