The matrix version of Hamburger’s theorem

Dyukarev, Yu. M.; Choque-Rivero, Abdon E.

doi:10.1134/s0001434612030236

Cited by 9 publications

(7 citation statements)

References 4 publications

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“…In view of (10.1) and (10.10) the application of Theorem 10.1 completes the proof.2 Note that formula (10.1) in the proof of Corollary 10.2 could also be proved using Schur complements analogously to the proof of formula (10.10) in the proof of Corollary 10.2.It should be mentioned that in[26] a classical result due to Hamburger[38][39][40] is generalized to the matrix case. Hamburger characterizes those sequences (s j ) ∞ j=0 ∈ K > [n,2n−1] H † 1,n−1 y [n+1,2n] , if n ≥ 1 For all n ∈ N 0 with 2n − 1 ≤ κ, let…”

mentioning

confidence: 83%

On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials

Choque-Rivero

2015

Linear Algebra and its Applications

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Dedicated to my teacher at Kharkov University Professor Yu.M. Dyukarev MSC: 44A60 33C45 30B70 Keywords: Non-degenerate truncated matricial Stieltjes moment problem Dyukarev's resolvent matrix Orthogonal matrix polynomials Non-degenerate truncated matricial Hamburger moment problem Kovalishina's resolvent matrix Matrix continued fractionsThe main result of this paper is a new representation of Yu.M. Dyukarev's resolvent matrix for the non-degenerate truncated matricial version of the classical Stieltjes moment (TMCSM) problem. This new representation is based on the use of a quadruple of q × q matrix polynomials which are characterized by certain orthogonality properties. In this way, useful new interrelations between the moment problems of Stieltjes and Hamburger which are associated with a Stieltjes positive definite sequence (s j ) 2n j=0 of complex q × q matrices are found. This includes an explicit formula connecting the resolvent matrices used by Yu.M. Dyukarev and I.V. Kovalishina, respectively. Furthermore, the coefficients in the recurrence formulas for the orthogonal polynomials with respect to a Stieltjes positive definite sequence are expressed in terms of the Dyukarev-Stieltjes parameters. Additionally, we obtain two different representations for each extremal solution of the TMCSM problem via matrix continued fraction expansions.

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confidence: 83%

On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials

Choque-Rivero

2015

Linear Algebra and its Applications

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“…Further investigations of OMP on the real line were made by Kovalishina [29], Aptekarev and Nikishin [1], Dym [20], Durán and coauthors [16][17][18][19], Dette and coauthors [11][12][13][14], Damanik/Pushnitski/Simon [10] and the references therein. See also [25][26][27][28]34].…”

Section: Statement Of the Truncated Hausdorff Matrix Moment Problemmentioning

confidence: 96%

From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem

Choque-Rivero

2015

Bol. Soc. Mat. Mex.

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In their book The Markov Moment Problem and Extremal Problems, published in 1977, M. G. Krein and A. A. Nudel'man presented a complete solution of the truncated Hausdorff moment problem via orthogonal polynomials on a finite interval [a, b]. By using the Potapov schema the matrix version of this moment problem was studied by the author, Yu. M. Dyukarev, B. Fritzsche and B. Kirstein.In the present work, we obtain the matrix generalisation of the above-mentioned Krein-Nudel'man representation. We also obtain explicit relations between four families of orthogonal matrix polynomials on [a, b] and their second kind polynomials, which are associated with the matrix version of the truncated Hausdorff moment problem. Mathematics Subject Classification

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“…) for m = 2n − 2 (m = 2n + 1) are introduced in ( 27), (A.1), ( 28) and (31). Equalities ( 8) and ( 9) are the consequence of [9,Equality (6.26)] and [10,Equalities (6.26), (6.27)].…”

Section: Introductionmentioning

confidence: 99%

“…In [31], by using a decomposition of the RM of the TSMM problem, the following were demonstrated: necessary and sufficient conditions for the TSMM problem to have a unique solution and infinitely many solutions for the Hamburger moment problem with the same moments. Note that in [47] and [14] the operator approach was employed to solve the THMM problem.…”

Section: Introductionmentioning

confidence: 99%

A Multiplicative Representation of the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem via New Dyukarev-Stieltjes Parameters

2017

MAMM

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The matrix version of Hamburger’s theorem

Cited by 9 publications

References 4 publications

On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials

On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials

From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem

A Multiplicative Representation of the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem via New Dyukarev-Stieltjes Parameters

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