Approximation in spaces of analytic functions

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 theorem to D-omega boolean AND lambda, where D-omega denotes superharmonically weighted Dirichlet spaces (including standard Dirichlet spaces). As a consequence, we give a complete description of closed ideals of the Banach algebras lambda(alpha) and D-omega boolean AND lambda(alpha). This extends previous results obtained by A. Matheson and B. Bouya.