Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Higher-Order Szego Theorems With Two Singular Points   We consider probability measures, $d\mu=w(\theta) \f{d\theta}{2\pi}
+d\mu_\s$, on the unit circle, $\partial\bbD$, with Verblunsky coefficients,
$\{\alpha_j\}_{j=0}^\infty$. We prove for $\theta_1\neq\theta_2$ in $[0,2\pi)$
and $(\delta\beta)_j=\beta_{j+1}$ that \[ \int
[1-\cos(\theta-\theta_1)][1-\cos(\theta-\theta_2)] \log w(\theta)
\f{d\theta}{2\pi} >-\infty \] if and only if \[ \sum_{j=0}^\infty
\bigl|\bigl\{(\delta -e^{-i\theta_2}) (\delta -e^{-i\theta_1})
\alpha\bigr\}_j\bigr|^2 +\abs{\alpha_j}^4 <\infty \] We also prove that \[ \int
(1-\cos\theta)^2 \log w(\theta) \f{d\theta}{2\pi} >-\infty \] if and only if \[
\sum_{j=0}^\infty \abs{\alpha_{j+2}-2\alpha_{j+1} +\alpha_j}^2 +
\abs{\alpha_j}^6 <\infty \]

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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex