On Some Hermitian Forms Associated with Two Given Curves of the Complex Plane

Szegő, G.

doi:10.2307/1989634

Cited by 42 publications

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“…Since H N is positive definite, then all its eigenvalues are positive. The large N asymptotics of the smallest eigenvalue, denoted as λ N , of the Hankel matrix H N has been studied in papers by Szegő [11], Widom and Wilf [13], Chen and Lawrence [6]. See also the monograph by Wilf [14].…”

Section: Dα(x)mentioning

confidence: 99%

Small eigenvalues of large Hankel matrices: The indeterminate case

Berg¹,

Chen²,

Ismail³

2002

MATH. SCAND.

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Section: Dα(x)mentioning

confidence: 99%

Small eigenvalues of large Hankel matrices: The indeterminate case

Berg¹,

Chen²,

Ismail³

2002

MATH. SCAND.

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“…In these sections we show, following [8], by an appropriate choice of the vector {c j }, that the lower bound given by (1.13) is in fact an asymptotic estimate for large N. By a simple application of Laplace's method, N j=0 K jj is estimated. Thus the asymptotic form of λ N follows.…”

Section: Introductionmentioning

confidence: 93%

“…Let λ N denote the smallest eigenvalue. Szegö also investigated the behaviour of λ N for large N [8]. He studied the cases for which J can either be a finite or infinite interval with special choices for w. If w(x) = 1, x ∈ (−1, 1) and w(x) = 1, x ∈ (0, 1), then the respective smallest eigenvalues are for large N Widom and Wilf [11] generalised Szegö's results to a kind of "universal" law.…”

Section: Introductionmentioning

confidence: 99%

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Small eigenvalues of large Hankel matrices

Chen¹,

Lawrence²

1999

J. Phys. A: Math. Gen.

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In this paper we investigate the smallest eigenvalue, denoted as λ N , of a (N +1)×(N +1) Hankel or moments matrix, associated with the weight, w(x) = exp(−x β ), x > 0, β > 0, in the large N limit. Using a previous result, the asymptotics for the polynomials, P n (z), z / ∈ [0, ∞), orthonormal with respect to w, which are required in the determination of λ N are found. Adopting an argument of Szegö the asymptotic behaviour of λ N , for β > 1/2 where the related moment problem is determinate, is derived. This generalises the result given by Szegö for β = 1. It is shown that for β > 1/2 the smallest eigenvalue of the infinite Hankel matrix is zero, while for 0 < β < 1/2 it is greater then a positive constant. This shows a phase transition in the corresponding Hermitian random matrix model as the parameter β varies with β = 1/2 identified as the critical point. The smallest eigenvalue at this point is conjectured.

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The smallest eigenvalue of large Hankel matrices generated by a deformed Laguerre weight

Zhu

Emmart

Chen

et al. 2019

Math Methods in App Sciences

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We study the asymptotic behavior of the smallest eigenvalue, λN, of the Hankel (or moments) matrix denoted by HN=()μm+n0≤m,n≤N, with respect to the weight wfalse(xfalse)=xαnormale−xβ,3.0235ptx∈false[0,∞false),3.0235ptα>−1,3.0235ptβ>12. An asymptotic expression of the polynomials orthogonal with w(x) is established. Using this, we obtain the specific asymptotic formulas of λN in this paper. Applying a parallel numerical algorithm, we get a variety of numerical results of λN corresponding to our theoretical calculations.

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On Some Hermitian Forms Associated with Two Given Curves of the Complex Plane

Cited by 42 publications

References 0 publications

Small eigenvalues of large Hankel matrices: The indeterminate case

Small eigenvalues of large Hankel matrices: The indeterminate case

Small eigenvalues of large Hankel matrices

The smallest eigenvalue of large Hankel matrices generated by a deformed Laguerre weight

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