A Finer Aspect of Eigenvalue Distribution of Selfadjoint Band Toeplitz Matrices

Zizler, Peter; Zuidwijk, Rob; Taylor, Keith F.; Arimoto, Shigeru

doi:10.1137/s089547989834915x

Cited by 21 publications

(18 citation statements)

References 22 publications

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“…If a is real-valued, then the matrices T n (a) are all Hermitian, and in this case a number of results on the asymptotics of the eigenvalues of T n (a) is known; see, for example, [6], [7], [13], [16], [18], [20], [21], [23], [24], [26], [27], [29], [30]. We here consider genuinely complex-valued symbols, in which case the overall picture is less complete.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

Bogoya

Böttcher

Grudsky

2012

Recent Progress in Operator Theory and Its Applications

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Abstract. We study the asymptotic behavior of individual eigenvalues of the n-by-n truncations of certain infinite Hessenberg Toeplitz matrices as n goes to infinity. The generating function of the Toeplitz matrices is supposed to be of the form a(t) = t, where α is a positive real number but not an integer and f is a smooth function in H ∞ . The classes of generating functions considered here and in a recent paper by Dai, Geary, and Kadanoff are overlapping, and in the overlapping cases, our results imply in particular a rigorous justification of an asymptotic formula which was conjectured by Dai, Geary, and Kadanoff on the basis of numerical computations.MSC 2010. Primary 47B35. Secondary 15A15, 15A18, 47N50, 65F15

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

Bogoya

Böttcher

Grudsky

2012

Recent Progress in Operator Theory and Its Applications

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“…This work is also outlined on pages 256 to 259 of [7]. Recent work on and applications of extreme eigenvalues include the papers [8][9][10][11][12][13][14][15]. The purpose of this paper is to explore the behavior of λ (n) j inside the set of the eigenvalues, for example, the asymptotics of λ (n) j as n → ∞ and j/n → x ∈ (0, 1).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…for all sufficiently large n. Note that, however, our assumptions on a are stronger than those required in [15].…”

Section: Remark 52mentioning

confidence: 97%

Inside the eigenvalues of certain Hermitian Toeplitz band matrices

Böttcher¹,

Grudsky²,

Maksimenko³

2010

Journal of Computational and Applied Mathematics

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MSC:15A18 41A25 47B35 65F15 Keywords:Toeplitz matrix Eigenvalue Asymptotic expansions a b s t r a c t While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for 1 ≤ j ≤ n. The realvalued generating function of the matrices is assumed to increase strictly from its minimum to its maximum, and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. The results, which are of interest in numerical analysis, probability theory, or statistical physics, for example, are illustrated and underpinned by numerical examples.

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“…At this moment, we are able to prove this fact in the case where a is a Laurent polynomial (and the corresponding Toeplitz matrices T n (a) are banded). In this case, it is a consequence of a result by Zizler, Zuidwijk, Taylor, and Arimoto [18]. …”

Section: Introduction and Main Resultsmentioning

confidence: 73%

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

Bogoya

Böttcher

Grudsky

et al. 2015

Journal of Approximation Theory

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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).

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A Finer Aspect of Eigenvalue Distribution of Selfadjoint Band Toeplitz Matrices

Cited by 21 publications

References 22 publications

Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

Inside the eigenvalues of certain Hermitian Toeplitz band matrices

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

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