Perturbations of orthogonal polynomials with periodic recursion coefficients

David Damanik; Rowan Killip; Barry Simon

doi:http://doi.org/10.4007/annals.2010.171.1931

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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@article {DS02, MRKEY = {1936144}, AUTHOR = {Dette, Holger and Studden, William J.}, TITLE = {A note on the matrix valued q-d algorithm and matrix orthogonal polynomials on {$[0,1]$} and {$[0,\infty)$}}, JOURNAL = {J. Comput. Appl. Math.}, FJOURNAL = {Journal of Computational and Applied Mathematics}, VOLUME = {148}, YEAR = {2002}, NUMBER = {2}, PAGES = {349--361}, ISSN = {0377-0427}, CODEN = {JCAMDI}, MRCLASS = {42C05 (15A99 33F05)}, MRNUMBER = {2004g:42029}, DOI = {10.1016/S0377-0427(02)00555-1}, ZBLNUMBER = {1052.42022}, }
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@article {Dur99, MRKEY = {1714953}, AUTHOR = {Dur{á}n, Antonio J. }, TITLE = {Ratio asymptotics for orthogonal matrix polynomials}, JOURNAL = {J. Approx. Theory}, FJOURNAL = {Journal of Approximation Theory}, VOLUME = {100}, YEAR = {1999}, NUMBER = {2}, PAGES = {304--344}, ISSN = {0021-9045}, CODEN = {JAXTAZ}, MRCLASS = {42C05 (33C45)}, MRNUMBER = {2000j:42038}, MRREVIEWER = {Walter Van Assche}, DOI = {10.1006/jath.1999.3351}, ZBLNUMBER = {0944.42015}, }
[DD02] $Go to document$ A. J. Durán and E. Defez, "Orthogonal matrix polynomials and quadrature formulas," Linear Algebra Appl., vol. 345, pp. 71-84, 2002.

