A note on Gabor frames

Wilson frames ${psi_j^k :w_0,w_{-1}in L^2(mathbb{R})}_{jinmathbb{Z}atop {k inmathbb{N}_0}}$ 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