On two resolvent matrices of the truncated Hausdorff matrix moment problem
Abstract
We consider the truncated Hausdorff matrix moment problem (THMM) in case of a finite number of even moments to be called non degenerate if two block Hankel matrices constructed via the moments are both positive definite matrices. The set of solutions of the THMM problem in case of a finite number of even moments is given with the help of the block matrices of the so-called resolvent matrix. The resolvent matrix of the THMM problem in the non degenerate case for matrix moments of dimension $q\times q$, is a $2q\times 2q$ matrix polynomial constructed via the given moments.
In 2001, in [Yu.M. Dyukarev, A.E. Choque Rivero, Power moment problem on compact intervals, Mat. Sb.-2001. -69(1-2). -P.175-187], the resolvent matrix $V^{(2n+1)}$ for the
mentioned THMM problem was proposed for the first time. In 2006, in [A. E. Choque Rivero, Y. M. Dyukarev, B. Fritzsche and B. Kirstein, A truncated matricial moment problem on a finite interval,
Interpolation, Schur Functions and Moment Problems. Oper. Theory: Adv. Appl. -2006. - 165. - P. 121-173], another resolvent matrix $U^{(2n+1)}$ for the same problem was given.
In this paper, we prove that there is an explicit relation between these two resolvent matrices of the form $V^{(2n+1)}=A U^{(2n+1)}B$, where $A$ and $B$ are constant matrices. We also focus on the following difference:
For the definition of the resolvent matrix $V^{(2n+1)}$, one requires an additional condition when compared with the resolvent matrix $U^{(2n+1)}$ which only requires that two block Hankel matrices be positive definite.
In 2015, in [A. E. Choque Rivero, From the Potapov to the Krein-Nudel'man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259], a representation of the resolvent matrix of 2006 via matrix orthogonal polynomials was given. In this work, we do not relate the resolvent matrix $V^{(2n+1)}$ with the results of [A. E. Choque Rivero, From the Potapov to the Krein-Nudel'man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259]. The importance of the relation between $U^{(2n+1)}$ and $V^{(2n+1)}$ is explained by the fact that new relations among orthogonal matrix polynomials, Blaschke-Potapov factors, Dyukarev-Stieltjes parameters, and matrix continued fraction can be found. Although in the present work algebraic identities are used, to prove the relation between $U^{(2n+1)}$ and $V^{(2n+1)}$, the analytic justification of both resolvent matrices relies on the V.P. Potapov method. This approach was successfully developed in a number of works concerning
interpolation matrix problems in the Nevanlinna class of functions and matrix moment problems.
Downloads
References
A. E. Choque Rivero, Y. M. Dyukarev, B. Fritzsche and B. Kirstein, A truncated matricial moment problem on a finite interval, Interpolation, Schur Functions and Moment Problems. Oper. Theory: Adv. Appl. - 2006. - 165. - P. 121-173. DOI: 10.1007/3-7643-7547-7_4.
A. E. Choque Rivero, Y. M. Dyukarev, B. Fritzsche and B. Kirstein, A truncated matricial moment problem on a finite interval. The case of an odd number of prescribed moments, Interpolation, Schur Functions and Moment Problems. Oper. Theory: Adv. Appl. - 2007. - 176. - P. 99-174. DOI: 10.1007/978-3-7643-8137-0_2.
A. E. Choque Rivero, V. I. Korobov, G. M. Sklyar, The admissible control problem from the moment problem point of view, Appl. Math. Lett. -- 2010. -- 23(1). -- P. 58--63. DOI: 10.1016/j.aml.2009.06.030.
A. E. Choque Rivero and Yu. Karlovich, The time optimal control as an interpolation problem, Commun. Math. Anal. - 2011. - 3. - P. 1-11.
A. E. Choque Rivero, Multiplicative structure of the resolvent matrix for the truncated Hausdorff matrix moment problem, Operator Theory: Advances and Applications. -- 2012. -- 226. -- P. 193--210. DOI: 10.1007/978-3-0348-0428-8_4.
A. E. Choque Rivero, Decompositions of the Blaschke-Potapov factors of the truncated Hausdorff matrix moment problem. The case of even number of moments, Commun. Math. Anal. - 2014. - 17(2). - P. 82-97.
A. E. Choque Rivero, On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials, Linear Algebra and its Applications - 2015. - 474. - P. 44-109. DOI: 10.1016/j.laa.2015.01.027.
A. E. Choque Rivero, From the Potapov to the Krein-Nudel'man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259. https://doi.org/10.1007/s40590-015-0060-z.
