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