Valeur propre minimale d'une matrice de Toeplitz et d'un produit de matrices de Toeplitz

TL;DR

This paper investigates the minimal eigenvalue of Toeplitz matrices generated by specific functions, deriving asymptotic behavior and bounds, and explores eigenvalues of products of such matrices.

Contribution

It provides new asymptotic formulas and bounds for the minimal eigenvalues of Toeplitz matrices with singular symbols and their products, extending prior spectral analysis.

Findings

Asymptotic behavior of minimal eigenvalues for Toeplitz matrices with singular symbols.

Bounds for the constants involved in eigenvalue asymptotics.

Asymptotic analysis of eigenvalues for products of Toeplitz matrices.

Abstract

This paper is essentially devoted to the study of the minimal eigenvalue $\lambda_{N,\alpha}$ of the Toepllitz matrice $T_N(\varphi_{\alpha})$ where $\varphi_{\alpha}(e^{i \theta})=|1- e^{i \theta} |^{2\alpha} c_{1}(e^{i \theta})$ with $c_{1}$ a positive sufficiently smooth function and $0<\alpha<\frac{1}{2}$. We obtain $\lambda_{N,\alpha}\sim c_{\alpha}N^{-2\alpha}c_{1}(1)$ when $N$ goes to the infinity and we have the bounds of $c_{\alpha}$. To obtain the asymptotic of $\lambda_{N,\alpha}$ we give a theorem which suggests that the entries of $T_N^{-1}(\varphi_{\alpha})$ and $T_N (\varphi^{-1}_\alpha)$ are closely related. If $\alpha_1 + \alpha_2 > \frac{1}{2}$ we obtain the asymptotic of the minimal eigenvalue of $T_N (\varphi_{\alpha_1}) T_N (\varphi_{\alpha_2}).$