# Stable recovery of signals by $\ell_1$ optimization

Suppose the received vector $$y$$ is generated from a vector $$x^*$$ as $$y = { D}x^* + z$$ for some dictionary" matrix $${D}$$ and noise vector $$z$$ s.t for some $$\epsilon >0$$ we have, $$\Vert z \Vert_2 \leq \epsilon$$.

Now we solve the so-called $$P_1^\epsilon$$ optimization question to get, $$\hat{x} := \min \Vert x \Vert_1 \text{ s.t } \Vert {D}x - y \Vert _2 \leq \epsilon$$.

• Then under certan RIP conditions on D and some constant $$C$$ it follows that, $$\Vert x^* - \hat{x} \Vert \leq C \epsilon$$. Can someone kindly reference me the exact theorem and proof of this statement?

Are there more general forms of such a result? I would be glad to get some references in that direction too.