The null space property of the weighted ℓr − ℓ1 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 paper studies NSP of the weighted [Formula: see text] ([Formula: see text]) minimization. Several versions of NSP of the weighted [Formula: see text] minimization including the weighted [Formula: see text] NSP, the weighted [Formula: see text] stable NSP, the weighted [Formula: see text] robust NSP and the [Formula: see text] weighted [Formula: see text] robust NSP for [Formula: see text], are proposed, as well as the associating considerable results are derived. Under these NSPs, sufficient conditions for the recovery of (sparse) signals with the weighted [Formula: see text] minimization are established. Furthermore, we show that to some extent, the weighted [Formula: see text] stable NSP is weaker than the restricted isometric property (RIP). And the RIP condition we obtained is better than that of Zhou (2022).