Asmptotic of the eigenvalues of Toeplitz matrices with even symbol

Abstract

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 $f(\theta )\in [f(\theta \_1,f(\theta \_2)]⟺\theta \in [\theta \_1,\theta \_2]$. 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. $[f(\theta \_2,f(\theta \_1)]$).