Maximum norm versions of the Szegő and Avram–Parter theorems for Toeplitz matrices

Abstract

The collective behavior of the singular values of large Toeplitz matrices is described by the Avram-Parter theorem. In the case of Hermitian matrices, the Avram-Parter theorem is equivalent to Szegő's theorem on the eigenvalues. The Avram-Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the singular values and the appropriately ordered absolute values of the symbol. The purpose of this paper is twofold. Under natural hypotheses, we first strengthen the known estimates in the mean to estimates in the maximum norm, thus turning from collective results on the singular values to results on individual singular values. Secondly, we want to emphasize that the use of the quantile function eases the proofs and statements of results significantly and provides a promising language for forthcoming research into higher order asymptotics for individual singular values. (J.M. Bogoya), aboettch@mathematik.tu-chemnitz.de (A. Böttcher), grudsky@math.cinvestav.mx (S.M. Grudsky), maximenko@esfm.ipn.mx (E.A. Maximenko).

A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices

“…In the case where u = 1 identically, it was already formulated in [13]. In the case where u = 1 identically and α = 0, it can be formally proved by adapting the argument used by Bogoya, Böttcher, Grudsky, and Maximenko in the proof of [7,Theorem 1.6]. …”

On the Asymptotic Equivalence of Circulant and Toeplitz Matrices

“…1) C N : Bogoya et al [15] proved that the samples of the symbol h are the main asymptotic terms of the eigenvalues of the Toeplitz matrix H N . Given only H N , one practical strategy for estimating the eigenvalues is to first approximate h by the (N − 1) th partial Fourier sum…”

“…for all k ≥ k 0 , where the last line follows from (14) and (15). Note that the above equation is equivalent to…”

Cofactors and eigenvectors of banded Toeplitz matrices: Trench formulas via skew Schur polynomials

“…Trench [29,30] discovered another equivalent formula for the determinants and exact formulas for the inverse matrices and eigenvectors. Among many recent investigations on Toeplitz matrices and their generalizations we mention [2,5,6,10,[15][16][17][18]. See also the books [8,11,12,20] which employ an analytic approach and contain asymptotic results on Toeplitz determinants, inverse matrices, eigenvalue distribution, etc.It is obvious from the Jacobi-Trudi formulas that there is a simple connection between Toeplitz minors and skew Schur polynomials.…”

“…Let a 2 = 1, n = 7, ξ = (3, 6) and η = (3, 7). Then d = 2, m = 5, ρ = (1, 2, 4, 5, 7) and σ =(1,2,4,5,6). By striking out the rows ξ and the columns η in the Toeplitz matrix T 7 (a) we obtain the minordet(T 7 (a) ρ,σ ) = a 0 a −1 a −3 0 0 a 1 a 0 a −2 a −3 0 0 a 2 a 0 a −1 a −2 0 By Vieta's formulas the a k are expressed through e 2−k : det(T 7 (a) ρ,σ ) = −2 −e −1 −e 1 e 2 −e 3 e −4 −e −3 −e −1 e 0 −e Using (2.5), we have λ = (5 2 , 6 − 2, 3 − 1) = (5 2 , 4, 2) and µ = (7 − 2, 3 − 1) = (5, 2), then det(T n (a) ρ,σ ) = (−1) 10+19+18 s (5,5,4,2)/(5,2) = −s (5,4,2)/(2) .…”