The Smallest Eigenvalue of Hankel Matrices

Berg, Christian; Szwarc, Ryszard

doi:10.1007/s00365-010-9109-4

Cited by 49 publications

(61 citation statements)

References 24 publications

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“…The following result is essentially contained in [2,6] in the case of positive measures on R. The same result is true when positive measures on C are considered. We include it for the sake of completeness: (1) lim n→∞ λ n > 0.…”

Section: Remarkmentioning

confidence: 52%

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Small eigenvalues of large Hermitian moment matrices

Escribano

Gonzalo

Torrano

2011

Journal of Mathematical Analysis and Applications

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We consider an infinite Hermitian positive definite matrix M which is the moment matrix associated with a measure μ with infinite and compact support on the complex plane.We prove that if the polynomials are dense in L 2 (μ) then the smallest eigenvalue λ n of the truncated matrix M n of M of size (n + 1) × (n + 1) tends to zero when n tends to infinity. In the case of measures in the closed unit disk we obtain some related results.

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Section: Remarkmentioning

confidence: 52%

“…We finish this section with a result which relates the behaviour of the smallest eigenvalue λ n with the norm of the monic polynomials, based in the results in [5]:…”

Section: Remarkmentioning

confidence: 98%

Small eigenvalues of large Hermitian moment matrices

Escribano

Gonzalo

Torrano

2011

Journal of Mathematical Analysis and Applications

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“…Is shifted to Section 5. h Remark 3. Part of assertion (i) namely the statement P T n P n ¼ M À1 n and assertion (iv) were shown in [3]. We presented these statements for the completeness of the paper.…”

Section: Cholesky Decomposition and Its Consequencesmentioning

confidence: 99%

A few remarks on orthogonal polynomials

Szabłowski

2015

Applied Mathematics and Computation

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Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n$, two sequences of coefficients of the 3-term recurrence of the family of $\left\{ p_{n}\right\} _{n\geq 0}$, the so called "linearization coefficients" i.e. coefficients of expansion of $% p_{n}p_{m}$ in the series of $p_{j},$ $j\leq m+n.$\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0},$ we express with their help: coefficients of the power series expansion of $p_{n}$, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n,$ moments of the distribution that makes polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ orthogonal. \newline Further having two different families of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ and $\left\{ q_{n}\right\} _{n\geq 0}$ and knowing for each of them sequences of the 3-term recurrences, we give sequence of the so called "connection coefficients" between these two families of polynomials. That is coefficients of the expansions of $p_{n}$ in the series of $q_{j},$ $j\leq n.$\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials $\left\{ p_{j}\left( x)\right) \right\} _{j=0}^{n}$ as a linear transformation of the vector $\left\{ x^{j}\right\} _{j=0}^{n}$ by some lower triangular $(n+1)\times (n+1)$ matrix $\mathbf{\Pi }_{n}.$Comment: 18 page

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“…This result in the case of Hankel matrices going back to A.C. Aitken, cf. [10] has been recovered several times see [5,8] (also [12] in the context of moment Hermitian matrices). In particular, if A n = B n B t n , it follows that…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

On the inversion of infinite moment matrices

Escribano

Gonzalo

Torrano

2015

Linear Algebra and its Applications

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Motivated by [8] we study the existence of the inverse of an infinite Hermitian positive definite matrix (in short, HPD matrix) from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections. We prove a sufficient condition to assure the inversion of an HPD matrix with square summable rows. For infinite Toeplitz matrices we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of an infinite Toeplitz positive definite matrix is asymptotic Toeplitz. Such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials. We apply these results in the context of the theory of orthogonal polynomials. In particular, we show that for measures on the unit circle T verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures.

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The Smallest Eigenvalue of Hankel Matrices

Cited by 49 publications

References 24 publications

Small eigenvalues of large Hermitian moment matrices

Small eigenvalues of large Hermitian moment matrices

A few remarks on orthogonal polynomials

On the inversion of infinite moment matrices

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