Asmptotic of the eigenvalues of Toeplitz matrices with even symbol

TL;DR

This paper derives higher order asymptotic formulas for the eigenvalues of Toeplitz matrices generated by smooth, even, periodic functions, focusing on eigenvalues within specific spectral intervals as matrix size grows.

Contribution

It provides a novel higher order asymptotic expansion for all eigenvalues of Toeplitz matrices with even symbols, extending existing spectral analysis results.

Findings

Higher order asymptotic formulas for eigenvalues

Eigenvalues within specific spectral intervals

Asymptotic behavior as matrix size tends to infinity

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 $f(\theta) \in [f(\theta\_{1}, f(\theta\_{2})] \iff \theta \in [\theta\_{1}, \theta\_{2}]$. 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. $[f(\theta\_{2}, f(\theta\_1)]$).