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J. M. Bogoya, A. Böttcher, S. M. Grudsky: Asymptotics of individual eigenvalues of a class of large Hessenberg Toeplitz matrices

J. M. Bogoya, A. Böttcher, S. M. Grudsky: Asymptotics of individual eigenvalues of a class of large Hessenberg Toeplitz matrices

Author(s):
J. M. Bogoya
A. Böttcher
S. M. Grudsky
Title:
Asymptotics of individual eigenvalues of a class of large Hessenberg Toeplitz matrices
Electronic source:
application/pdf
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 8, 2010
Mathematics Subject Classification:
47B35 [Toeplitz operators, Hankel operators, Wiener-Hopf operators ]
15A15 [Determinants, permanents, other special matrix functions ]
15A18 [Eigenvalues, singular values, and eigenvectors ]
47N50 [Applications in the physical sciences ]
65F15 [Eigenvalues, eigenvectors]
Abstract: We study the asymptotic behavior of individual eigenvalues of the $n$-by-$n$ truncations of certain infinite Hessenberg Toeplitz matrices as $n$ goes to infinity. The generating function of the Toeplitz matrices is supposed to be of the form $a(t)=t^{-1}(1-t)^{\alpha}f(t)$ ($t \in \mathbb{T}$), where $\alpha$ is a positive real number but not an integer and $f$ is a smooth function in $H^\infty$. The classes of generating functions considered here and in a recent paper by Dai, Geary, and Kadanoff are overlapping, and in the overlapping cases, our results imply in particular a rigorous justification of an asymptotic formula which was conjectured by Dai, Geary, and Kadanoff on the basis of numerical computations.
Keywords:
Toeplitz matrix, eigenvalue, Fourier integral, asymptotic expansion
Language:
English
Publication time:
06/2010