# On the Interplay of Basis Smoothness and Range Conditions in $\ell^1$-Regularization

Bernd Hofmann^{*}

In the past years Tikhonov regularization for the stable approximate solution of ill-posed operator equations in Banach spaces based on $\ell^1$-norm penalties was of relevant interest in case of sparsity and also if the sparsity assumptions are narrowly missed. For obtaining convergence rates, range conditions of the unit elements with respect to the adjoint of the forward operator occur which at first glance preferably refer to situations close to a singular value decomposition. However, it will be shown that such conditions are also natural if the forward operator is continuously invertible with respect to a weaker topology in the context of Gelfand triples. The Radon transform and linear integral operators of Volterra type prove to be examples of practical relevance for such approach. This is joint work with Stephan Anzengruber (Chemnitz) and Ronny Ramlau (Linz).

Mathematics Subject Classification: 65J20 47A52 44A12 49J40

Keywords: Ill-posed problems ; regularization ; sparsity ; range condition ; smoothness

Minisymposion: New Trends in Regularization Theory and Methods for Geomathematical Problems