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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<b< \frac{2N}{2N+1}$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b>\frac{2N}{2N+1}$\end{document} then a dual window supported on [ − N, N] does not exist. In the limit case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b=\frac{2N}{2N+1}$\end{document} a dual window supported on [ − N, N] might exist, but cannot be continuous.