The impact of a curious type of smoothness conditions on convergence rates in `-regularization

Tikhonov-type regularization of linear and nonlinear ill-posed problems in abstract spaces under sparsity constraints gained relevant attention in the past years. Since under some weak assumptions all regularized solutions are sparse if the `1-norm is used as penalty term, the `1-regularization was studied by numerous authors although the non-reflexivity of the Banach space `1 and the fact that such penalty functional is not strictly convex lead to serious difficulties. We consider the case that the sparsity assumption is narrowly missed. This means that the solutions may have an infinite number of nonzero but fast decaying components. For that case we formulate and prove convergence rates results for the `1-regularization of nonlinear operator equations. In this context, we outline the situations of Hölder rates and of an exponential decay of the solution components. MSC2010 subject classification: 47J06, 65J20, 47A52, 49J40