Geodesic
Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2017), pp. 66-79.

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Let $\{c_j\}_{j=0}^{n}$ be a sequence of matrix moments associated with a matrix of measures supported on the unit circle, and let $\{P_j\}_{j=0}^{n}$ be its corresponding sequence of monic matrix orthogonal polynomials. In this contribution, we consider a perturbation on the moments and find an explicit relation for the perturbed orthogonal polynomials in terms of $\{P_j\}_{j=0}^{n}$. We also obtain an expression for the corresponding second kind polynomials.
Keywords: matrix orthogonal polynomials on the unit circle, moment problem, second kind polynomials, Toeplitz matrices. 
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A. E. Choke Rivero; L. E. Garza Gaona. Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2017), pp. 66-79. http://geodesic.mathdoc.fr/item/IVM_2017_12_a6/

[1] Kovalishina I. V., “Analiticheskaya teoriya odnogo klassa interpolyatsionnykh zadach”, Izv. AN SSSR, Ser. matem., 47:3 (1983), 455–497 | MR

[2] Miranian L., “Matrix valued orthogonal polynomials on the unit circle: some extensions to the classical theory”, Canad. Math. Bull., 52:1 (2009), 95–104 | DOI | MR | Zbl

[3] Krein M. G., “Osnovnye polozheniya teorii predstavleniya ermitovykh operatorov s indeksom defekta $(m,m)$”, Ukr. matem. zhurn., 1:2 (1949), 3–66

[4] Krein M. G., “Beskonechnye T-matritsy i matrichnaya problema momentov”, DAN SSSR, 69 (1949), 125–128 | MR | Zbl

[5] Aptekarev A. I., Nikishin E. M., “Zadacha rasseyaniya dlya diskretnogo operatora Shturma–Liuvillya”, Matem. sb., 121 (1983), 327–358 | Zbl

[6] Dette H., Studden W. J., “Matrix measures, moment spaces and Favard's theorem on the interval $[0,1]$ and $[0,\infty)$”, Linear. Alg. Appl., 345 (2002), 169–193 | DOI | MR | Zbl

[7] Durán A. J., “A generalization of Favard's theorem for polynomials satisfying a recurrence relation”, J. Approx. Theory, 74 (1997), 7818–7829 | MR

[8] Durán A. J., “On orthogonal polynomials with respect to positive definite matrix of measures”, Can. J. Math., 47 (1995), 88–112 | DOI | MR | Zbl

[9] Durán A. J., “Markov's theorem for orthogonal matrix polynomials”, Can. J. Math., 48 (1996), 1180–1195 | DOI | MR | Zbl

[10] Durán A. J., “Rodrigues' formulas for orthogonal matrix polynomials satisfying higher-order differential equations”, Experimental Math., 20:1 (2011), 15–24 | DOI | MR | Zbl

[11] Dyukarev Yu. M., “Geometricheskie i operatornye mery vyrozhdennosti mnozhestva reshenii matrichnoi problemy momentov Stiltesa”, Matem. sb., 207:4 (2016), 47–64 | DOI | Zbl

[12] Grünbaum F. A., “Matrix-valued Jacobi polynomials”, Bull. Sci. Math., 127 (2003), 207–214 | DOI | MR | Zbl

[13] Marcellán F., Sansigre G., “On a class of matrix orthogonal polynomials on the real line”, Linear Algebra Appl., 181 (1993), 97–109 | DOI | MR | Zbl

[14] Damanik D., Pushnitski A., Simon B., “The analytic theory of matrix orthogonal polynomials”, Surv. Approx. Theory, 4 (2008), 1–85 | MR | Zbl

[15] Sinap A., Van Assche W., “Orthogonal matrix polynomials and applications”, J. Comput. Appl. Math., 66 (1996), 27–52 | DOI | MR | Zbl

[16] Choque-Rivero A. E., “The resolvent matrix for the matricial Hausdorff moment problem expressed by orthogonal matrix polynomials”, Complex Anal. Oper. Theory, 7:4 (2013), 927–944 | DOI | MR | Zbl