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@article {DD02, MRKEY = {1883268}, AUTHOR = {Dur{á}n, Antonio J. and Defez, Emilio}, TITLE = {Orthogonal matrix polynomials and quadrature formulas}, JOURNAL = {Linear Algebra Appl.}, FJOURNAL = {Linear Algebra and its Applications}, VOLUME = {345}, YEAR = {2002}, PAGES = {71--84}, ISSN = {0024-3795}, CODEN = {LAAPAW}, MRCLASS = {41A55 (42C05 65D32)}, MRNUMBER = {2002i:41042}, MRREVIEWER = {Sven Ehrich}, DOI = {10.1016/S0024-3795(01)00474-8}, ZBLNUMBER = {0990.42009}, }
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@article {DLR96, MRKEY = {1368729}, AUTHOR = {Dur{á}n, Antonio J. and L{ó}pez-Rodr{\'ı}guez, Pedro}, TITLE = {Orthogonal matrix polynomials: zeros and {B}lumenthal's theorem}, JOURNAL = {J. Approx. Theory}, FJOURNAL = {Journal of Approximation Theory}, VOLUME = {84}, YEAR = {1996}, NUMBER = {1}, PAGES = {96--118}, ISSN = {0021-9045}, CODEN = {JAXTAZ}, MRCLASS = {42C05 (47A58)}, MRNUMBER = {97b:42039}, MRREVIEWER = {Walter Van Assche}, DOI = {10.1006/jath.1996.0007}, ZBLNUMBER = {0861.42016}, }
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@incollection {DLR01, MRKEY = {1882637}, AUTHOR = {Dur{á}n, Antonio J. and L{ó}pez-Rodr{\'ı}guez, Pedro}, TITLE = {The matrix moment problem}, BOOKTITLE = {Margarita Mathematica}, PAGES = {333--348}, PUBLISHER = {Univ. La Rioja, Logroño}, YEAR = {2001}, MRCLASS = {44A60 (42C05 47A57 47B36)}, MRNUMBER = {2003c:44010}, }
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@incollection {DLR04, MRKEY = {2085851}, AUTHOR = {Dur{á}n, Antonio J. and L{ó}pez-Rodr{\'ı}guez, Pedro}, TITLE = {Orthogonal matrix polynomials}, BOOKTITLE = {Laredo {L}ectures on {O}rthogonal {P}olynomials and {S}pecial {F}unctions}, SERIES = {Adv. Theory Spec. Funct. Orthogonal Polynomials}, PAGES = {13--44}, PUBLISHER = {Nova Sci. Publ.}, ADDRESS = {Hauppauge, NY}, YEAR = {2004}, MRCLASS = {42C05 (33C47)}, MRNUMBER = {2006b:42032}, MRREVIEWER = {Pascal Maroni}, ZBLNUMBER = {1100.42018}, }
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@article {DLRS99, MRKEY = {1714465}, AUTHOR = {Dur{á}n, Antonio J. and L{ó}pez-Rodr{\'ı}guez, Pedro and Saff, E. B.}, TITLE = {Zero asymptotic behaviour for orthogonal matrix polynomials}, JOURNAL = {J. Anal. Math.}, FJOURNAL = {Journal d'Analyse Mathématique}, VOLUME = {78}, YEAR = {1999}, PAGES = {37--60}, ISSN = {0021-7670}, CODEN = {JOAMAV}, MRCLASS = {42C05 (33C45)}, MRNUMBER = {2000j:42037}, MRREVIEWER = {Walter Van Assche}, ZBLNUMBER = {0945.42013}, }
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@article {FIT, MRKEY = {871819}, AUTHOR = {Finkel, Allan and Isaacson, Eli and Trubowitz, Eugene}, TITLE = {An explicit solution of the inverse periodic problem for {H}ill's equation}, JOURNAL = {{\rm SIAM} J. Math. Anal.}, FJOURNAL = {SIAM Journal on Mathematical Analysis}, VOLUME = {18}, YEAR = {1987}, NUMBER = {1}, PAGES = {46--53}, ISSN = {0036-1410}, CODEN = {SJMAAH}, MRCLASS = {34B30 (34B25 58F19)}, MRNUMBER = {88d:34037}, MRREVIEWER = {F. M. Arscott}, DOI = {10.1137/0518003}, ZBLNUMBER = {0622.34021}, }
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@article {Fu76, MRKEY = {0407652}, AUTHOR = {Fukushima, Masahisa}, TITLE = {A spectral representation on ordinary linear difference equation with operator-valued coefficients of the second order}, JOURNAL = {J. Mathematical Phys.}, FJOURNAL = {Journal of Mathematical Physics}, VOLUME = {17}, YEAR = {1976}, NUMBER = {6}, PAGES = {1064--1072}, ISSN = {0022-2488}, MRCLASS = {47B45 (82.47)}, MRNUMBER = {53 \#11424}, MRREVIEWER = {K. Kumar}, DOI = {10.1063/1.523018}, ZBLNUMBER = {0333.47015}, }
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@article {GI1, MRKEY = {1918787}, AUTHOR = {Georgescu, Vladimir and Iftimovici, Andrei}, TITLE = {Crossed products of {$C\sp \ast$}-algebras and spectral analysis of quantum {H}amiltonians}, JOURNAL = {Comm. Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}, VOLUME = {228}, YEAR = {2002}, NUMBER = {3}, PAGES = {519--560}, ISSN = {0010-3616}, CODEN = {CMPHAY}, MRCLASS = {81Q10 (46L55 47N50 81R15)}, MRNUMBER = {2003k:81053}, MRREVIEWER = {Alan L. Carey}, DOI = {10.1007/s002200200669}, ZBLNUMBER = {1005.81026}, }
