On the stability of wavelet and Gabor frames (Riesz bases)

if the sequence of functions {phi(j,k)} is a wavelet frame (Riesz basis) or Gabor frame (Riesz basis), we obtain its perturbation system {psi(j,k)} which is still a frame (Riesz basis) under very mild conditions. For example, we do not need to know that the support of phi or psi (<(phi)over cap> or <(psi)over cap>) is compact as in [14]. We also discuss the stability of irregular sampling problems. In order to arrive at some of our results, we set up a general multivariate version of Littlewood-Paley type inequality which was originally considered by Lemarie and Meyer [17], then by Chui and Shi [9], and Long [16].