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