# The Null Space Property of the Weighted $\ell_r-\ell_1$ Minimization

The null space property (NSP), which relies merely on the null space of the sensing matrix column space, has drawn numerous interests in sparse signal recovery. This article studies NSP of the weighted $\ell_r-\ell_1$ minimization. Several versions of NSP of the weighted $\ell_r-\ell_1$ minimization including the weighted $\ell_r-\ell_1$ NSP, the weighted $\ell_r-\ell_1$ stable NSP, the weighted $\ell_r-\ell_1$ robust NSP, and the $\ell_q$ weighted $\ell_r-\ell_1$ NSP for $1\leq q\leq2$, are proposed, as well as the associating considerable results are derived. Under these NSP, sufficient conditions for the recovery of (sparse) signals with the weighted $\ell_r-\ell_1$ minimization are established. Furthermore, we show that to some extent, the weighted $\ell_r-\ell_1$ stable NSP is weaker than the restricted isometric property (RIP). And the RIP condition we obtained is better than that of Zhou Z. (2022).

## Citation

South Xihe Road, Qinzhou District, Tianshui，741000，Gansu Province, P.R.China,4/2022

## Article

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