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