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@article {Ger81, MRKEY = {626123}, AUTHOR = {Geronimo, Jeffrey S.}, TITLE = {Matrix orthogonal polynomials on the unit circle}, JOURNAL = {J. Math. Phys.}, FJOURNAL = {Journal of Mathematical Physics}, VOLUME = {22}, YEAR = {1981}, NUMBER = {7}, PAGES = {1359--1365}, ISSN = {0022-2488}, CODEN = {JMAPAQ}, MRCLASS = {42C05}, MRNUMBER = {82k:42013}, MRREVIEWER = {A. E. Danese}, DOI = {10.1063/1.525073}, ZBLNUMBER = {0505.42019}, }
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@article {GBk1, MRKEY = {0061706}, AUTHOR = {Geronimus, Ya. L.}, TITLE = {Polynomials orthogonal on a circle and their applications}, JOURNAL = {Amer. Math. Soc. Translation}, FJOURNAL = {American Mathematical Society Translations}, VOLUME = {1954}, YEAR = {1954}, NUMBER = {104}, PAGES = {79}, ISSN = {0065-9290}, MRCLASS = {41.1X}, MRNUMBER = {15,869i}, ZBLNUMBER = {0056.10303}, }
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@incollection{Ger77, author = {Geronimus, Ya. L.}, TITLE = {Orthogonal polynomials}, BOOKTITLE={Two Papers on Special Functions}, SERIES={Amer. Math. Soc. Transl., Ser. 2}, NUMBER={108}, PAGES={37--130}, PUBLISHER={Amer. Math. Soc.}, ADDRESS={Providence, RI}, YEAR={1977}, NOTE={English translation of the appendix to the Russian translation of Szegő's book}, ZBLNUMBER={0365.42007}, }
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@article {GShpa, MRKEY = {1441597}, AUTHOR = {Gesztesy, Fritz and Simon, Barry}, TITLE = {Inverse spectral analysis with partial information on the potential. {I}. {T}he case of an a.c. component in the spectrum}, JOURNAL = {Helv. Phys. Acta}, FJOURNAL = {Helvetica Physica Acta. Physica Theoretica. Societatis Physicae Helveticae Commentaria Publica}, VOLUME = {70}, YEAR = {1997}, NUMBER = {1-2}, PAGES = {66--71}, ISSN = {0018-0238}, CODEN = {HPACAK}, MRCLASS = {81U40 (34A55 34L40 47E05 47N50)}, MRNUMBER = {98f:81347}, }
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@article {GStams, MRKEY = {1694291}, AUTHOR = {Gesztesy, Fritz and Simon, Barry}, TITLE = {Inverse spectral analysis with partial information on the potential. {II}. {T}he case of discrete spectrum}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {352}, YEAR = {2000}, NUMBER = {6}, PAGES = {2765--2787}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {34A55 (34B24 34L40 47E05)}, MRNUMBER = {2000j:34019}, MRREVIEWER = {Vjacheslav A. Yurko}, DOI = {10.1090/S0002-9947-99-02544-1}, ZBLNUMBER = {0948.34060}, }
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@article {GT2000, MRKEY = {1784638}, AUTHOR = {Gesztesy, Fritz and Tsekanovskii, Eduard}, TITLE = {On matrix-valued {H}erglotz functions}, JOURNAL = {Math. Nachr.}, FJOURNAL = {Mathematische Nachrichten}, VOLUME = {218}, YEAR = {2000}, PAGES = {61--138}, ISSN = {0025-584X}, MRCLASS = {47A57 (30E05 34L05 47E05)}, MRNUMBER = {2001j:47018}, MRREVIEWER = {Aad Dijksma}, DOI = {10.1002/1522-2616(200010)218:1%3C61::AID-MANA61%3E3.3.CO%3B2-4}, ZBLNUMBER = {0961.30027}, }
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@article {GKS98, MRKEY = {1631291}, AUTHOR = {Gohberg, I. and Kaashoek, M. A. and Sakhnovich, A. L.}, TITLE = {Pseudo-canonical systems with rational {W}eyl functions: explicit formulas and applications}, JOURNAL = {J. Differential Equations}, FJOURNAL = {Journal of Differential Equations}, VOLUME = {146}, YEAR = {1998}, NUMBER = {2}, PAGES = {375--398}, ISSN = {0022-0396}, CODEN = {JDEQAK}, MRCLASS = {34L05 (34A55 35P99 37K10 47E05 47N70)}, MRNUMBER = {2000c:34211}, MRREVIEWER = {Hendrik S. V. de Snoo}, DOI = {10.1006/jdeq.1998.3428}, ZBLNUMBER = {0917.34074}, }
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@misc{GSprep, author={Golinskii, L. and Simon, B.}, TITLE={{\rm Section 4.3 of} {\it Orthogonal Polynomials on the Unit Circle, Part~{\rm 1}: Classical Theory}}, SERIES={Amer. Math. Colloq. Ser.}, PUBLISHER={Amer. Math. Soc.}, ADDRESS={Providence, RI}, YEAR={2005}, }
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@book {GvL, MRKEY = {1417720}, AUTHOR = {Golub, Gene H. and Van Loan, Charles F.}, TITLE = {Matrix Computations}, SERIES = {Johns Hopkins Studies in the Mathematical Sciences}, EDITION = {Third}, PUBLISHER = {Johns Hopkins University Press}, ADDRESS = {Baltimore, MD}, YEAR = {1996}, PAGES = {xxx+698}, ISBN = {0-8018-5413-X; 0-8018-5414-8}, MRCLASS = {65-02 (65Fxx)}, MRNUMBER = {97g:65006}, ZBLNUMBER = {0865.65009}, }
[HS81] $Go to document$ 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.

