Spectral Asymptotics for Toeplitz Matrices Having Certain Piecewise Continuous Symbols

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues of Toeplitz matrices with piecewise continuous symbols having a single discontinuity, extending known results to more general complex-valued functions and providing explicit formulas for determinants and eigenvalue distributions.

Contribution

It extends spectral asymptotics for Toeplitz matrices to symbols with a single discontinuity and complex exponents, providing explicit determinant formulas and eigenvalue distribution results.

Findings

Determinant asymptotics involving the geometric mean and a constant E

Explicit formula for the constant E in terms of Fourier coefficients and an analytic function of β

Determination of limiting eigenvalue distributions for the matrices

Abstract

The limiting behavior of the eigenvalues of the Toeplitz matrices $T_{n}[\sigma]=(\hat{\sigma}(i-j))$, where $0\leq i,j \leq n$, as $n \to \infty$, is investigated in the case of complex valued functions $\sigma$ defined on the unit circle $\mathbb{T}$ and having exactly one point of discontinuity. It is found that if $\sigma(z)=(-z)^{\beta}\tau(z)$, $\beta$ not an integer and $\tau$ satisfying certain smoothness conditions, then $\det T_{n}[\sigma]=\mathbf{G}[\tau]^{n+1}n^{-\beta^{2}}E[\tau,\beta](1+o(1))$ as $n \to \infty$, where $\mathbf{G}[\tau]$ denotes the geometric mean of $\tau$ and $E$ is a constant independent of $n$. A value for $E$ is found in terms of the Fourier coefficients of $\tau$ and an analytic function of $\beta$. These results were known previously in the case that $\Re \beta$, the real part of $\beta$, was sufficiently small. A corollary of this result is a determination of the limiting set and limiting distributions for the eigenvalues of $T_{n}[\sigma]$.