Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols

Abstract

This paper is devoted to the asymptotic behavior of all eigenvalues of Symmetric (in general non Hermitian) Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the complex plane line as the dimension of the matrices increases to infinity. The main result describes the asymptotic structure of all eigenvalues. The constructed expansion is uniform with respect to the number of eigenvalues.

A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

“…In [7, pp. 1329], [8], and then [18], an algorithm was proposed to approximate the functions c k (θ ), which was subsequently extended and studied, see [2][3][4][15][16][17], to cover also other types of Toeplitz-like matrices A n , possessing an asymptotic expansion such as (8). We call this type of methods matrix-less, since they do not need to construct the large matrix A n to approximate its eigenvalues; indeed, they approximate the functions c k (θ ) from α small matrices A n 1 , .…”

Asymptotics for the eigenvalues of Toeplitz matrices with a symbol having a power singularity

“…The class $$$ {\mathrm{SL}}^{\alpha } $$$ was introduced for the first time in Reference 20 but the seminal paper is Reference 21. The so called simple‐loop (SL) method was then investigated further in a number of articles including References 21‐25. For a symbol $$$ a\in {\mathrm{SL}}^{\alpha } $$$, let $$$ {\lambda}_j\left({T}_n(a)\right) $$$ $$$ \left(j=1,\dots, n\right) $$$ be the eigenvalues of $$$ {T}_n(a) $$$.…”

“…The simple‐loop method is a tool for obtaining individual asymptotic expressions for $$$ {\lambda}_j\left({T}_n(a)\right) $$$ and consists in employing certain techniques of operator theory, algebra, and asymptotic analysis to obtain a recursive equation for each $$$ {\lambda}_j\left({T}_n(a)\right) $$$ (see for instance (6)), and then solve it by an iterative procedure. The method started with Laurent polynomials, 21 and then it was extended to real‐valued symbols 22‐25 …”

“…The method started with Laurent polynomials, 21 and then it was extended to real-valued symbols. [22][23][24][25] The case 𝛼 ⩾ 4 was developed in Reference 20 while the more general case 𝛼 ⩾ 1 was studied in Reference 24 where it was necessary for the symbol a to satisfy the additional relations a(t) = (t − 1)q 1 (t) and a(t) − a(e i𝜎 0 ) = (t − e i𝜎 0 )q 2 (t), for some functions q 1 , q 2 ∈ W 𝛼 , generating the so-called modified simple-loop class. It is worth noticing that every function in W 𝛼 with 𝛼 ⩾ 2 satisfies those relations.…”

Asymptotics of the eigenvalues of seven-diagonal Toeplitz matrices of a special form

“…It should be said that the symbol under consideration has specific properties: it is a real, symmetric function, and the first and second derivatives of the symbol vanish at the points t = ±1. The last condition, namely the vanishing of the second derivative, significantly complicates the problem of finding an asymptotic formula for the eigenvalues, since in this case the general research methods developed in the work [10] are inapplicable (see also works [11], [12], [13], [14], [15], which present general approaches to finding the asymptotics of the eigenvalues for various classes of Toeplitz matrices). In addition, the case we are considering is more complicated than that considered in the work [16].…”