  The Tikhonov regularization of linear ill-posed problems with an $\ell^1$
penalty is considered. We recall results for linear convergence rates and
results on exact recovery of the support. Moreover, we derive conditions for
exact support recovery which are especially applicable in the case of ill-posed
problems, where other conditions, e.g. based on the so-called coherence or the
restricted isometry property are usually not applicable. The obtained results
also show that the regularized solutions do not only converge in the
$\ell^1$-norm but also in the vector space $\ell^0$ (when considered as the
strict inductive limit of the spaces $\R^n$ as $n$ tends to infinity).
Additionally, the relations between different conditions for exact support
recovery and linear convergence rates are investigated.
  With an imaging example from digital holography the applicability of the
obtained results is illustrated, i.e. that one may check a priori if the
experimental setup guarantees exact recovery with Tikhonov regularization with
sparsity constraints.
