×

zbMATH — the first resource for mathematics

Perturbations of orthogonal polynomials with periodic recursion coefficients. (English) Zbl 1194.47031
The authors develop the spectral theory of asymptotically periodic Jacobi and CMV matrices with the focus set on the extension of some basic results concerning asymptotically constant matrices of the above classes. The main difficulty for the asymptotically periodic setting is that now, instead of a single unperturbed operator, one has a manifold (the isospectral thorus) of unperturbed operators. The spectral measures that are close to those of the isospectral thorus correspond to coefficients that approach the isospectral thorus, in general without converging to any particular point therein.
One of the major new results of the paper is the following Theorem 1.2. Let \(J_0\) be a two-sided periodic Jacobi matrix and \(J\) a one-sided Jacobi matrix with Jacobi parameters \(\{a_m,b_m\}_{m=1}^\infty\). If \(\sigma_{ess}(J)=\sigma(J_0)\) and \(\sigma_{ac}(J)=\sigma(J_0)\), then \(d_m((a,b), T_{J_0})\to 0\). Here, \(T_{J_0}\) is the isospectral torus, and \(d\) is a suitable distance between pairs of sequences.

MSC:
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] N. I. Aheizer and M. Krein, Some Questions in the Theory of Moments, Providence, R.I.: Amer. Math. Soc., 1962.
[2] M. P. Alfaro, M. Bello Hernández, J. M. Montaner, and J. L. Varona, ”Some asymptotic properties for orthogonal polynomials with respect to varying measures,” J. Approx. Theory, vol. 135, iss. 1, pp. 22-34, 2005. · Zbl 1080.42022
[3] D. Alpay and I. Gohberg, ”Inverse spectral problems for difference operators with rational scattering matrix function,” Integral Eqs Oper. Theory, vol. 20, iss. 2, pp. 125-170, 1994. · Zbl 0816.47006
[4] D. Alpay and I. Gohberg, ”Inverse spectral problem for differential operators with rational scattering matrix functions,” J. Differential Equations, vol. 118, iss. 1, pp. 1-19, 1995. · Zbl 0819.47008
[5] A. J. Antony and M. Krishna, ”Almost periodicity of some Jacobi matrices,” Proc. Indian Acad. Sci. Math. Sci., vol. 102, iss. 3, pp. 175-188, 1992. · Zbl 0769.60061
[6] A. I. Aptekarev and E. M. Nikishin, ”The scattering problem for a discrete Sturm-Liouville operator,” Mat. Sb., vol. 121(163), iss. 3, pp. 327-358, 1983. · Zbl 0527.34024
[7] J. Avron and B. Simon, ”Almost periodic Schrödinger operators. I. Limit periodic potentials,” Comm. Math. Phys., vol. 82, iss. 1, pp. 101-120, 1981/82. · Zbl 0484.35069
[8] M. Bakonyi and T. Constantinescu, ”Schur’s algorithm and several applications,” Pitman Research Notes in Math., vol. 261, 1992. · Zbl 0797.47011
[9] D. Barrios Rolan’ia, B. de la Calle Ysern, and G. López Lagomasino, ”Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures,” J. Approx. Theory, vol. 139, iss. 1-2, pp. 223-256, 2006. · Zbl 1100.42014
[10] M. Bello Hernández and G. López Lagomasino, ”Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle,” J. Approx. Theory, vol. 92, iss. 2, pp. 216-244, 1998. · Zbl 0897.42016
[11] S. V. Belyi and E. R. Tsekanovskii, ”Classes of operator \(R\)-functions and their realization by conservative systems,” Dokl. Akad. Nauk SSSR, vol. 321, iss. 3, pp. 441-445, 1991.
