Perturbations of orthogonal polynomials with periodic recursion coefficients

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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      AUTHOR = {Nevai, Paul G. and Totik, Vilmos},
      TITLE = {Orthogonal polynomials and their zeros},
      JOURNAL = {Acta Sci. Math. $($Szeged$)$},
      FJOURNAL = {Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum},
      VOLUME = {53},
      YEAR = {1989},
      NUMBER = {1-2},
      PAGES = {99--104},
      ISSN = {0001-6969},
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      JOURNAL = {J. Approx. Theory},
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      VOLUME = {127},
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      AUTHOR = {Pastur, Leonid and Figotin, Alexander},
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      SERIES = {Grundl. Math. Wissen.},
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      ISBN = {3-540-50622-5},
      MRCLASS = {47B80 (34L40 35P25 47E05 47F05 60H25 82B10 82C10)},
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      VOLUME = {89},
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      ISSN = {0021-7670},
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      AUTHOR = {Rabinovich, V. S.},
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      JOURNAL = {Mat. Zametki},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Matematicheskie Zametki},
      VOLUME = {51},
      YEAR = {1992},
      NUMBER = {5},
      PAGES = {90--101, 158},
      ISSN = {0025-567X},
      MRCLASS = {47B37 (39A12 43A15 45E10 47A53 47B35)},
      MRNUMBER = {93j:47041},
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      author={Rakhmanov, E. A.},
      TITLE={On the asymptotics of the ratio of orthogonal polynomials},
      JOURNAL={Math. {\rm USSR} Sb.},
      VOLUME={32},
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      author={Rakhmanov, E. A.},
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      JOURNAL={Math. {\rm USSR} Sb.},
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      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
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      ISSN = {0143-3857},
      MRCLASS = {34B25 (34A55 58F19 58G25)},
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      JOURNAL = {Comm. Math. Phys.},
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      MRCLASS = {47E05 (34L05 47A70)},
      MRNUMBER = {92c:47056},
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      FJOURNAL = {Journal of Functional Analysis},
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      CODEN = {JFUAAW},
      MRCLASS = {47B36 (30D30 30D50 39A70)},
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      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
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      ISSN = {0001-5962},
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      NUMBER = {54.1},
      PUBLISHER = {Amer. Math. Soc.},
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      YEAR = {2005},
      PAGES = {xxvi+466},
      ISBN = {0-8218-3446-0},
      MRCLASS = {42-02 (30C85 33C45 42C05 47B36 47N50)},
      MRNUMBER = {2006a:42002a},
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      AUTHOR = {Simon, Barry},
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      NUMBER = {54.2},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      PAGES = {i--xxii and 467--1044},
      ISBN = {0-8218-3675-7},
      MRCLASS = {42-02 (30C85 33C45 42C05 47B36 47N50)},
      MRNUMBER = {2006a:42002b},
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      ZBLNUMBER = {1082.42021},
      }
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      AUTHOR = {Simon, Barry},
      TITLE = {Sturm oscillation and comparison theorems},
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      PAGES = {29--43},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2005},
      MRCLASS = {34C10 (34C11 39A11 47E05)},
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      JOURNAL = {Funktsional. Anal. i Prilozhen.},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Funktsional'nyĭAnaliz i ego Prilozheniya},
      VOLUME = {41},
      YEAR = {2007},
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      PAGES = {78--92, 112},
      ISSN = {0374-1990},
      MRCLASS = {47B36 (30C10 39A10 42C05)},
      MRNUMBER = {2345042},
      MRREVIEWER = {Stanislav Kupin},
      DOI = {10.1007/s10688-007-0013-z},
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      }
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      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},
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      }
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      JOURNAL = {Comm. Math. Phys.},
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      ISSN = {0010-3616},
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      MRNUMBER = {2005f:42057},
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      ISSN = {1050-6926},
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      FJOURNAL = {Mathematische Annalen},
      VOLUME = {76},
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      ISSN = {0025-5831},
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Authors

David Damanik

Department of Mathematics
Rice University
Houston, TX 77005
United States

Rowan Killip

Department of Mathematics
University of California Los Angeles
Box 951555
Los Angeles, CA 90095-1555
United States

Barry Simon

Mathematics 253-37
California Institute of Technology
Pasadena, CA 91125
United States