Expression asymptotique des valeurs propres d'une matrice de Toeplitz \`a symbole r\'eel

TL;DR

This paper derives asymptotic formulas for the minimal eigenvalues of Toeplitz matrices with real, periodic, even, and differentiable symbols, and characterizes the inverse of Toeplitz band matrices with non-vanishing symbols.

Contribution

It provides new asymptotic expressions for eigenvalues and inverse entries of Toeplitz matrices with real symbols, extending existing inversion formulas.

Findings

Asymptotic expression for minimal eigenvalues of Toeplitz matrices.

Proof that Toeplitz band matrices with non-zero symbols are invertible.

Asymptotic estimation of inverse matrix entries.

Abstract

This work provides two results obtained as a consequence of an inversion formula for Toeplitz matrices with real symbol. First we obtain an asymptotic expression for the minimal eigenvalues of a Toeplitz matrix with a symbolwhich 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 Toeplitz matrix with a regular symbol.