[12] S. V. Belyi and E. R. Tsekanovskii, ”Realization theorems for operator-valued \(R\)-functions,” in New Results in Operator Theory and its Applications, Basel: Birkhäuser, 1997, pp. 55-91. · Zbl 0894.47006
[13] O. Blumenthal, ”Über die Entwicklung einer willkürlichen Funktion nach den Nennern des Kettenbruches für \(\int_{-\infty}^0 \frac{\varphi (\xi)\, d\xi}{z-\xi}\),” PhD Thesis , Göttingen, 1898. · JFM 29.0364.01
[14] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies, , 1998, vol. 135. · Zbl 0906.35099
[15] M. Buys and A. Finkel, ”The inverse periodic problem for Hill’s equation with a finite-gap potential,” J. Differential Equations, vol. 55, iss. 2, pp. 257-275, 1984. · Zbl 0508.34013
[16] M. J. Cantero, M. P. Ferrer, L. Moral, and L. Velázquez, ”A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line,” J. Comput. Appl. Math., vol. 154, iss. 2, pp. 247-272, 2003. · Zbl 1016.42011
[17] K. M. Case, ”Orthogonal polynomials from the viewpoint of scattering theory,” J. Mathematical Phys., vol. 15, pp. 2166-2174, 1974. · Zbl 0288.42009
[18] K. M. Case, ”Orthogonal polynomials. II,” J. Mathematical Phys., vol. 16, pp. 1435-1440, 1975. · Zbl 0304.42015
[19] V. A. Chulaevskiui, ”An inverse spectral problem for limit-periodic Schrödinger operators,” Funktsional. Anal. i Prilozhen., vol. 18, iss. 3, pp. 63-66, 1984. · Zbl 0608.34025
[20] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, New York: McGraw-Hill Book Company, 1955. · Zbl 0064.33002
[21] D. Damanik, A. Pushnitski, and B. Simon, ”The analytic theory of matrix orthogonal polynomials,” Surv. Approx. Theory, vol. 4, pp. 1-85, 2008. · Zbl 1193.42097
[22] D. Damanik and B. Simon, ”Jost functions and Jost solutions for Jacobi matrices. I. A necessary and sufficient condition for Szeg\Ho asymptotics,” Invent. Math., vol. 165, iss. 1, pp. 1-50, 2006. · Zbl 1122.47029
[23] D. Damanik and B. Simon, ”Jost functions and Jost solutions for Jacobi matrices. II. Decay and analyticity,” Int. Math. Res. Not., p. I, 2006. · Zbl 1122.47030
[24] E. Defez, L. Jódar, A. Law, and E. Ponsoda, ”Three-term recurrences and matrix orthogonal polynomials,” Util. Math., vol. 57, pp. 129-146, 2000. · Zbl 0962.05064
[25] P. Delsarte and Y. V. Genin, ”On a generalization of the Szeg\Ho-Levinson recurrence and its application in lossless inverse scattering,” IEEE Trans. Inform. Theory, vol. 38, iss. 1, pp. 104-110, 1992. · Zbl 0744.65022
[26] P. Delsarte, Y. V. Genin, and Y. G. Kamp, ”Orthogonal polynomial matrices on the unit circle,” IEEE Trans. Circuits and Systems, vol. CAS-25, iss. 3, pp. 149-160, 1978. · Zbl 0408.15018
[27] P. Delsarte, Y. V. Genin, and Y. G. Kamp, ”Planar least squares inverse polynomials. I. Algebraic properties,” IEEE Trans. Circuits and Systems, vol. 26, iss. 1, pp. 59-66, 1979. · Zbl 0404.94002
[28] P. Delsarte, Y. V. Genin, and Y. G. Kamp, ”Schur parametrization of positive definite block-Toeplitz systems,” SIAM J. Appl. Math., vol. 36, iss. 1, pp. 34-46, 1979. · Zbl 0417.42013
