Smallest eigenvalues of Hankel matrices for exponential weights

Chen, Yang; Lubinsky, D. S.

doi:10.1016/j.jmaa.2004.01.032

Cited by 19 publications

(16 citation statements)

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“…Let λ N denote the smallest eigenvalue of

H_{N}

. The asymptotic behavior of λ N for large N has been investigated . Also see Beckermann and Lubinsky,in which the authors have studied the behavior of the condition number

κ ((), H_{N}) : = \frac{Λ_{N}}{λ_{N}}

, where Λ N denotes the largest eigenvalue of

H_{N}

.…”

Section: Introductionmentioning

confidence: 99%

“…Also, in the same paper, they obtained a lower bound of λ N for large N . Chen and Lubinsky obtained the behavior of λ N when

w false(x false) = {normale}^{- false| x |^{α}}, x \in double-struckR, α > 1

. Berg and Szwarc proved that λ N has exponential decay to zero for any measure that has compact support.…”

Section: Introductionmentioning

confidence: 99%

“…The remainder of this paper is organized in five sections. In Section , we reproduce some known results that will be applied to find the estimation of λ N . In Section , by adopting a previous result, we obtain the asymptotic formula for the polynomials orthonormal with respect to

w false(x false) = x^{α} {normale}^{- x^{β}}, x \in false[0, \infty false), α > - 1, β > \frac{1}{2}

, which is then employed in Sections and for the determination of the large N behavior of λ N .…”

Section: Introductionmentioning

confidence: 99%

“…The asymptotic behavior of N for large N has been investigated. [5][6][7][8][9][10][11][12][13][14] Also see Beckermann 15 and Lubinsky, 16 in which the authors have studied the behavior of the condition number ( N ) ∶=…”

Section: Introductionmentioning

confidence: 99%

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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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“…Let λ N denote the smallest eigenvalue of

H_{N}

. The asymptotic behavior of λ N for large N has been investigated . Also see Beckermann and Lubinsky,in which the authors have studied the behavior of the condition number

κ ((), H_{N}) : = \frac{Λ_{N}}{λ_{N}}

, where Λ N denotes the largest eigenvalue of

H_{N}

.…”

Section: Introductionmentioning

confidence: 99%

“…Also, in the same paper, they obtained a lower bound of λ N for large N . Chen and Lubinsky obtained the behavior of λ N when

w false(x false) = {normale}^{- false| x |^{α}}, x \in double-struckR, α > 1

. Berg and Szwarc proved that λ N has exponential decay to zero for any measure that has compact support.…”

Section: Introductionmentioning

confidence: 99%

w false(x false) = x^{α} {normale}^{- x^{β}}, x \in false[0, \infty false), α > - 1, β > \frac{1}{2}

, which is then employed in Sections and for the determination of the large N behavior of λ N .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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show abstract

“…Our focus here is to provide matching upper and lower bounds for κ(H n ) when W 2 = e −2Q is an exponential weight. In an earlier paper, the author and Y. Chen [4] obtained upper and lower bounds for λ n . Our main task here is then to obtain matching upper and lower bounds for Λ n .…”

mentioning

confidence: 99%

Condition numbers of Hankel matrices for exponential weights

Lubinsky

2006

Journal of Mathematical Analysis and Applications

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The Smallest Eigenvalue of Hankel Matrices

Berg

Szwarc

2010

Constr Approx

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Let H N = (s n+m ), n, m ≤ N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue λ N of H N . It is proved that λ N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of λ N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where λ N is known to be bounded below, we prove that the limit of the n'th smallest eigenvalue of H N for N → ∞ tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

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Smallest eigenvalues of Hankel matrices for exponential weights

Abstract: We obtain the rate of decay of the smallest eigenvalue of the Hankel matricesfor a general class of even exponential weights W 2 = exp(−2Q) on an interval I . More precise asymptotics for more special weights have been obtained by many authors.  2004 Elsevier Inc. All rights reserved.

Cited by 19 publications

References 10 publications

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

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

Condition numbers of Hankel matrices for exponential weights

The Smallest Eigenvalue of Hankel Matrices

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