Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein
  Algebras   Let $\alpha,\beta\in(0,1)$ and \[ K^{\alpha,\beta}:=\left\{a\in L^\infty(\T):
\sum_{k=1}^\infty |\hat{a}(-k)|^2 k^{2\alpha}<\infty, \sum_{k=1}^\infty
|\hat{a}(k)|^2 k^{2\beta}<\infty \right\}. \] Mark Krein proved in 1966 that
$K^{1/2,1/2}$ forms a Banach algebra. He also observed that this algebra is
important in the asymptotic theory of finite Toeplitz matrices. Ten years
later, Harold Widom extended earlier results of Gabor Szeg\H{o} for scalar
symbols and established the asymptotic trace formula \[
\operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1) \quad\text{as}\
n\to\infty \] for finite Toeplitz matrices $T_n(a)$ with matrix symbols $a\in
K^{1/2,1/2}_{N\times N}$. We show that if $\alpha+\beta\ge 1$ and $a\in
K^{\alpha,\beta}_{N\times N}$, then the Szeg\H{o}-Widom asymptotic trace
formula holds with $o(1)$ replaced by $o(n^{1-\alpha-\beta})$.

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The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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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: cs
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: econ