Inversion des matrices de Toeplitz dont le symbole admet un zéro d’ordre rationnel positif, valeur propre minimale

Rambour, Philippe; Seghier, Abdellatif

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 1, pp. 173-211.

Cet article présente trois résultats distincts. Dans une première partie nous donnons l’asymptotique quand $N$ tend vers l’infini des coefficients des polynômes orthogonaux de degré $N$ associés au poids ${\varphi }_{\alpha }\left(\theta \right)={|1-e^{i\theta }|}^{2\alpha }{f}_1\left(e^{i\theta }\right)$, où $f_1$ est une fonction strictement positive suffisamment régulière et $\alpha >\frac{1}{2},\alpha \in ℝ\setminus \mathbb{N}$. Nous en déduisons l’asymptotique des éléments de l’inverse de la matrice de Toeplitz $T_N\left({\varphi }_{\alpha }\right)$ au moyen d’un noyau intégral $G_{\alpha }.$ Nous prolongeons ensuite un résultat de A. Böttcher et H. Windom relatif à l’asymptotique de la valeur propre minimale des matrices de Toepliz de symbole ${\varphi }_{\alpha }$. On sait que dans ce cas la plus petite valeur propre de cette matrice admet une asymptotique, quand $N$ tend vers l’infini, de la forme $\frac{c_{\alpha }}{N^{2\alpha }}{f}_1\left(1\right)$. Pour $\alpha \in {\mathbb{N}}^{*}$ A. Böttcher et H. Windom obtiennent une asymptotique de $c_{\alpha }$ quand $\alpha$ tend vers l’infini, et un encadrement de $c_{\alpha }$ dans les autres cas. Nous obtenons ici le même type de résultat, mais pour $\alpha \in ]\frac{1}{2},+\infty [\setminus \mathbb{N}$.

Three results are stated in this paper. The first one is devoted to the study of the orthogonal polynomial with respect of the weight ${\varphi }_{\alpha }\left(\theta \right)={|1-e^{i\theta }|}^{2\alpha }{f}_1\left(e^{i\theta }\right)$, with $\alpha >\frac{1}{2}$ and $\alpha \in ℝ\setminus \mathbb{N}$, 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 ${\left(T_N\left({\varphi }_{\alpha }\right)\right)}^{-1}$ where $T_N\left({\varphi }_{\alpha }\right)$ 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 $T_N\left({\varphi }_{\alpha }\right)$. For $N$ goes to the infinity it is well known that this minimal eigenvalue admits as asymptotic $\frac{c_{\alpha }}{N^{2\alpha }}{f}_1\left(1\right)$. 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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