A Multiplicative Representation of the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem via New Dyukarev-Stieltjes Parameters.
Keywords:
Orthogonal matrix polynomial, Dyukarev-Stieltjes parameter, Resolvent matrix, Continued fractions
Abstract
A new multiplicative decomposition of the resolvent matrix of the truncated Hausdorff matrix moment (THMM) problem in the case of an odd and even number of moments via new Dyukarev-Stieltjes matrix (DSM) parameters is attained. Additionally, we derive Blaschke-Potapov factors of auxiliary resolvent matrices; each factor is decomposed with the help of the DSM parameters.Downloads
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References
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Choque Rivero A.E., Decompositions of the Blaschke-Potapov factors of the truncated Hausdorff matrix moment problem. The case of odd number of moments, / Commun. Math. Anal., 2014. – 17(2). – P. 66–81.
Choque Rivero A.E., 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.
Choque Rivero A.E., From the Potapov to the Krein-Nudel’man
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Choque Rivero A.E., On Dyukarev’s resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials, / Lin. Alg. and Appl., 2015. – 474. – P. 44–109.
Choque Rivero A.E., On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials, / Lin. Alg. and Appl., 2015. – 476. – P. 56–84.
Choque Rivero A.E., Dyukarev-Stieljtes parameters of the truncated Hausdorff matrix moment problem, / Boletin Soc. Mat. Mexicana, 2016. – P. 1–28. DOI: 10.1007/s40590-015-0083-5.
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Choque Rivero A.E., M¨adler C., On Hankel positive definite perturbations of Hankel positive definite sequences and interrelations to orthogonal matrix polynomials, / Complex Anal. Oper. Theory, 2014. – 8(8). – P. 121–173.
Choque Rivero A.E., M¨adler C., On resolvent matrix, Dyukarev–Stieltjes parameters and orthogonal matrix polynomials via [0, +∞)-Stieltjes transformed sequences, / Complex Anal. Oper. Theory, 2017. – P. 1–44, DOI
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Dyukarev Yu.M., Indeterminacy criteria for the Stieltjes matrix moment problem, / Math. Notes, 2004. – 75(1-2). – P. 66–82.
Dyukarev Yu.M., Indeterminacy of interpolation problems in the Stieltjes class, Math. Sb., 2005. – 196(3). – P. 61–88.
Dyukarev Yu.M., A Generalized Stieltjes Criterion for the Complete Indeterminacy of Interpolation Problems, / Math. Notes, 2008. – 84(1). –P. 23–39.
Dyukarev Yu.M., Criterion for complete indeterminacy of limiting interpolation problem of Stieltjes type in terms of orthonormal matrix functions, / Russian Mathematics (Iz. VUZ), 2015. – 59(4). – P. 61–88.
Dyukarev Yu.M., Geometric and operator measures of degeneracy for the set of solutions to the Stieltjes matrix moment problem, Mat. Sb., 2016. – 207(4). – P. 47–64.
Dyukarev Yu.M. ,Choque Rivero A.E., Power moment problem on compact intervals, / Mat. Sb., 2001. – 69(1-2). – P. 175–187.
Dyukarev Yu.M. ,Choque Rivero A.E., A matrix version of one Hamburger theorem, / Mat. Sb., 2012. – 91(4). – P. 522–529.
Dyukarev Yu.M., Serikova I.Yu., Complete indeterminacy of the Nevanlinna Pick problem in the class S[a,b], / Russian Math. (Iz. VUZ), 2007. – 51(11). – P. 17–29.
Dyukarev Yu.M., Serikova I.Yu., Step-by-step solving of ordered interpolational problem for Stieltjes functions, / Russian Math. (Iz.VUZ), 2017. – 6. – P. 18–32.
Fritzsche B., Kirstein B., M¨adler C., On Hankel nonegative definite sequences, the canonical Hankel parametrization, and orthogonal matrix polynomials, / Compl. Anal. Oper. Theory, 2011. – 5(2). – P. 447–511.
Fritzsche B., Kirstein B., M¨adler C., Transformations of matricial α-Stieltjes non-negative definite sequences, / Lin. Alg. and Appl., 2013. – 439. – P. 3893–3933.
Geronimo J.S., Scattering theory and matrix orthogonal polynomials on the real line, / Circuits Systems Sigmal Process, 1982. – 1(3-4). – P. 471–495.
I.V. Kovalishina, New aspects of the classical moment problems. Second doctoral thesis (in Russian), 1986. – Institute of Railway-Transport Engineers.
Kovalishina I.V., Analytic theory of a class of interpolation problems, / Izv. Math., 1983. – 47(3). – P. 455–497.
Krein M.G., Fundamental aspects of the representation theory of hermitian operators with deficiency (m, m), / Ukrain. Mat. Zh., 1949. – 1(2). – P. 3–66.
Krein M.G., Infinite J-matrices and a matrix moment problem, / Dokl. Akad. Nauk, 1949. – 69(2). – P. 125–128.
