Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

This paper addresses the regularization by sparsity constraints by means of weighted l p penalties for 0 <= p <= 2. For 1 <= p <= 2 special attention is payed to convergence rates in norm and to source conditions. As main results it is proven that one gets a convergence rate of root delta in the 2-norm for 1 < p <= 2 and in the 1-norm for p = 1 as soon as the unknown solution is sparse. The case p = 1 needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed. For p < 1 only preliminary results are shown. These results indicate that, different from p >= 1, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for p = 0 shows that regularization need not to happen.