On $\ell^1$-regularization in light of Nashed's ill-posedness concept

Based on the powerful tool of variational inequalities, in recent papers convergence rates results on $\ell^1$-regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in $\ell^1$. In the present paper we improve those convergence rates results and apply them to the Ces\'aro operator equation in $\ell^2$ and to specific denoising problems. Moreover, we formulate in this context relationships between Nashed's types of ill-posedness and mapping properties like compactness and strict singularity.