  Variational sparsity regularization based on $\ell^1$-norms and other
nonlinear functionals has gained enormous attention recently, both with respect
to its applications and its mathematical analysis. A focus in regularization
theory has been to develop error estimation in terms of regularization
parameter and noise strength. For this sake specific error measures such as
Bregman distances and specific conditions on the solution such as source
conditions or variational inequalities have been developed and used.
  In this paper we provide, for a certain class of ill-posed linear operator
equations, a convergence analysis that works for solutions that are not
completely sparse, but have a fast decaying nonzero part. This case is not
covered by standard source conditions, but surprisingly can be treated with an
appropriate variational inequality. As a consequence the paper also provides
the first examples where the variational inequality approach, which was often
believed to be equivalent to appropriate source conditions, can indeed go
farther than the latter.
