2018/07/04 by R.A. Libby, Libby, Richard A.
Computer Science · Mathematics · #47B35 #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1807.01441
openalex publication_date 2018/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The limiting behavior of the eigenvalues of the Toeplitz matrices Tn[σ]=(σ(i-j)), where 0≤ i,j ≤ n, as n → ∞, is investigated in the case of complex valued functions σ defined on the unit circle \mathbbT 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 Tn[σ]=G[τ]n+1n^-β2E[τ,β](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 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 Tn[σ].