  In $\ell^1$-regularization, which is an important tool in signal and image
processing, one usually is concerned with signals and images having a sparse
representation in some suitable basis, e.g. in a wavelet basis. Many results on
convergence and convergence rates of sparse approximate solutions to linear
ill-posed problems are known, but rate results for the $\ell^1$-regularization
in case of lacking sparsity had not been published until 2013. In the last two
years, however, two articles appeared providing sufficient conditions for
convergence rates in case of non-sparse but almost sparse solutions. In the
present paper, we suggest a third sufficient condition, which unifies the
existing two and, by the way, also incorporates the well-known restricted
isometry property.