[17] 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., 474 (2015), 44–109 | DOI | MR | Zbl

[18] Choque-Rivero A. E., Maedler C., “On Hankel positive definite perturbations of Hankel positive definite sequences and interrelations to orthogonal matrix polynomials”, Complex Anal. Oper. Theory, 8:8 (2014), 121–173 | DOI | MR

[19] Choque-Rivero A. E., “On matrix Hurwitz type polynomials and its interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials”, Lin. Alg. and Appl., 476 (2015), 56–84 | DOI | MR | Zbl

[20] Youla D.C, Kazanijan N. N., “Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle”, IEEE Transactions on Circuits and Systems, 25:2 (1978), 57–69 | DOI | MR | Zbl

[21] Delsarte P., Genin Y., Kamp Y., “Orthogonal polynomials matrices on the unit circle”, IEEE Transactions on Circuits and Systems, 25:3 (1978), 149–160 | DOI | MR | Zbl

[22] Dubovoj V. K., Fritzsche B., Kirstein B., Matricial version of the classical Schur problem, Teubner-Texte zur Math., 129, B.G. Teubner, Stuttgart–Leipzig, 1992 | MR | Zbl

[23] Geronimo J. S., “Matrix orthogonal polynomials on the unit circle”, J. Math. Phys., 22 (1981), 1359–1365 | DOI | MR | Zbl

[24] Fritzsche B., Kirstein B., “An extension problem for non-negative Hermitian block Teoplitz matrices. III”, Math. Nachr., 135 (1988), 319–341 | DOI | MR | Zbl

[25] Marcellán F., Gonzáles I. R., “A class of matrix orthogonal polynomials on the unit circle”, Linear Algebra Appl., 121 (1989), 233–241 | DOI | MR | Zbl

[26] Choque-Rivero A. E., Garza L. E., “Moment perturbation of matrix polynomials”, Integral Transf. Special Func., 26 (2015), 177–191 | DOI | MR | Zbl

[27] Castillo K., Dimitrov D. K., Garza L. E., Rafaeli F. R., “Perturbations on the antidiagonals of Hankel matrices”, Appl. Math. Comput., 221 (2013), 444–452 | DOI | MR | Zbl

[28] Castillo K., Garza L. E., Marcellán F., “Perturbations on the subdiagonals of Toeplitz matrices”, Linear Alg. Appl., 434 (2011), 1563–1579 | DOI | MR | Zbl

[29] Yakhlef H. O., Marcellán F., “Relative asymptotics for orthogonal matrix polynomials with respect to a perturbed matrix measure on the unit circle”, Approx. Theory Appl., 18 (2002), 1–19 | DOI | MR | Zbl

[30] Kim D. H., Kwon K. H., “Quasi-definiteness of generalized Uvarov transforms of moment functionals”, J. Appl. Math., 1 (2001), 69–90 | DOI | MR | Zbl

[31] Spiridonov V., Vinet L., Zhedanov A., “Spectral transformations, selfsimilar reductions and orthogonal polynomials”, J. Phys. A: Math. Gen., 30 (1997), 7621–7637 | DOI | MR | Zbl

[32] Zhedanov A., “Rational spectral transformations and orthogonal polynomials”, J. Comput. Appl. Math., 85 (1997), 67–86 | DOI | MR | Zbl

[33] Efimov A. V., Potapov V. P., “$J$-rastyagivayuschie matritsy-funktsii i ikh rol v analiticheskoi teorii elektricheskikh tsepei”, UMN, 28:1 (1973), 65–130 | MR | Zbl

[34] Simon B., Orthogonal polynomials on the unit circle, 2 vols., AMS Coll. Publ. Ser., 54, Providence, RI, 2005 | MR | Zbl

[35] Szegö G., Orthogonal polynomials, AMS Colloq. Publ. Ser., 23, 4th Ed., Providence, RI, 1975 | MR