This article presents novel results on recovering signals from undersampled data when the signals are not sparse in an orthonormal basis or incoherent dictionary but in a truly redundant dictionary. The authors introduce a condition on the measurement matrix, a generalization of the restricted isometry property, which guarantees accurate recovery of signals that are nearly sparse in highly overcomplete and coherent dictionaries. This condition imposes no incoherence restriction on the dictionary, and the results are the first of their kind. The paper discusses practical examples and the implications of these results on various applications, demonstrating the potential of $\ell_1$-analysis for such problems. The main theorem shows that the solution to the $\ell_1$-analysis problem is very accurate if the coefficients of $D^*f$ decay rapidly, where $D$ is the dictionary and $f$ is the signal. Numerical experiments confirm the effectiveness of $\ell_1$-analysis in recovering signals represented in redundant dictionaries, and the robustness of this method with respect to noise. The paper also explores alternatives such as $\ell_1$-synthesis and proposes a modification called "Split-analysis" to exploit sparsity in different components of the signal. Finally, it discusses fast transforms that satisfy the D-RIP, providing a viable approach for practical applications.This article presents novel results on recovering signals from undersampled data when the signals are not sparse in an orthonormal basis or incoherent dictionary but in a truly redundant dictionary. The authors introduce a condition on the measurement matrix, a generalization of the restricted isometry property, which guarantees accurate recovery of signals that are nearly sparse in highly overcomplete and coherent dictionaries. This condition imposes no incoherence restriction on the dictionary, and the results are the first of their kind. The paper discusses practical examples and the implications of these results on various applications, demonstrating the potential of $\ell_1$-analysis for such problems. The main theorem shows that the solution to the $\ell_1$-analysis problem is very accurate if the coefficients of $D^*f$ decay rapidly, where $D$ is the dictionary and $f$ is the signal. Numerical experiments confirm the effectiveness of $\ell_1$-analysis in recovering signals represented in redundant dictionaries, and the robustness of this method with respect to noise. The paper also explores alternatives such as $\ell_1$-synthesis and proposes a modification called "Split-analysis" to exploit sparsity in different components of the signal. Finally, it discusses fast transforms that satisfy the D-RIP, providing a viable approach for practical applications.