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Asmptotic of the eigenvalues of Toeplitz matrices with even symbol

Philippe Rambour

TL;DR

This work analyzes the eigenvalues of real symmetric Toeplitz matrices T_N(f) with an even, differentiable, 2π-periodic symbol f, focusing on a monotone subinterval [θ1, θ2] within (0, π). It derives a higher-order asymptotic expansion for eigenvalues lying in [f(θ1), f(θ2)], giving an explicit three-term formula with coefficients expressed through a Wiener-Hopf‑derived function ρ and its derivatives, and proves uniform O((N+2)^{-3}) accuracy. The authors employ a Toeplitz inversion formula together with predictor polynomials and Wiener–Hopf techniques to reduce the problem to a scalar equation and to control error terms, yielding eigenvalue separation results and a locality aspect of Toeplitz spectra. They also extend the results to the complementary monotone side via the A^− class and discuss implications for symbol regularity and connections to prior eigenvalue expansions for Toeplitz matrices. The findings advance precise spectral asymptotics for structured matrices and have potential relevance to numerical analysis and mathematical physics where Toeplitz operators arise.

Abstract

In this paper we consider an interval $[θ\_{1}, θ\_{2}] \subset [0, π]$ and $f$ a differentiable, periodic and even function sufficiently smooth such that $f(θ) \in [f(θ\_{1}, f(θ\_{2})] \iff θ\in [θ\_{1}, θ\_{2}]$. 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 $[f(θ\_{1}, f(θ\_{2})]$ (resp. $[f(θ\_{2}, f(θ\_1)]$).

Asmptotic of the eigenvalues of Toeplitz matrices with even symbol

TL;DR

This work analyzes the eigenvalues of real symmetric Toeplitz matrices T_N(f) with an even, differentiable, 2π-periodic symbol f, focusing on a monotone subinterval [θ1, θ2] within (0, π). It derives a higher-order asymptotic expansion for eigenvalues lying in [f(θ1), f(θ2)], giving an explicit three-term formula with coefficients expressed through a Wiener-Hopf‑derived function ρ and its derivatives, and proves uniform O((N+2)^{-3}) accuracy. The authors employ a Toeplitz inversion formula together with predictor polynomials and Wiener–Hopf techniques to reduce the problem to a scalar equation and to control error terms, yielding eigenvalue separation results and a locality aspect of Toeplitz spectra. They also extend the results to the complementary monotone side via the A^− class and discuss implications for symbol regularity and connections to prior eigenvalue expansions for Toeplitz matrices. The findings advance precise spectral asymptotics for structured matrices and have potential relevance to numerical analysis and mathematical physics where Toeplitz operators arise.

Abstract

In this paper we consider an interval and a differentiable, periodic and even function sufficiently smooth such that . Then we obtain an higher order asymptotic formula for all the eigenvalues of the Toeplitz matrix as which belong to (resp. ).

Paper Structure

This paper contains 8 sections, 12 theorems, 121 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Proof of Theorem \ref{['THEO1ter']}
  3. Preliminaries
  4. Equation for the eigenvalues
  5. Limit of the sequence $(\rho_{N})_{(N\in \mathbb N)}$
  6. Derivation and solutions of the equation for the eigenvalues
  7. Proof of Theorem \ref{['THEO3']}
  8. Appendix

Key Result

Theorem 1

Let $f$ in $C^4[0, 2\pi]$ be such that $f \in \mathcal{A}^+\left([\theta_{1}, \theta_{2}]\right)$ for an interval $[\theta_{1}, \theta_{2}] \subset]0, \pi[$. Then for all interval $[a,b]\subset ]\theta_{1}, \theta_{2}[$ and for a sufficiently large integer $N$ we have the two following statements.

Theorems & Definitions (23)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 3
  • ...and 13 more