  The convergence rates results in $\ell^1$-regularization when the sparsity
assumption is narrowly missed, presented by Burger et al. (2013 Inverse
Problems 29 025013), are based on a crucial condition which requires that all
basis elements belong to the range of the adjoint of the forward operator.
Partly it was conjectured that such a condition is very restrictive. In this
context, we study sparsity-promoting varieties of Tikhonov regularization for
linear ill-posed problems with respect to an orthonormal basis in a separable
Hilbert space using $\ell^1$ and sublinear penalty terms. In particular, we
show that the corresponding range condition is always satisfied for all basis
elements if the problems are well-posed in a certain weaker topology and the
basis elements are chosen appropriately related to an associated Gelfand
triple. The Radon transform, Symm's integral equation and linear integral
operators of Volterra type are examples for such behaviour, which allows us to
apply convergence rates results for non-sparse solutions, and we further extend
these results also to the case of non-convex $\ell^q$-regularization with
0<q<1.
