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Asymptotics of eigenvalues for Toeplitz matrices with rational symbols that have a minimum of the 4th order
Complex Variables and Elliptic Equations ( IF 0.6 ) Pub Date : 2021-08-12 , DOI: 10.1080/17476933.2021.1963711
Mauricio Barrera 1 , Sergei M. Grudsky 1, 2
Affiliation  

In Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. we have considered the problem about asymptotic formulas for all eigenvalues of Tn(a), as n goes to infinity, assuming that a is a specific model symbol with a unique zero of order 4. In this paper, we continue our investigation and we explore the case where a is a more general real-valued rational symbol with a unique zero of order 4. It should be noted that we apply a different method than the one used in Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. This method coming from works Bogoya JM, Böttcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols. J Math Anal Appl. 2015;422(2):1308–1334 and Bogoya JM, Böttcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices generated by simple-loop symbols with relaxed smoothness. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017. p. 179–212, where it is considered the class of all symbols having zeros of second order and one can reduce the problem to asymptotic analysis of a nonlinear equation. As well, we construct uniform asymptotic expansions for all eigenvalues, which allow us to precise the classical results of Widom and Parter for first and very last eigenvalues.



中文翻译:

具有最小四阶有理符号的 Toeplitz 矩阵的特征值的渐近

在 Barrera M,Grudsky SM。五对角对称 Toeplitz 矩阵的特征值的渐近。在:大型截断 Toeplitz 矩阵、toeplitz 算子和相关主题。算子理论:进展与应用卷。259,伯克豪瑟,湛。2017;页。51-77。我们已经考虑了关于所有特征值的渐近公式的问题n(一种),当 n 趋于无穷大时,假设a是具有唯一 4 阶零的特定模型符号。在本文中,我们继续研究并探讨a是一个更一般的实值有理符号,具有唯一的 4 阶零。应该注意的是,我们应用的方法与 Barrera M, Grudsky SM 中使用的方法不同。五对角对称 Toeplitz 矩阵的特征值的渐近。在:大型截断 Toeplitz 矩阵、Toeplitz 算子和相关主题。算子理论:进展与应用卷。259,伯克豪瑟,湛。2017;页。51-77。这种方法来自 Bogoya JM、Böttcher A、Grudsky SM 等人的作品。具有平滑单环符号的 Hermitian Toeplitz 矩阵的特征值。J数学肛门应用程序。2015;422(2):1308–1334 和 Bogoya JM、Böttcher A、Grudsky SM 等。Hermitian Toeplitz 矩阵的特征值由具有松弛平滑度的简单循环符号生成。在:大型截断 Toeplitz 矩阵、Toeplitz 算子和相关主题。算子理论:进展与应用卷。259,伯克豪瑟,湛。2017 年。179-212,其中它被认为是具有二阶零的所有符号的类别,并且可以将问题简化为非线性方程的渐近分析。同样,我们为所有特征值构造了一致的渐近展开,这使我们能够精确 Widom 和 Parter 的第一个和最后一个特征值的经典结果。

更新日期:2021-08-12
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