Miranian L., Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory, / J. Phys. A: Math. Gen., 2005. – 38. –P. 5731–5749.
Potapov V.P., The multiplicative structure of J-nonexpansive matrix functions, / Trudy Moskov. Mat. Ob., 1955. – 4. – P. 125–236.
Serikova I. Yu., Indeterminacy criteria for the Nevanlinna-Pick interpolation problem in class R[a, b], / Zb. Pr. Inst. Mat. NAN Ukr., 2006. – 3(4). – P. 126–142.
Simon B., The classical problem as a self-adjoint finite difference operator, / Adv. Math., 1998. – 137. – P. 82–203.
Sinap A., Van Assche W., Orthogonal matrix polynomials and applications, / J. Comput. Appl. Math., 1996. – 66(1-2). – P. 27–52.
H. Thiele, Beitr¨age zu matriziellen otenzmomentenproblemen, PhD Thesis, 2006. – Leipzig University.
Zagorodnyuk S., The truncated matrix Hausdorff moment problem, / Methods Appl. Anal., 2012. – 19(1). – P. 021–042.
Choque Rivero A.E., The resolvent matrix for the matricial Hausdorff moment problem expressed by orthogonal matrix polynomials, / Complex Anal. Oper. Theory, 2013. – 7(4). – P. 927–944.
Choque Rivero A.E., Decompositions of the Blaschke-Potapov factors of the truncated Hausdorff matrix moment problem. The case of odd number of moments, / Commun. Math. Anal., 2014. – 17(2). – P. 66–81.
Choque Rivero A.E., 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.
Choque Rivero A.E., 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.
Choque Rivero A.E., On Dyukarev’s resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials, / Lin. Alg. and Appl., 2015. – 474. – P. 44–109.
Choque Rivero A.E., On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials, / Lin. Alg. and Appl., 2015. – 476. – P. 56–84.
Choque Rivero A.E., Dyukarev-Stieljtes parameters of the truncated Hausdorff matrix moment problem, / Boletin Soc. Mat. Mexicana, 2016. – P. 1–28. DOI: 10.1007/s40590-015-0083-5.
Choque Rivero A.E., Dyukarev Yu.M., Fritzsche B. and Kirstein B., A truncated matricial moment problem on a finite interval. The case of an odd number of prescribed moments, / System Theory, Schur Algorithm and Multidimensional Analysis. Oper. Theory: Adv. Appl., 2007. – 176. – P. 99–174.
Choque Rivero A.E., Dyukarev Yu.M., Fritzsche B. and Kirstein B., A truncated matricial moment problem on a finite interval. // Interpolation, Schur Functions and Moment Problems. Oper. Theory: Adv. Appl., 2006. – 165. – P. 121–173.
Choque Rivero A.E., Garza L.E, Moment perturbation of matrix polynomials, / Integral Transforms Spec. Funct., 2015. – 26. – P. 177–191.
Choque Rivero A.E., M¨adler C., On Hankel positive definite perturbations of Hankel positive definite sequences and interrelations to orthogonal matrix polynomials, / Complex Anal. Oper. Theory, 2014. – 8(8). – P. 121–173.
Choque Rivero A.E., M¨adler C., On resolvent matrix, Dyukarev–Stieltjes parameters and orthogonal matrix polynomials via [0, +∞)-Stieltjes transformed sequences, / Complex Anal. Oper. Theory, 2017. – P. 1–44, DOI
10.1007/s11785-017-0655-7.
Choque Rivero A.E., Zagorodnyuk S., An algorithm for the truncated matrix Hausdorff moment problem, / Commun. Math. Anal., 2014. – 17(2). – P. 108–130.
Damanik D., Pushnitski A. and Simon B., The analytic theory of matrix orthogonal polynomials, / Surv. Approx. Theory, 2008. – 4. – P. 1–85.
Dette H., Studden W.J., Matrix measures, moment spaces and Favard’s theorem on the interval [0, 1] and [0, ∞), / Lin. Alg. and Appl., 2002. – 345. – P. 169–193.
Duran A., Markov’s theorem for orthogonal matrix polynomials, / Can. J. Math., 1996. – 48(6). – P. 1180–1195.
Dur´an A.J., Rodrigues’s formulas for orthogonal matrix polynomials satisfying higher-order differential equations, / Exp. Math., 2011. – 20(1). – P. 15–24.
Dur´an A.J., Gr¨unbaum F.A., Matrix differential equations and scalarpolynomials satisfying higher order recursions, / J. Math. Anal. Appl., 2009. – 354. – P. 1–11.
Dur´an A.J., De la Iglesia M.D., Some examples of orthogonal matrix polynomials satisfying odd order differential equations, / J. Approx. Theory, 2008. – 150. – P. 153–174.
