Gabor windows supported on [ -aEuro parts per thousand 1, 1] and dual windows with small support

Abstract

Consider a continuous function g aaEuro parts per thousand L (2)(a"e) that is supported on [ -aEuro parts per thousand 1, 1] and generates a Gabor frame with translation parameter 1 and modulation parameter 0 < b < 2N/2N+1 for some N epsilon N center dot. 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.