Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: B-spline approximations of the Gaussian, their Gabor frame properties,
  and approximately dual frames   We prove that Gabor systems generated by certain scaled B-splines can be
considered as perturbations of the Gabor systems generated by the Gaussian,
with a deviation within an arbitrary small tolerance whenever the order $N$ of
the B-spline is sufficiently large. As a consequence we show that for any
choice of translation/modulation parameters $a,b>0$ with $ab<1,$ the scaled
version of $B_N$ generates Gabor frames for $N$ sufficiently large. Considering
the Gabor frame decomposition generated by the Gaussian and a dual window, the
results lead to estimates of the deviation from perfect reconstruction that
arise when the Gaussian is replaced by a scaled B-spline, or when the dual
window of the Gaussian is replaced by certain explicitly given and compactly
supported linear combinations of the B-splines. In particular, this leads to a
family of approximate dual windows of a very simple form, leading to "almost
perfect reconstruction" within any desired error tolerance whenever the product
$ab$ is sufficiently small. In contrast, the known (exact) dual windows have a
very complicated form. A similar analysis is sketched with the scaled B-splines
replaced by certain truncations of the Gaussian. As a consequence of the
approach we prove (mostly known) convergence results for the considered scaled
B-splines to the Gaussian in the $L^p$-spaces, as well in the time-domain as in
the frequency domain.

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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys