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

Abstract

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.

A favard type theorem for Hurwitz polynomials

“…The following remark permits to write the sequence (f j ) 5 j=1 in terms of the coefficients A k,j and B k,j for k = 1, 2.…”

“…5, Remark E.4] and[5, Lemma E.11]. Clearly, by using a given sequence of infinite or finite polynomials(p 1,j , q 1,j ) Ξ j=0 or (p 2,j , q 2,j ) Ξj=0 , one can generate a sequence of Hurwitz polynomials.…”

The Kharitonov theorem and robust stabilization via orthogonal polynomials

“…Remark 1 [10, Lemma 3.1] Let f n be a real polynomial of degree n, and let h n , g n be as in ( 6) and (7). The Markov parameter sequence (s j ) 2m j=0 (resp.…”

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

“…Kovalishina [37], [38], H. Dym [22], B. Simon [44], Damanik/Pushnitski/-Simon [15] and the references therein. See also [17], [18], [19], [20], [21], [34], [16], [41], [45], [31], [12], [13], [11], [6] and [7].…”