Stable reconstructions for the analysis formulation of $\ell^p$-minimization using redundant systems

Abstract: In compressed sensing sparse solutions are usually obtained by solving an $\ell^1$-minimization problem. Furthermore, the sparsity of the signal does need not be directly given. In fact, it is sufficient to have a signal that is sparse after an application of a suitable transform. In this paper we consider the stability of solutions obtained from $\ell^p$-minimization for arbitrary $0<p \leq1$. Further we suppose that the signals are sparse with respect to general redundant transforms associated to not necessarily tight frames. Since we are considering general frames the role of the dual frame has to be additionally discussed. For our stability analysis we will introduce a new concept of so-called \emph{frames with identifiable duals}. Further, we numerically highlight a gap between the theory and the applications of compressed sensing for some specific redundant transforms.