On the Null Space Property oflq-Minimization for0

The paper discusses the relationship between the null space property (NSP) and thelq-minimization in compressed sensing. Several versions of the null space property, that is, thelqstable NSP, thelqrobust NSP, and thelq,probust NSP for0<p≤q<1based on the standardlqNSP, are proposed, and their equivalent forms are derived. Consequently, reconstruction results for thelq-minimization can be derived easily under the NSP condition and its equivalent form. Finally, thelqNSP is extended to thelq-synthesis modeling and the mixedl2/lq-minimization, which deals with the dictionary-based sparse signals and the block sparse signals, respectively.