  The first part of this paper is devoted to the study of the orthogonal
polynomial on the circle, with respect of a weight of type $f_\alpha (\theta) =
(2\cos \theta- 2\cos \theta_0)^{2\alpha} c_1$ with $\theta_0 \in ]0,\pi[$, -1/2
<\alpha<1/2 and c_1 a sufficiently smooth function. In a second part of the
paper we obtain an asymptotic of the entries $(T_N f_\alpha)^{-1}_{k+1,l+1}$
for sufficiently large values of $k,l$, that provides a lower bound on the
eigenvalues of this matrix.
