Spectral Asymptotics for Toeplitz Matrices Having Certain Piecewise Continuous Symbols

Abstract

AbstractThe limiting behavior of the eigenvalues of the Toeplitz matrices \(T_{n}[\sigma ]=(\hat {\sigma }(i-j))\), where 0 ≤ i, j ≤ n, as n →∞, 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)β τ(z), β not an integer and τ 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 →∞, where 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 \(\Re \beta \), 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].KeywordsToeplitz operatorToeplitz determinantSpectral asymptotics