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 $0<b< \frac{2N}{2N+1}$ 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 $b>\frac{2N}{2N+1}$ then a dual window supported on [ − N, N] does not exist. In the limit case $b=\frac{2N}{2N+1}$ a dual window supported on [ − N, N] might exist, but cannot be continuous.