In this talk, we give an elementary proof of the approximation theorem established by A. Matheson for analytic Lipschitz spaces $\lambda _\alpha$. Our approach allows us to extend this approximation theorem to $\cD _\omega \cap \lambda _\alpha$, where $\cD _\omega$ denotes superharmonically weighted Dirichlet spaces (including standard Dirichlet spaces). As a consequence, we give a complete description of closed ideals of the Banach algebra $\cD _\omega \cap \lambda _\alpha$. This extends previous results obtained A. Matheson and B. Bouya.