{"title":"截断Hausdorff矩阵矩问题的两个可解矩阵","authors":"A. E. Choque-Rivero, B. E. Medina-Hernandez","doi":"10.26565/2221-5646-2022-95-01","DOIUrl":null,"url":null,"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.\n\nIn 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\nmentioned 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,\nInterpolation, 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.\nIn 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:\nFor 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.\n\nIn 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\ninterpolation matrix problems in the Nevanlinna class of functions and matrix moment problems.","PeriodicalId":33522,"journal":{"name":"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Ekonomika","volume":"30 5","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On two resolvent matrices of the truncated Hausdorff matrix moment problem\",\"authors\":\"A. E. Choque-Rivero, B. E. Medina-Hernandez\",\"doi\":\"10.26565/2221-5646-2022-95-01\",\"DOIUrl\":null,\"url\":null,\"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.\\n\\nIn 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\\nmentioned 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,\\nInterpolation, 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.\\nIn 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:\\nFor 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.\\n\\nIn 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\\ninterpolation matrix problems in the Nevanlinna class of functions and matrix moment problems.\",\"PeriodicalId\":33522,\"journal\":{\"name\":\"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Ekonomika\",\"volume\":\"30 5\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-07-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Ekonomika\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.26565/2221-5646-2022-95-01\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Ekonomika","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26565/2221-5646-2022-95-01","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
摘要
考虑有限偶数矩为非退化的情况下截断Hausdorff矩阵矩问题(THMM),如果由这些矩构成的两个块Hankel矩阵都是正定矩阵。利用分解矩阵的分块矩阵,给出了有限偶矩情况下THMM问题的解集。对于维数为$q\ * q$的矩阵矩,在非退化情况下,THMM问题的解矩阵是一个由给定矩构造的$2q\ * 2q$矩阵多项式。2001年,在[u. m .]Choque Rivero, A.E. Dyukarev,紧区间的幂矩问题,Mat. s1 -2001。-69(1 - 2)。- p。[175-187],首次提出了上述THMM问题的求解矩阵$V^{(2n+1)}$。2006年,在[A]。E. Choque Rivero, Y. M. Dyukarev, B. Fritzsche和B. Kirstein,有限区间上的截断质点矩问题,插值,Schur函数和矩问题。③。理论:广告应用。-2006年。- 165。[P. 121-173],给出了同样问题的另一个可解矩阵$U^{(2n+1)}$。本文证明了这两个解析矩阵之间存在显式关系,其形式为$V^{(2n+1)}=A U^{(2n+1)}B$,其中$A$和$B$是常数矩阵。对于求解矩阵$V^{(2n+1)}$的定义,与求解矩阵$U^{(2n+1)}$相比,需要一个附加条件,即只要求两个块汉克尔矩阵是正定的。2015年,在[A]。E. Choque Rivero,截断Hausdorff矩阵矩问题的解矩阵的Potapov到Krein-Nudel'man表示,[j]。Soc。垫,墨西哥。——2015年。- 21(2)。[P. 233—259],通过矩阵正交多项式给出了2006年的求解矩阵的表示。在这项工作中,我们没有将求解矩阵$V^{(2n+1)}$与[A]的结果联系起来。E. Choque Rivero,截断Hausdorff矩阵矩问题的解矩阵的Potapov到Krein-Nudel'man表示,[j]。Soc。垫,墨西哥。——2015年。- 21(2)。——第233—259页]。通过发现正交矩阵多项式、Blaschke-Potapov因子、Dyukarev-Stieltjes参数和矩阵连分式之间的新关系,说明了$U^{(2n+1)}$与$V^{(2n+1)}$之间关系的重要性。虽然在本工作中使用了代数恒等式,但为了证明$U^{(2n+1)}$与$V^{(2n+1)}$之间的关系,两个解析矩阵的解析证明都依赖于V.P. Potapov方法。这一方法在许多关于Nevanlinna函数类和矩阵矩问题的插值矩阵问题的著作中得到了成功的发展。
On two resolvent matrices of the truncated Hausdorff matrix moment problem
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.