Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improved bounds for the eigenvalues of prolate spheroidal wave functions
  and discrete prolate spheroidal sequences   The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal
sequences in $\ell_2(\mathbb{Z})$ which are strictly bandlimited to a frequency
band $[-W,W]$ and maximally concentrated in a time interval $\{0,\ldots,N-1\}$.
The timelimited DPSSs (sometimes referred to as the Slepian basis) are an
orthonormal set of vectors in $\mathbb{C}^N$ whose discrete time Fourier
transform (DTFT) is maximally concentrated in a frequency band $[-W,W]$. Due to
these properties, DPSSs have a wide variety of signal processing applications.
The DPSSs are the eigensequences of a timelimit-then-bandlimit operator and the
Slepian basis vectors are the eigenvectors of the so-called prolate matrix. The
eigenvalues in both cases are the same, and they exhibit a particular
clustering behavior -- slightly fewer than $2NW$ eigenvalues are very close to
$1$, slightly fewer than $N-2NW$ eigenvalues are very close to $0$, and very
few eigenvalues are not near $1$ or $0$. This eigenvalue behavior is critical
in many of the applications in which DPSSs are used. There are many asymptotic
characterizations of the number of eigenvalues not near $0$ or $1$. In
contrast, there are very few non-asymptotic results, and these don't fully
characterize the clustering behavior of the DPSS eigenvalues. In this work, we
establish two novel non-asymptotic bounds on the number of DPSS eigenvalues
between $\epsilon$ and $1-\epsilon$. Also, we obtain bounds detailing how close
the first $\approx 2NW$ eigenvalues are to $1$ and how close the last $\approx
N-2NW$ eigenvalues are to $0$. Furthermore, we extend these results to the
eigenvalues of the prolate spheroidal wave functions (PSWFs), which are the
continuous-time version of the DPSSs. Finally, we present numerical experiments
demonstrating the quality of our non-asymptotic bounds on the number of DPSS
eigenvalues between $\epsilon$ and $1-\epsilon$.

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The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
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The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int