Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices

Basor, Estelle; Ehrhardt, Torsten

doi:10.1002/1522-2616(200108)228:1<5::aid-mana5>3.0.co;2-e

Cited by 36 publications

(66 citation statements)

References 12 publications

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“…We need some results about Toeplitz operators and Hankel operators (see [5] and [11] for the general theory). First of all, in addition to the projections P n , and Q n = I − P n we define…”

Section: Additional Operator Theoretic Resultsmentioning

confidence: 99%

“…For example, if we let a be in L 1 (T) and denote the kth Fourier coefficients of a by a k then understanding the behavior of det (a j−k + a j+k+1 ) j,k=0,...,n−1 as n → ∞ is important in random matrix theory. It has been shown in [5] that the above determinant behaves asymptotically like G n E with certain explicitly given constants G and * 2010 MSC: 47B35; keywords: Toeplitz operator, Hankel operator, Toeplitz plus Hankel operator † ebasor@aimath.org ‡ tehrhard@ucsc.edu…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices

Basor

Ehrhardt

2017

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

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We compute the asymptotics of the determinants of certain n×n Toeplitz + Hankel matrices T n (a) + H n (b) as n → ∞ with symbols of Fisher-Hartwig type. More specifically we consider the case where a has zeros and poles and where b is related to a in specific ways. Previous results of Deift, Its and Krasovsky dealt with the case where a is even. We are generalizing this in a mild way to certain non-even symbols.E if a is a sufficiently well-behaved function. Such a result is an analogue of the classical Szegö-Widom limit theorem [17] for Toeplitz determinants.The above determinant is a special case of more general determinants, det (a j−k + b j+k+1 ) j,k=0,...,n−1,

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Section: Additional Operator Theoretic Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices

Basor

Ehrhardt

2017

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

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“…Asymptotic behavior of variable-coefficient Toeplitz determinants was studied by T. Ehrhardt and B. Shao [9]. Asymptotic of the determinants of a sum of finite Toeplitz and Hankel matrices was examined by E. Basor and T. Ehrhardt in [1,2]. Very recently T. Ehrhardt [8] suggests a new algebraic approach to the Szegő-Widom limit theorem and gives a new proof of it.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

On Asymptotics of Toeplitz Determinants with Symbols of Nonstandard Smoothness

Santos¹

2004

J Fourier Anal Appl

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We prove Szegő's strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary N-functions both satisfying the 0 2 -condition and by weight sequences satisfying some regularity and compatibility conditions.

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“…Finally, the discrete analogue of computing Toeplitz + Hankel determinants, has been recently investigated by the authors and results have been generalized to the case where the symbol is discontinuous [3] (see also [2,4]). …”

Section: Introductionmentioning

confidence: 99%

Asymptotics of Determinants of Bessel Operators

Basor¹,

Ehrhardt²

2003

Communications in Mathematical Physics

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For a ∈ L ∞ (R + ) ∩ L 1 (R + ) the truncated Bessel operator B τ (a) is the integral operator acting on L 2 [0, τ ] with the kernelwhere J ν stands for the Bessel function with ν > −1. In this paper we determine the asymptotics of the determinant det(I + B τ (a)) as τ → ∞ for sufficiently smooth functions a for which a(x) = 1 for all x ∈ [0, ∞). The asymptotic formula is of the form det(I + B τ (a)) ∼ G τ E with certain constants G and E, and thus similar to the well-known Szegö-Akhiezer-Kac formula for truncated Wiener-Hopf determinants.

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Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices

Cited by 36 publications

References 12 publications

Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices

Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices

On Asymptotics of Toeplitz Determinants with Symbols of Nonstandard Smoothness

Asymptotics of Determinants of Bessel Operators

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