Gabor windows supported on [ - 1, 1] and dual windows with small support

Consider a continuous function g ∈ L2(R) that is supported on [−1, 1] and generates a Gabor frame with translation parameter 1 and modulation parameter 0 < b < 2N 2N+1 for some N ∈ N. Under an extra condition on the zeroset of the window g we show that there exists a continuous dual window supported on [−N, N]. We also show that this result is optimal: indeed, if b > 2N 2N+1 then a dual window supported on [−N, N] does not exist. In the limit case b = 2N 2N+1 a dual window supported on [−N, N] might exist, but cannot be continuous.