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Variational sparsity regularization based on $\ell^1$-norms and othernonlinear functionals has gained enormous attention recently, both with respectto its applications and its mathematical analysis. A focus in regularizationtheory has been to develop error estimation in terms of regularizationparameter and noise strength. For this sake specific error measures such asBregman distances and specific conditions on the solution such as sourceconditions or variational inequalities have been developed and used.  In this paper we provide, for a certain class of ill-posed linear operatorequations, a convergence analysis that works for solutions that are notcompletely sparse, but have a fast decaying nonzero part. This case is notcovered by standard source conditions, but surprisingly can be treated with anappropriate variational inequality. As a consequence the paper also providesthe first examples where the variational inequality approach, which was oftenbelieved to be equivalent to appropriate source conditions, can indeed gofarther than the latter.

Convergence rates in $\mathbf{\ell^1}$-regularization if the sparsity assumption fails