Irregular wavelet/Gabor frames in Sobolev space

The theory of regular wavelet/Gabor frames has been extensively investigated, including the sufficient and necessary conditions for regular wavelet/Gabor systems to be frames and the perturbation theorem. This paper addresses the irregular wavelet/Gabor systems in Sobolev space Hs(R). The sufficient and necessary conditions for irregular wavelet/Gabor systems to be frames are presented. As corollaries, under certain restrictions on the support, we also give the characterization of irregular wavelet/Gabor systems to be frames. Finally, we discuss the perturbation theorem of irregular wavelet/Gabor frames.