Injectivity and weak*-to-weak continuity suffice for convergence rates in l1-regularization

We show that the convergence rate of l(1)-regularization for linear ill-posed equations is always O(delta) if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied.