Regularity of Dual Gabor Windows

Abstract

We present a construction of dual windows associated with Gabor frames with compactly supported windows. The size of the support of the dual windows is comparable to that of the given window. Under certain conditions, we prove that there exist dual windows with higher regularity than the canonical dual window. On the other hand, there are cases where no differentiable dual window exists, even in the overcomplete case. As a special case of our results, we show that there exists a common smooth dual window for an interesting class of Gabor frames. In particular, for any value of $K\in \mathrm{\mathbb{N},}$ there is a smooth function $h$ which simultaneously is a dual window for all B-spline generated Gabor frames ${\{E_{mb}{T}_n{B}_N(x/2)\}}_{m,n\in \mathbb{N}}$ for B-splines $B_N$ of order $N=1,\dots ,2K+1$ with a fixed and sufficiently small value of $b$.