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@article {HS81, MRKEY = {621248}, AUTHOR = {Hinton, D. B. and Shaw, J. K.}, TITLE = {On {T}itchmarsh-{W}eyl {$M(\lambda )$}-functions for linear {H}amiltonian systems}, JOURNAL = {J. Differential Equations}, FJOURNAL = {Journal of Differential Equations}, VOLUME = {40}, YEAR = {1981}, NUMBER = {3}, PAGES = {316--342}, ISSN = {0022-0396}, CODEN = {JDEQAK}, MRCLASS = {34B20}, MRNUMBER = {82h:34018}, MRREVIEWER = {H. Hochstadt}, DOI = {10.1016/0022-0396(81)90002-4}, }
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@article {HL, MRKEY = {0470319}, AUTHOR = {Hochstadt, Harry and Lieberman, Burton}, TITLE = {An inverse {S}turm-{L}iouville problem with mixed given data}, JOURNAL = {{\rm SIAM} J. Appl. Math.}, FJOURNAL = {SIAM Journal on Applied Mathematics}, VOLUME = {34}, YEAR = {1978}, NUMBER = {4}, PAGES = {676--680}, ISSN = {0036-1399}, MRCLASS = {34B25}, MRNUMBER = {57 \#10077}, MRREVIEWER = {Shinichi Kotani}, DOI = {10.1137/0134054}, }
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@article {HunS, MRKEY = {1928259}, AUTHOR = {Hundertmark, Dirk and Simon, Barry}, TITLE = {Lieb-{T}hirring inequalities for {J}acobi matrices}, JOURNAL = {J. Approx. Theory}, FJOURNAL = {Journal of Approximation Theory}, VOLUME = {118}, YEAR = {2002}, NUMBER = {1}, PAGES = {106--130}, ISSN = {0021-9045}, CODEN = {JAXTAZ}, MRCLASS = {39A70 (15A45)}, MRNUMBER = {2003h:39016}, MRREVIEWER = {Fozi M. Dannan}, DOI = {10.1006/jath.2002.3704}, ZBLNUMBER = {1019.39013}, }
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@article {Iwa, MRKEY = {932784}, AUTHOR = {Iwasaki, Katsunori}, TITLE = {Inverse problem for {S}turm-{L}iouville and {H}ill equations}, JOURNAL = {Ann. Mat. Pura Appl.}, FJOURNAL = {Annali di Matematica Pura ed Applicata. Serie Quarta}, VOLUME = {149}, YEAR = {1987}, PAGES = {185--206}, ISSN = {0003-4622}, CODEN = {ANLMAE}, MRCLASS = {34B25 (34A55 34B30)}, MRNUMBER = {89d:34053}, MRREVIEWER = {Zhaojun Liang}, DOI = {10.1007/BF01773933}, ZBLNUMBER = {0641.34012}, }
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@article {JDP96, MRKEY = {1465566}, AUTHOR = {J{ó}dar, L. and Defez, E. and Ponsoda, E.}, TITLE = {Orthogonal matrix polynomials with respect to linear matrix moment functionals: theory and applications}, JOURNAL = {Approx. Theory Appl.}, FJOURNAL = {Approximation Theory and its Applications. New Series}, VOLUME = {12}, YEAR = {1996}, NUMBER = {1}, PAGES = {96--115}, ISSN = {1000-9221}, MRCLASS = {42C05 (41A10)}, MRNUMBER = {98f:42022}, ZBLNUMBER = {0853.42022}, }
[Jo87] $Go to document$ R. A. Johnson, "$m$-functions and Floquet exponents for linear differential systems," Ann. Mat. Pura Appl., vol. 147, pp. 211-248, 1987.

