Perturbation of frame sequences and its applications to shift-invariant spaces

We generalize the main result in [O. Christensen, H.O. Kim, R.Y. Kim, J.K. Lim, Perturbation of frame sequences in shift-invariant spaces, J. Geom. Anal. 15 (2005) 181–191] in order to make it comparable with existing results. Then we compare the special cases of the three results in the literature in the setting of the perturbation of the generating sets of finitely generated shift-invariant spaces of L2(Rd).