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 ${\lambda }_{N,\alpha }$ of the Toepllitz matrice $T_N({\phi }_{\alpha })$ where ${\phi }_{\alpha }(e^{i\theta })=|1-e^{i\theta }{|}^{2\alpha }{c}_1(e^{i\theta })$ 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 $T_N^{-1}({\phi }_{\alpha })$ and $T_N({\phi }_{\alpha }^{-1})$ are closely related. If ${\alpha }_1+{\alpha }_2>\frac{1}{2}$ we obtain the asymptotic of the minimal eigenvalue of $T_N({\phi }_{{\alpha }_1})T_N({\phi }_{{\alpha }_2}).$

References in corpus (1)

• Trace et valeurs propres extrêmes d'un produit de matrices de Toeplitz. Le cas singulier

Cited by in corpus (1)

• Asymptotic of the terms of the Gegenbauer polynomial on the unit circle and applications to the inverse of Toeplitz matrices