Gabor windows supported on [−1,1] and compactly supported dual windows

Consider a bounded function g supported on [−1,1] and a modulation parameter b∈]1/2,1[ for which the Gabor system {EmbTng}m,n∈Z is a frame. We show that such a frame always has a compactly supported dual window. More precisely, we show that if for some N∈N, it is possible to find a dual window supported on [−N,N]. Under the additional assumption that g is continuous and only has a finite number of zeros on ]−1,1[, we characterize the frame property of {EmbTng}m,n∈Z. As a consequence we obtain easily verifiable criteria for a function g to generate a Gabor frame with a dual window having compact support of prescribed size.