[29] P. Delsarte, Y. V. Genin, and Y. G. Kamp, ”The Nevanlinna-Pick problem for matrix-valued functions,” SIAM J. Appl. Math., vol. 36, iss. 1, pp. 47-61, 1979. · Zbl 0421.41002
[30] P. Delsarte, Y. V. Genin, and Y. G. Kamp, ”Generalized Schur representation of matrix-valued functions,” SIAM J. Algebraic Discrete Methods, vol. 2, iss. 2, pp. 94-107, 1981. · Zbl 0497.15018
[31] S. A. Denisov, ”On Rakhmanov’s theorem for Jacobi matrices,” Proc. Amer. Math. Soc., vol. 132, iss. 3, pp. 847-852, 2004. · Zbl 1050.47024
[32] H. Dette and W. J. Studden, ”Matrix measures, moment spaces and Favard’s theorem for the interval \([0,1]\) and \([0,\infty)\),” Linear Algebra Appl., vol. 345, pp. 169-193, 2002. · Zbl 1022.42016
[33] H. Dette and W. J. Studden, ”A note on the matrix valued q-d algorithm and matrix orthogonal polynomials on \([0,1]\) and \([0,\infty)\),” J. Comput. Appl. Math., vol. 148, iss. 2, pp. 349-361, 2002. · Zbl 1052.42022
[34] A. J. Durán, ”Ratio asymptotics for orthogonal matrix polynomials,” J. Approx. Theory, vol. 100, iss. 2, pp. 304-344, 1999. · Zbl 0944.42015
[35] A. J. Durán and E. Defez, ”Orthogonal matrix polynomials and quadrature formulas,” Linear Algebra Appl., vol. 345, pp. 71-84, 2002. · Zbl 0990.42009
[36] A. J. Durán and P. López-Rodr’iguez, ”Orthogonal matrix polynomials: zeros and Blumenthal’s theorem,” J. Approx. Theory, vol. 84, iss. 1, pp. 96-118, 1996. · Zbl 0861.42016
[37] A. J. Durán and P. López-Rodr’iguez, ”The matrix moment problem,” in Margarita Mathematica, Univ. La Rioja, Logroño, 2001, pp. 333-348. · Zbl 1253.44010
[38] A. J. Durán and P. López-Rodr’iguez, ”Orthogonal matrix polynomials,” in Laredo Lectures on Orthogonal Polynomials and Special Functions, Hauppauge, NY: Nova Sci. Publ., 2004, pp. 13-44. · Zbl 1100.42018
[39] A. J. Durán, P. López-Rodr’iguez, and E. B. Saff, ”Zero asymptotic behaviour for orthogonal matrix polynomials,” J. Anal. Math., vol. 78, pp. 37-60, 1999. · Zbl 0945.42013
[40] M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Edinburgh: Scottish Academic Press, 1975. · Zbl 0287.34016
[41] A. Finkel, E. Isaacson, and E. Trubowitz, ”An explicit solution of the inverse periodic problem for Hill’s equation,” SIAM J. Math. Anal., vol. 18, iss. 1, pp. 46-53, 1987. · Zbl 0622.34021
[42] M. Fukushima, ”A spectral representation on ordinary linear difference equation with operator-valued coefficients of the second order,” J. Mathematical Phys., vol. 17, iss. 6, pp. 1064-1072, 1976. · Zbl 0333.47015
[43] V. Georgescu and A. Iftimovici, ”Crossed products of \(C^\ast\)-algebras and spectral analysis of quantum Hamiltonians,” Comm. Math. Phys., vol. 228, iss. 3, pp. 519-560, 2002. · Zbl 1005.81026
[44] J. S. Geronimo, ”Matrix orthogonal polynomials on the unit circle,” J. Math. Phys., vol. 22, iss. 7, pp. 1359-1365, 1981. · Zbl 0505.42019
[45] Y. L. Geronimus, ”Polynomials orthogonal on a circle and their applications,” Amer. Math. Soc. Translation, vol. 1954, iss. 104, p. 79, 1954. · Zbl 0056.10303
