On ℓ1-Regularization in Light of Nashed's Ill-Posedness Concept

Abstract Based on the powerful tool of variational inequalities, in recent papers convergence rates results on ℓ1-regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in ℓ1. In the present paper, we improve those convergence rates results and apply them to the Cesáro operator equation in ℓ2 and to specific denoising problems. Moreover, we formulate in this context relationships between Nashed's types of ill-posedness and mapping properties like compactness and strict singularity.

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