How to Mathematically Recover Sparse Signal (Least Squares with $ {L}_{1} $ Regularization - LASSO)?

Question

I got this equation for signal recovery by using LASSO:

$$\alpha_y =\mathrm{argmin} \left(\| Y - D\alpha \|_2^2 + \lambda^* \|\alpha\|_1\right) $$

Here $D$ is a Dictionary, $Y$ is noisy data signal and $\alpha$ is sparse signal.

My question is that How can I mathematically recover $\alpha_y$ ?