Perturbation of Frames from the Weyl-Heisenberg Group and the Extended Affine Group

Using a contraction between the affine group and the Weyl-Heisenberg group, we derive several new sufficient conditions, with explicit frame bounds, for perturbation of Gabor frames and wavelet frames. Firstly, we give perturbation of Gabor frames with respect to the translation parameter, modulation parameter, and window functions. An equivalent criteria for frame conditions in terms of a given frame and its perturbed sequence is given. We give necessary and sufficient conditions for the existence of wavelet frames from the extended affine group. Sufficient conditions for perturbation of frames from the extended affine group are derived. Finally, we give an interplay between perturbation of frames from the Weyl-Heisenberg group and extended affine group.