Expression asymptotique des valeurs propres d'une matrice de Toeplitz à symbole positif

In thi work we give an asymptotic of the eigenvalues of a Toeplitz matrix $T_N \left( \frac{\vert P\vert^2}{\vert Q\vert^2}\right)$ with $P$ et $Q$ are two complex polynomials. As a consequence of the proof we obtain also the inverse of a class of Toeplitz band matrices. Our main tool is an inversion formula of Toeplitz matrices with a real symbol, and not only with a positive symbol, which has been stated in a previous paper.