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The convergence rates results in $\ell^1$-regularization when the sparsityassumption is narrowly missed, presented by Burger et al. (2013 InverseProblems 29 025013), are based on a crucial condition which requires that allbasis elements belong to the range of the adjoint of the forward operator.Partly it was conjectured that such a condition is very restrictive. In thiscontext, we study sparsity-promoting varieties of Tikhonov regularization forlinear ill-posed problems with respect to an orthonormal basis in a separableHilbert space using $\ell^1$ and sublinear penalty terms. In particular, weshow that the corresponding range condition is always satisfied for all basiselements if the problems are well-posed in a certain weaker topology and thebasis elements are chosen appropriately related to an associated Gelfandtriple. The Radon transform, Symm's integral equation and linear integraloperators of Volterra type are examples for such behaviour, which allows us toapply convergence rates results for non-sparse solutions, and we further extendthese results also to the case of non-convex $\ell^q$-regularization with0<q<1.

On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization