This is an announcement for the paper “Injectivity and weak$^*$-to-weak continuity suffice for convergence rates in $\ell_1$-regularization” by Jens Flemminghttps://arxiv.org/find/math/1/au:+Flemming_J/0/1/0/all/0/1, Daniel Gerthhttps://arxiv.org/find/math/1/au:+Gerth_D/0/1/0/all/0/1.
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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.03460