Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gabor windows supported on $[-1,1]$ and construction of compactly
  supported dual windows with optimal frequency localization   We consider Gabor frames $\{e^{2\pi i bm \cdot} g(\cdot-ak)\}_{m,k \in
\mathbb{Z}}$ 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.

37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph