  Sparsity promoting regularization is an important technique for signal
reconstruction and several other ill-posed problems. Theoretical investigation
typically bases on the assumption that the unknown solution has a sparse
representation with respect to a fixed basis. We drop this sparsity assumption
and provide error estimates for non-sparse solutions. After discussing a result
in this direction published earlier by one of the authors and coauthors we
prove a similar error estimate under weaker assumptions. Two examples
illustrate that this set of weaker assumptions indeed covers additional
situations which appear in applications.
