<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Adamyan, V</style></author><author><style face="normal" font="default" size="100%">Berezansky, Y</style></author><author><style face="normal" font="default" size="100%">Gohberg, I</style></author><author><style face="normal" font="default" size="100%">Gorbachuk, M</style></author><author><style face="normal" font="default" size="100%">Gorbachuk, V</style></author><author><style face="normal" font="default" size="100%">Kochubei, A</style></author><author><style face="normal" font="default" size="100%">Langer, H</style></author><author><style face="normal" font="default" size="100%">Popov, G</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Asymptotics of Toeplitz matrices with symbols in some generalized Krein algebras</style></title><secondary-title><style face="normal" font="default" size="100%">Modern Analysis and Applications: Mark Krein Centenary Conference, Vol. 1. Operator Theory Advances and Applications, 190</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2009</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-7643-9919-1_21</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">341-359</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\alpha,\beta\in(0,1)\) and&lt;br /&gt;
\[&lt;br /&gt;
K^{\alpha,\beta}:=\left\{a\in L^\infty(\mathbb{T}):\&lt;br /&gt;
\sum_{k=1}^\infty |\widehat{a}(-k)|^2 k^{2\alpha}&amp;lt;\infty,\&lt;br /&gt;
\sum_{k=1}^\infty |\widehat{a}(k)|^2 k^{2\beta}&amp;lt;\infty&lt;br /&gt;
\right\}.&lt;br /&gt;
\]&lt;br /&gt;
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&lt;br /&gt;
earlier results of Gabor Szegö for scalar symbols and established the asymptotic trace formula&lt;br /&gt;
\[&lt;br /&gt;
\operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1)&lt;br /&gt;
\quad\text{as}\ n\to\infty&lt;br /&gt;
\]&lt;br /&gt;
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 Szegö-Widom asymptotic trace formula holds with \(o(1)\) replaced by \(o(n^{1-\alpha-\beta})\).&lt;/p&gt;
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