[46] Y. L. Geronimus, ”Orthogonal polynomials,” in Two Papers on Special Functions, Providence, RI: Amer. Math. Soc., 1977, pp. 37-130. · Zbl 0365.42007
[47] F. Gesztesy and B. Simon, ”Inverse spectral analysis with partial information on the potential. I. The case of an a.c. component in the spectrum,” Helv. Phys. Acta, vol. 70, iss. 1-2, pp. 66-71, 1997. · Zbl 0870.34017
[48] F. Gesztesy and B. Simon, ”Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum,” Trans. Amer. Math. Soc., vol. 352, iss. 6, pp. 2765-2787, 2000. · Zbl 0948.34060
[49] F. Gesztesy and E. Tsekanovskii, ”On matrix-valued Herglotz functions,” Math. Nachr., vol. 218, pp. 61-138, 2000. · Zbl 0961.30027
[50] I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, ”Pseudo-canonical systems with rational Weyl functions: explicit formulas and applications,” J. Differential Equations, vol. 146, iss. 2, pp. 375-398, 1998. · Zbl 0917.34074
[51] L. Golinskii and B. Simon, Section 4.3 of Orthogonal Polynomials on the Unit Circle, Part  1: Classical TheoryProvidence, RI: Amer. Math. Soc., 2005.
[52] G. H. Golub and C. F. Van Loan, Matrix Computations, Third ed., Baltimore, MD: Johns Hopkins University Press, 1996. · Zbl 0865.65009
[53] D. B. Hinton and J. K. Shaw, ”On Titchmarsh-Weyl \(M(\lambda )\)-functions for linear Hamiltonian systems,” J. Differential Equations, vol. 40, iss. 3, pp. 316-342, 1981. · Zbl 0472.34014
[54] H. Hochstadt and B. Lieberman, ”An inverse Sturm-Liouville problem with mixed given data,” SIAM J. Appl. Math., vol. 34, iss. 4, pp. 676-680, 1978. · Zbl 0418.34032
[55] D. Hundertmark and B. Simon, ”Lieb-Thirring inequalities for Jacobi matrices,” J. Approx. Theory, vol. 118, iss. 1, pp. 106-130, 2002. · Zbl 1019.39013
[56] K. Iwasaki, ”Inverse problem for Sturm-Liouville and Hill equations,” Ann. Mat. Pura Appl., vol. 149, pp. 185-206, 1987. · Zbl 0641.34012
[57] L. Jódar, E. Defez, and E. Ponsoda, ”Orthogonal matrix polynomials with respect to linear matrix moment functionals: theory and applications,” Approx. Theory Appl., vol. 12, iss. 1, pp. 96-115, 1996. · Zbl 0853.42022
[58] R. A. Johnson, ”\(m\)-functions and Floquet exponents for linear differential systems,” Ann. Mat. Pura Appl., vol. 147, pp. 211-248, 1987. · Zbl 0652.34016
[59] R. Killip, ”Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum,” Int. Math. Res. Not., iss. 38, pp. 2029-2061, 2002. · Zbl 1021.34071
[60] R. Killip, ”Spectral theory via sum rules,” in Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Providence, RI: Amer. Math. Soc., 2007, pp. 907-930. · Zbl 1137.47001
[61] R. Killip and B. Simon, ”Sum rules for Jacobi matrices and their applications to spectral theory,” Ann. of Math., vol. 158, iss. 1, pp. 253-321, 2003. · Zbl 1050.47025
[62] S. Kotani, ”Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators,” in Stochastic Analysis (Katata/Kyoto, 1982), Amsterdam: North-Holland, 1984, pp. 225-247. · Zbl 0549.60058
[63] S. Kotani, ”Generalized Floquet theory for stationary Schrödinger operators in one dimension,” Chaos Solitons Fractals, vol. 8, iss. 11, pp. 1817-1854, 1997. · Zbl 0936.34074
