Toeplitz矩阵的逆矩阵不满足零阶正分数阶，值最小
Inversion des matrices de Toeplitz dont le symbole admet un zéro d'ordre fractionnaire positif, valeur propre minimale
摘要：
具有奇异符号的Toeplitz矩阵的逆。最小特征值。本文提出了三个结果。第一部分研究了关于权$φ_α (θ)=\vert 1- e^{i θ} \vert ^{2α} f_{1}(e^{i θ})$的正交多项式，其中$α> \demi$和$α\in \rr \setminus \nn $，以及$f_{1}$是正则函数。我们得到了这些多项式的系数的渐近展开式，我们推导出的$\left( T_{N} (φ_α)\right)^{-1}$，其中$T_{N} (φ_α)$是一个Toeplitz矩阵符号$φ_α$的条目的渐近。然后推广了A. Böttcher和H.得到了Toeplitz矩阵$T_{N}(φ_α)$的最小特征值的一个重要结果.对于$N$趋于无穷大，众所周知，这个最小特征值承认为渐近$\frac{c_α}{N^{2α}} f_{1}(1)$。当$α\in \nn$时，以前的作者得到了$c_α$对于$α$趋于无穷大的渐近解，并得到了对于其它情形$c_α$的界.在这里，我们获得了相同类型的结果，但$α$为正实数。
英文摘要：
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 $φ_α (θ)=\vert 1- e^{i θ} \vert ^{2α} f_{1}(e^{i θ})$, with $α> \demi$ and $α\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 $\left( T_{N} (φ_α)\right)^{-1}$ where $T_{N} (φ_α)$ is a Toeplitz matrix with symbol $φ_α$. Then we extend a result of A. Böttcher and H. Widom result related to the minimal eigenvalue of the Toeplitz matrix $T_{N}(φ_α)$. For $N$ goes to the infinity it is well known that this minimal eigenvalue admit as asymptotic $\frac{c_α}{N^{2α}} f_{1}(1)$. When $α\in \nn$ the previous authors obtain an asymptotic of $c_α$ for $α$ going to the infinity, and they have the bounds of $c_α$ for the other cases. Here we obtain the same type of results but for $α$ a positive real.
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