Abstract
The results of Denisov-Rakhmanov, Szegő-Shohat-Nevai, and Killip-Simon are extended from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.
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MRCLASS = {47B36 (39A70 42C05 47A10)},
MRNUMBER = {2008e:47081a},
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} -
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@article {Sak92, MRKEY = {1090913},
AUTHOR = {Sakhnovich, A. L.},
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JOURNAL = {Mat. Sb.},
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VOLUME = {181},
YEAR = {1990},
NUMBER = {11},
PAGES = {1510--1524},
ISSN = {0368-8666},
MRCLASS = {47E05 (34L05 47A70)},
MRNUMBER = {92c:47056},
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NOTE = {in Russian; translated in {\it Math. {\rm USSR} Sb.} {\bf 71} (1992), 355--369.},
} -
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[Sh] J. A. Shohat, "Théorie générale des polinomes orthogonaux de Tchebichef," Mémorial des Sciences Mathématiques, vol. 66, pp. 1-69, 1934.
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author={Shohat, J. A.},
TITLE={Théorie générale des polinomes orthogonaux de Tchebichef},
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VOLUME={66},
PAGES={1--69},
YEAR={1934},
} -
[S168]
B. Simon, "Kotani theory for one-dimensional stochastic Jacobi matrices," Comm. Math. Phys., vol. 89, iss. 2, pp. 227-234, 1983.
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JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
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YEAR = {1983},
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MRNUMBER = {85d:60122},
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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.
@article {Sim288, MRKEY = {2083307},
AUTHOR = {Simon, Barry},
TITLE = {A canonical factorization for meromorphic {H}erglotz functions on the unit disk and sum rules for {J}acobi matrices},
JOURNAL = {J. Funct. Anal.},
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VOLUME = {214},
YEAR = {2004},
NUMBER = {2},
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ISSN = {0022-1236},
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} -
[Sim303]
B. Simon, "Meromorphic Szegő functions and asymptotic series for Verblunsky coefficients," Acta Math., vol. 195, pp. 267-285, 2005.
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TITLE = {Meromorphic {S}zegő functions and asymptotic series for {V}erblunsky coefficients},
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MRCLASS = {30C10},
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ZBLNUMBER={1117.42005},
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[OPUC1] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1; Classical Theory, Providence, RI: Amer. Math. Soc., 2005.
@book {OPUC1, MRKEY = {2105088},
AUTHOR = {Simon, Barry},
TITLE = {Orthogonal Polynomials on the Unit Circle. {P}art 1; Classical Theory},
SERIES = {Amer. Math. Soc. Colloq. Ser.},
NUMBER = {54.1},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
PAGES = {xxvi+466},
ISBN = {0-8218-3446-0},
MRCLASS = {42-02 (30C85 33C45 42C05 47B36 47N50)},
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ZBLNUMBER = {1082.42020},
} -
[OPUC2] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2: Spectral theory, Providence, RI: Amer. Math. Soc., 2005.
@book {OPUC2, MRKEY = {2105089},
AUTHOR = {Simon, Barry},
TITLE = {Orthogonal Polynomials on the Unit Circle. {P}art 2: Spectral theory},
SERIES = {Amer. Math. Soc. Colloq. Ser.},
NUMBER = {54.2},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
PAGES = {i--xxii and 467--1044},
ISBN = {0-8218-3675-7},
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MRNUMBER = {2006a:42002b},
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[Sim_Sturm] B. Simon, "Sturm oscillation and comparison theorems," in Sturm-Liouville Theory, Basel: Birkhäuser, 2005, pp. 29-43.
@incollection {Sim_Sturm, MRKEY = {2145076},
AUTHOR = {Simon, Barry},
TITLE = {Sturm oscillation and comparison theorems},
BOOKTITLE = {Sturm-{L}iouville Theory},
PAGES = {29--43},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2005},
MRCLASS = {34C10 (34C11 39A11 47E05)},
MRNUMBER = {2145076},
ZBLNUMBER = {1117.39013},
} -
[Jost3]
B. Simon, "Meromorphic Jost functions and asymptotic expansions for Jacobi parameters," Funktsional. Anal. i Prilozhen., vol. 41, iss. 2, pp. 78-92, 112, 2007.
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JOURNAL = {Funktsional. Anal. i Prilozhen.},
FJOURNAL = {Rossiĭskaya Akademiya Nauk. Funktsional'nyĭAnaliz i ego Prilozheniya},
VOLUME = {41},
YEAR = {2007},
NUMBER = {2},
PAGES = {78--92, 112},
ISSN = {0374-1990},
MRCLASS = {47B36 (30C10 39A10 42C05)},
MRNUMBER = {2345042},
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ZBLNUMBER={pre05206865},
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[SimonNew] B. Simon, Szegő’s Theorem and Its Descendants : Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials, Princeton, NJ: Princeton Univ. Press, 2010.
@book{SimonNew,
author = {Simon, Barry},
TITLE={Szegő's Theorem and Its Descendants{\rm :} Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials},
SERIES={Porter Lecture Series},
VENUE={September 2006},
PUBLISHER={Princeton Univ. Press},
ADDRESS={Princeton, NJ},
NOTE={in press},
YEAR={2010},
} -
[SZ] B. Simon and A. Zlatovs, "Sum rules and the Szegő condition for orthogonal polynomials on the real line," Comm. Math. Phys., vol. 242, iss. 3, pp. 393-423, 2003.
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AUTHOR = {Simon, Barry and Zlato{š},
Andrej},
TITLE = {Sum rules and the {S}zegő condition for orthogonal polynomials on the real line},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {242},
YEAR = {2003},
NUMBER = {3},
PAGES = {393--423},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {42C05 (15A18 39A70 47B36)},
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[SY] 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.
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G. SzegHo, "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion," Math. Ann., vol. 76, iss. 4, pp. 490-503, 1915.
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@incollection {YM2001, MRKEY = {1842902},
AUTHOR = {Yakhlef, Hossain O. and Marcell{á}n, Francisco},
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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.
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@article {YK, MRKEY = {0469461},
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DOI = {10.1109/TCS.1978.1084443},
ZBLNUMBER={0417.93018},
}