Asmptotic of the eigenvalues of Toeplitz matrices with even symbol

In this paper we consider an interval $[\theta\_{1}, \theta\_{2}] \subset [0, \pi]$ and $f$ a differentiable, periodic and even function sufficiently smooth such that $f(\theta) \in [f(\theta\_{1}, f(\theta\_{2})] \iff \theta \in [\theta\_{1}, \theta\_{2}]$. Then we obtain an higher order asymptotic formula for all the eigenvalues of the Toeplitz matrix $T\_N(f)$ as $N \to + \infty$ which belong to $[f(\theta\_{1}, f(\theta\_{2})]$ (resp. $[f(\theta\_{2}, f(\theta\_1)]$).