Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

This paper addresses the regularization by sparsity constraints by means of weighted ${\mathrm{\ell }}^p$ penalties for $0\le p\le 2$. For $1\le p\le 2$ special attention is payed to convergence rates in norm and to source conditions. As main result it is proven that one gets a convergence rate in norm of $\sqrt{\delta }$ for $1\le p\le 2$ 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 $p\ge 1$, 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.