A. E. Choque Rivero, Dyukarev-Stieljtes parameters of the truncated Hausdorff matrix moment problem, Boletin Soc. Mat. Mexicana. - 2017. - 23(2). - P. 891--918. https://doi.org/10.1007/s40590-015-0083-5.
A. E. Choque Rivero, On the solution set of the admissible bounded control problem via orthogonal polynomials, IEEE Trans. Autom. Control. - 2017. -- 62(10). -- P. 5213--5219. https://doi: 10.1109/TAC.2016.2633820.
A. E. Choque Rivero, Relations between the orthogonal matrix polynomials on [a,b], Dyukarev-Stieltjes parameters, and Schur complements, Spec. Matrices. -- 2017. -- 5. -- P. 303--318. https://doi.org/10.1515/spma-2017-0023.
A. E. Choque Rivero, A multiplicative representation of the resolvent matrix of the truncated Hausdorff matrix moment problem via new Dyukarev-Stieltjes parameters, Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics - 2017. - 85. - P. 16-42. DOI: 10.26565/2221-5646-2017-85-02.
A. E. Choque Rivero, Three-term recurrence relation coefficients and continued fractions related to orthogonal matrix polynomials on the finite interval [a,b], Linear and Multilinear Algebra, 2020. - P. 1-20. https://doi.org/10.1080/03081087.2020.1747967.
A. E. Choque Rivero, S. M. Zagorodnyuk, An algorithm for the truncated matrix Hausdorff moment problem, Commun. Math. Anal. - 2014. - 17(2). - P. 108-130.
A. Dubovoj, B. Fritzsche and B. Kirstein, Matricial Version of the Classical Schur Problem, Teubner-Texte Math. (Teubner Texts in Mathematics), vol. 129, Teubner Verlagsgesellschaft mbH, Stuttgart, 1992.
Yu. M. Dyukarev, Indeterminacy criteria for the Stieltjes matrix moment problem, Math. Notes. - 2004. - 75(1-2). - P. 66-82. https://doi.org/10.1023/B:MATN.0000015022.02925.bd.
Yu. M. Dyukarev, Indeterminacy of interpolation problems in the Stieltjes class, Mat. Sb. - 2005. - 196(3). - P. 61-88.
https://doi.org/10.1070/SM2005v196n03ABEH000884.
Yu. M. Dyukarev, A Generalized Stieltjes Criterion for the Complete Indeterminacy of Interpolation Problems, Math. Notes. -- 2008. -- 84(1). -- P. 23--39. https://doi.org/10.1134/S000143460807002X.
Yu. M. Dyukarev and A. E. Choque Rivero, Power moment problem on compact intervals, Mat. Notes -- 2001. -- 69(1-2). -- P. 175--187. https://doi.org/10.1023/A:1002868117970.
Yu. M. Dyukarev and A. E. Choque Rivero, A matrix version of one Hamburger theorem, Mat. Sb. - 2012. -- 91(4). -- P. 522-529. https://doi.org/10.1134/S0001434612030236.
B. Fritzsche, B. Kirstein and C. Madler, On Hankel nonegative definite sequences, the canonical Hankel parametrization, and orthogonal matrix polynomials, Compl. Anal. Oper. Theory. -- 2011. -- 5(2). -- P. 447--511. https://doi.org/10.1007/s11785-010-0054-9.
B. Fritzsche, B. Kirstein and C. Madler, A Schur–Nevanlinna type algorithm for the truncated matricial Hausdorff moment problem, Compl. Anal. Oper. Theory. - 2021. - 15(25). - P. 1-129. https://doi.org/10.1007/s11785-020-01051-w.
I. V. Kovalishina, Analytic theory of a class of interpolation problems, Izv. Math. - 1983. - 47(3). - P. 455-497. https://doi.org/10.1070/IM1984v022n03ABEH001452.
E. Freitag and R. Busam. Complex analysis. 2005. Springer--Verlag.
H. Thiele. Beitr"age zu matriziellen Potenzmomentenproblemen, PhD Thesis. Leipzig University, 2006. In German.
S. M. Zagorodnyuk, The truncated matrix Hausdorff moment problem, Methods Appl. Anal. - 2012. - 19(1). - P. 021-042. DOI: 10.4310/MAA.2012.v19.n1.a2.
Copyright (c) 2022 A. E. Choque-Rivero, B. E. Medina-Hernandez
![Creative Commons License](http://i.creativecommons.org/l/by-nc-nd/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
The copyright holder is the author.
Authors who publish with this journal agree to the following terms:
1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal. (Attribution-Noncommercial-No Derivative Works licence).
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (see The Effect of Open Access).