Injectivity and weak*-to-weak continuity suffice for convergence rates in ${\ell }^1$-regularization

Abstract

We show that the convergence rate of ${\ell }^1$-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.

References in corpus (1)

• Convergence rates and source conditions for Tikhonov regularization with sparsity constraints