Dur´an A.J., L´opez–Rodr´iguez P., Structural formulas for orthogonal matrix polynomials satisfying second order differential equations, II, / Constr. Approx., 2007. – 26(1). – P. 29–47.
Dym H., On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy, / Integral Equations and Operator Theory, 1989. – 12. – P. 757–812.
Dyukarev Yu.M., The multiplicative structure of resolvent matrices of interpolation problems in the Stieltjes class, / Vestnik Kharkov Univ. Ser. Mat. Prikl. Mat. i Mekh., 1999. – 458. – P. 143–153.
Dyukarev Yu.M., Factorization of operator functions of multiplicative Stieltjes class (Russian), / Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh., 2000. – 9. – P. 23–26.
Dyukarev Yu.M., Indeterminacy criteria for the Stieltjes matrix moment problem, / Math. Notes, 2004. – 75(1-2). – P. 66–82.
Dyukarev Yu.M., Indeterminacy of interpolation problems in the Stieltjes class, Math. Sb., 2005. – 196(3). – P. 61–88.
Dyukarev Yu.M., A Generalized Stieltjes Criterion for the Complete Indeterminacy of Interpolation Problems, / Math. Notes, 2008. – 84(1). –P. 23–39.
Dyukarev Yu.M., Criterion for complete indeterminacy of limiting interpolation problem of Stieltjes type in terms of orthonormal matrix functions, / Russian Mathematics (Iz. VUZ), 2015. – 59(4). – P. 61–88.
Dyukarev Yu.M., Geometric and operator measures of degeneracy for the set of solutions to the Stieltjes matrix moment problem, Mat. Sb., 2016. – 207(4). – P. 47–64.
Dyukarev Yu.M. ,Choque Rivero A.E., Power moment problem on compact intervals, / Mat. Sb., 2001. – 69(1-2). – P. 175–187.
Dyukarev Yu.M. ,Choque Rivero A.E., A matrix version of one Hamburger theorem, / Mat. Sb., 2012. – 91(4). – P. 522–529.
Dyukarev Yu.M., Serikova I.Yu., Complete indeterminacy of the Nevanlinna Pick problem in the class S[a,b], / Russian Math. (Iz. VUZ), 2007. – 51(11). – P. 17–29.
Dyukarev Yu.M., Serikova I.Yu., Step-by-step solving of ordered interpolational problem for Stieltjes functions, / Russian Math. (Iz.VUZ), 2017. – 6. – P. 18–32.
Fritzsche B., Kirstein B., M¨adler C., On Hankel nonegative definite sequences, the canonical Hankel parametrization, and orthogonal matrix polynomials, / Compl. Anal. Oper. Theory, 2011. – 5(2). – P. 447–511.
Fritzsche B., Kirstein B., M¨adler C., Transformations of matricial α-Stieltjes non-negative definite sequences, / Lin. Alg. and Appl., 2013. – 439. – P. 3893–3933.
Geronimo J.S., Scattering theory and matrix orthogonal polynomials on the real line, / Circuits Systems Sigmal Process, 1982. – 1(3-4). – P. 471–495.
I.V. Kovalishina, New aspects of the classical moment problems. Second doctoral thesis (in Russian), 1986. – Institute of Railway-Transport Engineers.
Kovalishina I.V., Analytic theory of a class of interpolation problems, / Izv. Math., 1983. – 47(3). – P. 455–497.
Krein M.G., Fundamental aspects of the representation theory of hermitian operators with deficiency (m, m), / Ukrain. Mat. Zh., 1949. – 1(2). – P. 3–66.
Krein M.G., Infinite J-matrices and a matrix moment problem, / Dokl. Akad. Nauk, 1949. – 69(2). – P. 125–128.
Miranian L., Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory, / J. Phys. A: Math. Gen., 2005. – 38. –P. 5731–5749.
Potapov V.P., The multiplicative structure of J-nonexpansive matrix functions, / Trudy Moskov. Mat. Ob., 1955. – 4. – P. 125–236.
Serikova I. Yu., Indeterminacy criteria for the Nevanlinna-Pick interpolation problem in class R[a, b], / Zb. Pr. Inst. Mat. NAN Ukr., 2006. – 3(4). – P. 126–142.
Simon B., The classical problem as a self-adjoint finite difference operator, / Adv. Math., 1998. – 137. – P. 82–203.
Sinap A., Van Assche W., Orthogonal matrix polynomials and applications, / J. Comput. Appl. Math., 1996. – 66(1-2). – P. 27–52.
H. Thiele, Beitr¨age zu matriziellen otenzmomentenproblemen, PhD Thesis, 2006. – Leipzig University.
Zagorodnyuk S., The truncated matrix Hausdorff moment problem, / Methods Appl. Anal., 2012. – 19(1). – P. 021–042.
Published
2017-11-29
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How to Cite
Choque-Rivero, A. E. (2017). 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, 85, 16-42. https://doi.org/10.26565/2221-5646-2017-85-02
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