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Philippe Rambour; Abdellatif Seghier
Inversion des matrices de Toeplitz dont le symbole admet un zéro d’ordre rationnel positif, valeur propre minimale
Annales de la faculté des sciences de Toulouse Sér. 6, 21 no. 1 (2012), p. 173-211, doi: 10.5802/afst.1332
Article PDF | Reviews MR 2954108 | Zbl 1243.15017

Résumé - Abstract

Three results are stated in this paper. The first one is devoted to the study of the orthogonal polynomial with respect of the weight $\varphi _{\alpha } (\theta )=\vert 1- e^{i \theta } \vert ^{2\alpha } f_{1}(e^{i \theta })$, with $\alpha > \frac{1}{2}$ and $\alpha \in \mathbb{R}\setminus \mathbb{N}$, and $f_{1}$ a regular function. We obtain an asymptotic expansion of the coefficients of these polynomials, and we deduce an asymptotic of the entries of $\left( T_{N} (\varphi _{\alpha })\right)^{-1}$ where $T_{N} (\varphi _{\alpha })$ is the Toeplitz matrix with symbol $\varphi _{\alpha }$. Then we extend a result of A. Böttcher and H. Widom result related to the minimal eigenvalue of the Toeplitz matrix $T_{N}(\varphi _{\alpha })$. For $N$ goes to the infinity it is well known that this minimal eigenvalue admits as asymptotic $\frac{c_{\alpha }}{N^{2\alpha }} f_{1}(1)$. When $\alpha \in \mathbb{N}$ the previous authors obtain an asymptotic of $c_{\alpha }$ for $\alpha $ going to the infinity, and they have the bounds of $c_{\alpha }$ for the other cases. Here we obtain the same type of results but for $\alpha \in ]\frac{1}{2},+\infty [\setminus \mathbb{N}$.

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