[64] R. Kozhan, ”Equivalence classes of block Jacobi matrices,” , preprint. · Zbl 1217.15018
[65] A. M. Krall, ”\(M(\lambda)\) theory for singular Hamiltonian systems with one singular point,” SIAM J. Math. Anal., vol. 20, iss. 3, pp. 664-700, 1989. · Zbl 0683.34008
[66] M. G. Krein and I. E. Ovvcarenko, ”Inverse problems for \(Q\)-functions and resolvent matrices of positive Hermitian operators,” Soviet Math. Dokl., vol. 19, pp. 1131-1134, 1978. · Zbl 0443.47026
[67] S. Kupin, ”On sum rules of special form for Jacobi matrices,” C. R. Math. Acad. Sci. Paris, vol. 336, iss. 7, pp. 611-614, 2003. · Zbl 1057.47037
[68] A. Laptev, S. Naboko, and O. Safronov, ”On new relations between spectral properties of Jacobi matrices and their coefficients,” Comm. Math. Phys., vol. 241, iss. 1, pp. 91-110, 2003. · Zbl 1135.47303
[69] Y. Last and B. Simon, ”Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators,” Invent. Math., vol. 135, iss. 2, pp. 329-367, 1999. · Zbl 0931.34066
[70] Y. Last and B. Simon, ”The essential spectrum of Schrödinger, Jacobi, and CMV operators,” J. Anal. Math., vol. 98, pp. 183-220, 2006. · Zbl 1145.34052
[71] N. Levinson, ”The Wiener RMS (root mean square) error criterion in filter design and prediction,” J. Math. Phys. Mass. Inst. Tech., vol. 25, pp. 261-278, 1947.
[72] W. Magnus and S. Winkler, Hill’s Equation, New York: Interscience Publishers, John Wiley & Sons, 1966. · Zbl 0158.09604
[73] M. Muantoiu, ”\(C^\ast\)-algebras, dynamical systems at infinity and the essential spectrum of generalized Schrödinger operators,” J. Reine Angew. Math., vol. 550, pp. 211-229, 2002. · Zbl 1036.46052
[74] F. Marcellán and I. Rodr’iguez González, ”A class of matrix orthogonal polynomials on the unit circle,” Linear Algebra Appl., vol. 121, pp. 233-241, 1989. · Zbl 0681.33013
[75] F. Marcellán and G. Sansigre, ”On a class of matrix orthogonal polynomials on the real line,” Linear Algebra Appl., vol. 181, pp. 97-109, 1993. · Zbl 0769.15010
[76] F. Marcellán and H. O. Yakhlef, ”Recent trends on analytic properties of matrix orthonormal polynomials,” Electron. Trans. Numer. Anal., vol. 14, pp. 127-141, 2002. · Zbl 1033.42026
[77] V. A. Marchenko, Sturm-Liouville Operators and Applications, Basel: Birkhäuser, 1986. · Zbl 0592.34011
[78] A. Máté, P. Nevai, and V. Totik, ”Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle,” Constr. Approx., vol. 1, iss. 1, pp. 63-69, 1985. · Zbl 0582.42012
[79] H. P. McKean and E. Trubowitz, ”Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points,” Comm. Pure Appl. Math., vol. 29, iss. 2, pp. 143-226, 1976. · Zbl 0339.34024
[80] P. B. Nauiman, ”On the theory of periodic and limit-periodic Jacobian matrices,” Dokl. Akad. Nauk SSSR, vol. 143, pp. 277-279, 1962. · Zbl 0118.11003
[81] P. B. Nauiman, ”On the spectral theory of non-symmetric periodic Jacobi matrices,” Zap. Meh.-Mat. Fak. Har’kov. Gos. Univ. i Har’kov. Mat. Ob\vs\vc., vol. 30, pp. 138-151, 1964.
[82] F. Nazarov, F. Peherstorfer, A. Volberg, and P. Yuditskii, ”On generalized sum rules for Jacobi matrices,” Int. Math. Res. Not., iss. 3, pp. 155-186, 2005. · Zbl 1089.47025
[83] P. G. Nevai, ”Orthogonal polynomials,” Mem. Amer. Math. Soc., vol. 18, iss. 213, p. v, 1979. · Zbl 0405.33009
[84] P. G. Nevai, ”Orthogonal polynomials, recurrences, Jacobi matrices, and measures,” in Progress in Approximation Theory, New York: Springer-Verlag, 1992, pp. 79-104. · Zbl 0789.42016
[85] P. G. Nevai and V. Totik, ”Orthogonal polynomials and their zeros,” Acta Sci. Math. \((\)Szeged\()\), vol. 53, iss. 1-2, pp. 99-104, 1989. · Zbl 0691.42020
[86] P. G. Nevai and V. Totik, ”Denisov’s theorem on recurrence coefficients,” J. Approx. Theory, vol. 127, iss. 2, pp. 240-245, 2004. · Zbl 1061.42014
[87] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, New York: Springer-Verlag, 1992. · Zbl 0752.47002
[88] F. Peherstorfer and P. Yuditskii, ”Asymptotic behavior of polynomials orthonormal on a homogeneous set,” J. Anal. Math., vol. 89, pp. 113-154, 2003. · Zbl 1032.42028
[89] V. S. Rabinovich, ”Operator-valued discrete convolutions and some of their applications,” Mat. Zametki, vol. 51, iss. 5, pp. 90-101, 158, 1992. · Zbl 0807.47006
[90] E. A. Rakhmanov, ”On the asymptotics of the ratio of orthogonal polynomials,” Math. USSR Sb., vol. 32, pp. 199-213, 1977. · Zbl 0401.30033
[91] E. A. Rakhmanov, ”On the asymptotics of the ratio of orthogonal polynomials, II,” Math. USSR Sb., vol. 46, pp. 105-117, 1983. · Zbl 0515.30030
[92] J. Ralston and E. Trubowitz, ”Isospectral sets for boundary value problems on the unit interval,” Ergodic Theory Dynam. Systems, vol. 8\(^*\), iss. Charles Conley Memorial Issue, pp. 301-358, 1988. · Zbl 0678.34025
[93] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, New York: Academic Press, 1978. · Zbl 0401.47001
[94] C. Remling, The absolutely continuous spectrum of Jacobi matrices. · Zbl 1235.47032
[95] L. Rodman, ”Orthogonal matrix polynomials,” in Orthogonal Polynomials (Columbus, OH, 1989), Dordrecht: Kluwer Acad. Publ., 1990, pp. 345-362. · Zbl 0703.42022
[96] E. Ryckman, ”A strong Szeg\Ho theorem for Jacobi matrices,” Comm. Math. Phys., vol. 271, iss. 3, pp. 791-820, 2007. · Zbl 1133.47025
[97] A. L. Sakhnovich, ”Spectral functions of a second-order canonical system,” Mat. Sb., vol. 181, iss. 11, pp. 1510-1524, 1990. · Zbl 0718.34112
[98] A. Sebbar and T. Falliero, ”Capacities and Jacobi matrices,” Proc. Edinb. Math. Soc., vol. 46, iss. 3, pp. 719-745, 2003. · Zbl 1041.31003
[99] J. A. Shohat, ”Théorie générale des polinomes orthogonaux de Tchebichef,” Mémorial des Sciences Mathématiques, vol. 66, pp. 1-69, 1934. · JFM 60.1037.01
[100] B. Simon, ”Kotani theory for one-dimensional stochastic Jacobi matrices,” Comm. Math. Phys., vol. 89, iss. 2, pp. 227-234, 1983. · Zbl 0534.60057
[101] B. Simon, ”A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices,” J. Funct. Anal., vol. 214, iss. 2, pp. 396-409, 2004. · Zbl 1064.30030
[102] B. Simon, ”Meromorphic Szeg\Ho functions and asymptotic series for Verblunsky coefficients,” Acta Math., vol. 195, pp. 267-285, 2005. · Zbl 1117.42005
[103] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1; Classical Theory, Providence, RI: Amer. Math. Soc., 2005. · Zbl 1082.42020
[104] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2: Spectral theory, Providence, RI: Amer. Math. Soc., 2005. · Zbl 1082.42021
[105] B. Simon, ”Sturm oscillation and comparison theorems,” in Sturm-Liouville Theory, Basel: Birkhäuser, 2005, pp. 29-43. · Zbl 1117.39013
[106] B. Simon, ”Meromorphic Jost functions and asymptotic expansions for Jacobi parameters,” Funktsional. Anal. i Prilozhen., vol. 41, iss. 2, pp. 78-92, 112, 2007. · Zbl 1255.47031
[107] B. Simon, Szeg\Ho’s Theorem and Its Descendants : Spectral Theory for \(L^2\) Perturbations of Orthogonal Polynomials, Princeton, NJ: Princeton Univ. Press, 2010.
[108] B. Simon and A. Zlatovs, ”Sum rules and the Szeg\Ho condition for orthogonal polynomials on the real line,” Comm. Math. Phys., vol. 242, iss. 3, pp. 393-423, 2003. · Zbl 1046.42017
[109] M. Sodin and P. Yuditskii, ”Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions,” J. Geom. Anal., vol. 7, iss. 3, pp. 387-435, 1997. · Zbl 1041.47502
[110] G. SzegHo, ”Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion,” Math. Ann., vol. 76, iss. 4, pp. 490-503, 1915. · JFM 45.0518.02
[111] G. SzegHo, ”Beiträge zur Theorie der Toeplitzschen Formen,” Math. Z., vol. 6, iss. 3-4, pp. 167-202, 1920.
[112] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Providence, RI: Amer. Math. Soc., 2000. · Zbl 1056.39029
[113] M. Toda, Theory of Nonlinear Lattices, Second ed., New York: Springer-Verlag, 1989. · Zbl 0694.70001
[114] W. Van Assche, ”Rakhmanov’s theorem for orthogonal matrix polynomials on the unit circle,” J. Approx. Theory, vol. 146, iss. 2, pp. 227-242, 2007. · Zbl 1136.33005
[115] P. van Moerbeke, ”The spectrum of Jacobi matrices,” Invent. Math., vol. 37, iss. 1, pp. 45-81, 1976. · Zbl 0361.15010
[116] D. S. Watkins, Fundamentals of Matrix Computations, second ed., New York: Wiley-Interscience, 2002. · Zbl 1005.65027
[117] J. Weidmann, Spectral Theory of Ordinary Differential Operators, New York: Springer-Verlag, 1987. · Zbl 0647.47052
[118] H. Weyl, ”Über beschränkte quadratische Formen, deren Differenz vollstetig ist,” Rend. Circ. Mat. Palermo, vol. 27, pp. 373-392, 1909. · JFM 40.0395.01
[119] H. O. Yakhlef and F. Marcellán, ”Orthogonal matrix polynomials, connection between recurrences on the unit circle and on a finite interval,” in Approximation, Optimization and Mathematical Economics (Pointe-à-Pitre, 1999), Heidelberg: Physica, 2001, pp. 369-382. · Zbl 0985.15020
[120] H. O. Yakhlef, F. Marcellán, and M. A. Piñar, ”Relative asymptotics for orthogonal matrix polynomials with convergent recurrence coefficients,” J. Approx. Theory, vol. 111, iss. 1, pp. 1-30, 2001. · Zbl 1005.42014
[121] D. C. Youla and N. N. Kazanjian, ”Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle,” IEEE Trans. Circuits and Systems, vol. CAS-25, iss. 2, pp. 57-69, 1978. · Zbl 0417.93018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.