Asymptotic expansion of the eigenvalues of a Toeplitz matrix with a real symbol

TL;DR

This paper derives asymptotic formulas for eigenvalues of Toeplitz matrices with real symbols and proves invertibility conditions, providing estimates for inverse entries based on symbol properties.

Contribution

It introduces new asymptotic expressions for minimal eigenvalues and establishes invertibility and inverse structure for Toeplitz band matrices with regular symbols.

Findings

Asymptotic expression for minimal eigenvalues of specific Toeplitz matrices.

Proof of invertibility for Toeplitz band matrices with non-zero symbols on the unit circle.

Asymptotic estimation of inverse matrix entries based on symbol regularity.

Abstract

Asymptotic expansion of the eigenvalues of a Toeplitz matrix with real symbol. This work provides two results obtained as a consequence of an inversion formula for Toeplitz matrices with real symbol. First we obtain an symptotic expression for the minimal eigenvalues of a Toeplitz matrix with a symbol which is periodic, even and derivable on $[0, 2\pi[$. Next we prove that a Toeplitz band matrix with a symbol without zeros on the united circle is invertible with an inverse which is essentially a band matrix. As a consequence of this last statement we give an asymptotic estimation for the entries of the inverse of a Toplitz matrix with a regular symbol.