A Wiener-Tauberian theorem for Fourier-Jacobi transform

Let f∈L^p(R ⁺, Δ(t)dt), we consider conditions under which the space spanned by generalized translates of f is dense in L^p(R ⁺, Δ(t)dt) in terms of Fourier-Jacobi trans form of f. This generalizes and improves the earlier results of A. Sitaram on semi-simple Lie groups of rank one.