Gabor windows supported on $[-1,1]$ and construction of compactly supported dual windows with optimal frequency localization
We consider Gabor frames $\{e^{2πi bm \cdot} g(\cdot-ak)\}_{m,k \in \mathbb{Z}}$ with translation parameter $a=L/2$, modulation parameter $b \in (0,2/L)$ and a window function $g \in C^n(\mathbb{R})$ supported on $[x_0,x_0+L]$ and non-zero on $(x_0,x_0+L)$ for $L>0$ and $x_0\in \mathbb{R}$. The set of all dual windows $h \in L^2(\mathbb{R})$ with sufficiently small support is parametrized by $1$-periodic measurable functions $z$. Each dual window $h$ is given explicitly in terms of the function $z$ in such a way that desirable properties (e.g., symmetry, boundedness and smoothness) of $h$ are directly linked to $z$. We derive easily verifiable conditions on the function $z$ that guarantee, in fact, characterize, compactly supported dual windows $h$ with the same smoothness, i.e., $h \in C^n(\mathbb{R})$. The construction of dual windows is valid for all values of the smoothness index $n \in \mathbb{Z}_{\ge 0} \cup \{\infty\}$ and for all values of the modulation parameter $b<2/L$; since $a=L/2$, this allows for arbitrarily small redundancy $(ab)^{-1}>1$. We show that the smoothness of $h$ is optimal, i.e., if $g \notin C^{n+1}(\mathbb{R})$ then, in general, a dual window $h$ in $C^{n+1}(\mathbb{R})$ does not exist.