On model spaces and density of functions smooth on the boundary

We characterize the model spaces $K\_\Theta$ in which functions with smooth boundary extensions are dense. It is shown that such approximations possible if and only singular measure associated to inner factor of $\Theta$ concentrated on a countable union Beurling–Carleson sets. In fact, we use duality argument show there exists restriction does not assign positive any set, then even larger classes functions, as Hölder large collections analytic Sobolev spaces, fail be contrast earlier results density continuous related existence approximant obtained through constructive method.