  Our focus is on the stable approximate solution of linear operator equations
based on noisy data by using $\ell^1$-regularization as a sparsity-enforcing
version of Tikhonov regularization. We summarize recent results on situations
where the sparsity of the solution slightly fails. In particular, we show how
the recently established theory for weak*-to-weak continuous linear forward
operators can be extended to the case of weak*-to-weak* continuity. This might
be of interest when the image space is non-reflexive. We discuss existence,
stability and convergence of regularized solutions. For injective operators, we
will formulate convergence rates by exploiting variational source conditions.
The typical rate function obtained under an ill-posed operator is strictly
concave and the degree of failure of the solution sparsity has an impact on its
behavior. Linear convergence rates just occur in the two borderline cases of
proper sparsity, where the solutions belong to $\ell^0$, and of well-posedness.
For an exemplary operator, we demonstrate that the technical properties used in
our theory can be verified in practice. In the last section, we briefly mention
the difficult case of oversmoothing regularization where $x^\dag$ does not
belong to $\ell^1$.
