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A. Böttcher; S. M. Grudsky : Estimates for the condition numbers of large semi-definite Toeplitz matrices

A. Böttcher; S. M. Grudsky : Estimates for the condition numbers of large semi-definite Toeplitz matrices

Author(s) :
A. Böttcher; S. M. Grudsky
Title :
Estimates for the condition numbers of large semi-definite Toeplitz matrices
Electronic source :
[gzipped dvi-file] 54 kB
[gzipped ps-file] 136 kB
Preprint series
Technische Universität Chemnitz-Zwickau, Fakultät für Mathematik (Germany). Preprint 97-13, 1997
Mathematics Subject Classification :
47B35 [ Toeplitz operators, etc. ]
Abstract :
This paper is devoted to asymptotic estimates for the condition numbers

$\kappa(T_n(a))=||T_n(a)|| ||T_n^(-1)(a)||$

of large $n\cross n$ Toeplitz matrices $T_N(a)$ in the case where $\alpha \element L^\infinity$ and $Re \alpha \ge 0$ . We describe several classes of symbols $\alpha$ for which $\kappa(T_n(a))$ increases like $(log n)^\alpha, n^\alpha$ , or even $e^(\alpha n)$ . The consequences of the results for singular values, eigenvalues, and the finite section method are discussed. We also consider Wiener-Hopf integral operators and multidimensional Toeplitz operators.

Keywords :
Toeplitz operator, Toeplitz matrices, singular values, eigenvalues, Wiener-Hopf
Language :
english
Publication time :
5/1997