This paper is essentially devoted to the study of the minimal eigenvalue $\lambda_{ N,\alpha}$ of the Toepllitz matrice $T_N(\varphi_{ \alpha})$ where $\varphi_{ \alpha}(e^{ i \theta})=|1- e^{ i \theta} |^{ 2\alpha} c_{ 1}(e^{ i \theta})$ 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 $T_N^{ -1}(\varphi_{ \alpha})$ and $T_N (\varphi^{ -1}_\alpha)$ are closely related. If $\alpha_1 + \alpha_2 > \frac{ 1}{ 2}$ we obtain the asymptotic of the minimal eigenvalue of $T_N (\varphi_{ \alpha_1}) T_N (\varphi_{ \alpha_2}).$
Categories: math.SP