Inversion des matrices de Toeplitz dont le symbole admet un z\'ero d'ordre fractionnaire positif, valeur propre minimale

Abstract

Inversion of Toeplitz matrices with singular symbol. Minimal eigenvalues. 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 $\alpha> \demi$ and $\alpha \in \rr \setminus \nn $, 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 a Toeplitz matrix with symbol $\varphi_{\alpha}$. Then we extend a result of A. B\"ottcher 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 admit as asymptotic $\frac{c_{\alpha}}{N^{2\alpha}} f_{1}(1)$. When $\alpha\in \nn$ 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 $\alpha$ a positive real.