Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral analysis of the truncated Hilbert transform with overlap   We study a restriction of the Hilbert transform as an operator $H_T$ from
$L^2(a_2,a_4)$ to $L^2(a_1,a_3)$ for real numbers $a_1 < a_2 < a_3 < a_4$. The
operator $H_T$ arises in tomographic reconstruction from limited data, more
precisely in the method of differentiated back-projection (DBP). There, the
reconstruction requires recovering a family of one-dimensional functions $f$
supported on compact intervals $[a_2,a_4]$ from its Hilbert transform measured
on intervals $[a_1,a_3]$ that might only overlap, but not cover $[a_2,a_4]$. We
show that the inversion of $H_T$ is ill-posed, which is why we investigate the
spectral properties of $H_T$.
  We relate the operator $H_T$ to a self-adjoint two-interval Sturm-Liouville
problem, for which we prove that the spectrum is discrete. The Sturm-Liouville
operator is found to commute with $H_T$, which then implies that the spectrum
of $H_T^* H_T$ is discrete. Furthermore, we express the singular value
decomposition of $H_T$ in terms of the solutions to the Sturm-Liouville
problem. The singular values of $H_T$ accumulate at both $0$ and $1$, implying
that $H_T$ is not a compact operator. We conclude by illustrating the
properties obtained for $H_T$ numerically.

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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex