Finite gap Jacobi matrices. II: The Szegő class. (English) Zbl 1236.42021

The authors study Jacobi matrices \(J\) and asymptotics of the associated orthogonal polynomials, where \(\sigma_{\text{ess}}(J)\) is a finite union of disjoint closed intervals. They study Szegő’s theorem for the general finite gap case. They use Remling’s theorem about the approach to the isospectral torus together with an analysis of Jost functions to provide a new proof of Szegő asymptotics including \(L^{2}\) Szegő asymptotics on the spectrum.
For Part I, see ibid. 32, No. 1, 1–65 (2010; Zbl 1200.42012).


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
58J53 Isospectrality
14H30 Coverings of curves, fundamental group
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations


Zbl 1200.42012
Full Text: DOI arXiv


[1] Aptekarev, A.I.: Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains. Math. USSR Sb. 53, 233–260 (1986). Russian original in Mat. Sb. (N.S.) 125(167), 231–258 (1984) · Zbl 0608.42016
[2] Christiansen, J., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices, I. The isospectral torus, Constr. Approx., to appear · Zbl 1200.42012
[3] Christiansen, J., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices, III. Beyond the Szego class, in preparation · Zbl 1238.42009
[4] Damanik, D., Killip, R., Simon, B.: Perturbations of orthogonal polynomials with periodic recursion coefficients. Ann. Math., to appear · Zbl 1194.47031
[5] Damanik, D., Simon, B.: Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szego asymptotics. Invent. Math. 165, 1–50 (2006) · Zbl 1122.47029
[6] Denisov, S.A.: On Rakhmanov’s theorem for Jacobi matrices. Proc. Am. Math. Soc. 132, 847–852 (2004) · Zbl 1050.47024
[7] Frank, R., Simon, B., Weidl, T.: Eigenvalue bounds for perturbations of Schrödinger operators and Jacobi matrices with regular ground states. Commun. Math. Phys. 282, 199–208 (2008) · Zbl 1158.35021
[8] Garnett, J.B.: Bounded Analytic Functions. Pure and Applied Math., vol. 96. Academic Press, New York (1981) · Zbl 0469.30024
[9] Geronimus, Ya.L.: Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval. Consultants Bureau, New York (1961)
[10] Hundertmark, D., Simon, B.: Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices. J. Math. Anal. Appl. 340, 892–900 (2008) · Zbl 1135.47026
[11] Killip, R., Simon, B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158, 253–321 (2003) · Zbl 1050.47025
[12] Last, Y., Simon, B.: The essential spectrum of Schrödinger, Jacobi, and CMV operators. J. Anal. Math. 98, 183–220 (2006) · Zbl 1145.34052
[13] Nevai, P.: Orthogonal polynomials. Mem. Am. Math. Soc. 18(213), 1–183 (1979)
[14] Peherstorfer, F., Yuditskii, P.: Private communication
[15] Peherstorfer, F., Yuditskii, P.: Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. Proc. Am. Math. Soc. 129, 3213–3220 (2001) · Zbl 0976.42012
[16] Peherstorfer, F., Yuditskii, P.: Asymptotic behavior of polynomials orthonormal on a homogeneous set. J. Anal. Math. 89, 113–154 (2003) · Zbl 1032.42028
[17] Peherstorfer, F., Yuditskii, P.: Remark on the paper ”Asymptotic behavior of polynomials orthonormal on a homogeneous set”. arXiv:math.SP/0611856 · Zbl 1032.42028
[18] Remling, C.: The absolutely continuous spectrum of Jacobi matrices. Preprint · Zbl 1235.47032
[19] Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) · Zbl 0925.00005
[20] Shohat, J.A.: Théorie générale des polinomes orthogonaux de Tchebichef. Mém. Sci. Math. 66, 1–69 (1934)
[21] Simon, B.: A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices. J. Funct. Anal. 214, 396–409 (2004) · Zbl 1064.30030
[22] Simon, B.: OPUC on one foot. Bull. Am. Math. Soc. 42, 431–460 (2005) · Zbl 1108.42005
[23] Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. AMS Colloquium Publications, vol. 54.1. American Mathematical Society, Providence (2005) · Zbl 1082.42020
[24] Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. AMS Colloquium Publications, vol. 54.2. American Mathematical Society, Providence (2005) · Zbl 1082.42021
[25] Simon, B.: Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging 1, 713–772 (2007) · Zbl 1149.31004
[26] Simon, B.: Szego’s Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials. Princeton University Press (expected 2010)
[27] Simon, B., Zlatoš, A.: Sum rules and the Szego condition for orthogonal polynomials on the real line. Commun. Math. Phys. 242, 393–423 (2003) · Zbl 1046.42017
[28] Sodin, M., Yuditskii, P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7, 387–435 (1997) · Zbl 1041.47502
[29] Stahl, H., Totik, V.: General Orthogonal Polynomials. Encyclopedia of Mathematics and its Applications, vol. 43. Cambridge University Press, Cambridge (1992) · Zbl 0791.33009
[30] Szego, G.: Beiträge zur Theorie der Toeplitzschen Formen. Math. Z. 6, 167–202 (1920) · JFM 47.0391.04
[31] Szego, G.: Beiträge zur Theorie der Toeplitzschen Formen, II. Math. Z. 9, 167–190 (1921)
[32] Szego, G.: Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind. Math. Ann. 86, 114–139 (1922) · JFM 48.0378.02
[33] Szego, G.: Orthogonal Polynomials. AMS Colloquium Publications, vol. 23, American Mathematical Society, Providence (1939). 3rd edn in 1967
[34] Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969) · Zbl 0183.07503
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