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

Abstract

This paper is essentially devoted to the study of the minimal eigenvalue $λ_{N,α}$ of the Toepllitz matrice $T_N(φ_α)$ where $φ_α(e^{i θ})=|1- e^{i θ} |^{2α} c_{1}(e^{i θ})$ with $c_{1}$ a positive sufficiently smooth function and $0<α<\frac{1}{2}$. We obtain $λ_{N,α}\sim c_αN^{-2α}c_{1}(1)$ when $N$ goes to the infinity and we have the bounds of $c_α$. To obtain the asymptotic of $λ_{N,α}$ we give a theorem which suggests that the entries of $T_N^{-1}(φ_α)$ and $T_N (φ^{-1}_α)$ are closely related. If $α_1 + α_2 > \frac{1}{2}$ we obtain the asymptotic of the minimal eigenvalue of $T_N (φ_{α_1}) T_N (φ_{α_2}).$