A note on Gabor frames

Abstract

Wilson frames $\{\psi_j^k :w_0,w_{-1}\in L^2(\mathbb{R})\}_{j\in\mathbb{Z}\atop {k \in\mathbb{N}_0}}$ in $L^2(\mathbb{R})$ have been defined and a characterization of Wilson frames in terms of Gabor frames is given when $w_0=w_{-1}$. Also, under certain conditions a necessary condition for a Wilson system to be a Wilson Bessel sequence is given. We have also obtained sufficient conditions for a Wilson system to be a Wilson frame in terms of Gabor Bessel sequences. For $w_0=w_{-1}$, stability of Wilson frames is discussed. Also, under the same assumption a necessary and sufficient condition is given for a Wilson system to be a Wilson Bessel sequence in terms of a Wilson frame.