  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.