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@article {Jo87, MRKEY = {916710}, AUTHOR = {Johnson, Russell A.}, TITLE = {{$m$}-functions and {F}loquet exponents for linear differential systems}, JOURNAL = {Ann. Mat. Pura Appl.}, FJOURNAL = {Annali di Matematica Pura ed Applicata. Serie Quarta}, VOLUME = {147}, YEAR = {1987}, PAGES = {211--248}, ISSN = {0003-4622}, CODEN = {ANLMAE}, MRCLASS = {34B20 (58F05 58F19)}, MRNUMBER = {88m:34021}, MRREVIEWER = {Walter Eberhard}, DOI = {10.1007/BF01762419}, ZBLNUMBER = {0652.34016}, }
[Kil02] $Go to document$ R. Killip, "Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum," Int. Math. Res. Not., iss. 38, pp. 2029-2061, 2002.

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@article {Kil02, MRKEY = {1925875}, AUTHOR = {Killip, Rowan}, TITLE = {Perturbations of one-dimensional {S}chrödinger operators preserving the absolutely continuous spectrum}, JOURNAL = {Int. Math. Res. Not.}, FJOURNAL = {International Mathematics Research Notices}, YEAR = {2002}, NUMBER = {38}, PAGES = {2029--2061}, ISSN = {1073-7928}, MRCLASS = {34L40 (34B20 47A55 47E05)}, MRNUMBER = {2004e:34133}, MRREVIEWER = {Denis Krutikov}, DOI = {10.1155/S1073792802204250}, ZBLNUMBER={1021.34071}, }
[Kil-SFest] 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.

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@incollection {Kil-SFest, MRKEY = {2310217}, AUTHOR = {Killip, Rowan}, TITLE = {Spectral theory via sum rules}, BOOKTITLE = {Spectral Theory and Mathematical Physics: A {F}estschrift in Honor of {B}arry {S}imon's 60th Birthday}, SERIES = {Proc. Sympos. Pure Math.}, NUMBER = {76}, PAGES = {907--930}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2007}, MRCLASS = {47E05 (34L05 47A55 47B36 47B39 81Q10)}, MRNUMBER = {2008d:47086}, MRREVIEWER = {Christian Remling}, ZBLNUMBER = {1137.47001}, }
[KS] $Go to document$ 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.

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@article {KS, MRKEY = {1999923}, AUTHOR = {Killip, Rowan and Simon, Barry}, TITLE = {Sum rules for {J}acobi matrices and their applications to spectral theory}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {158}, YEAR = {2003}, NUMBER = {1}, PAGES = {253--321}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {47B36 (39A70 47A10 47A55)}, MRNUMBER = {2004f:47040}, MRREVIEWER = {Gerald Teschl}, DOI = {10.4007/annals.2003.158.253}, ZBLNUMBER = {1050.47025}, }
[Kot] 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.

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@incollection {Kot, MRKEY = {780760}, AUTHOR = {Kotani, Shinichi}, TITLE = {Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional {S}chrödinger operators}, BOOKTITLE = {Stochastic Analysis ({K}atata/{K}yoto, 1982)}, SERIES = {North-Holland Math. Library}, NUMBER = {32}, PAGES = {225--247}, PUBLISHER = {North-Holland}, ADDRESS = {Amsterdam}, YEAR = {1984}, MRCLASS = {60H25 (35P05 81C20 82A57)}, MRNUMBER = {86h:60117}, MRREVIEWER = {W. Kirsch}, ZBLNUMBER={0549.60058}, }
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@article {Kot97, MRKEY = {1477262}, AUTHOR = {Kotani, Shinichi}, TITLE = {Generalized {F}loquet theory for stationary {S}chrödinger operators in one dimension}, JOURNAL = {Chaos Solitons Fractals}, FJOURNAL = {Chaos, Solitons and Fractals}, VOLUME = {8}, YEAR = {1997}, NUMBER = {11}, PAGES = {1817--1854}, ISSN = {0960-0779}, CODEN = {CSFOEH}, MRCLASS = {34L40 (34A55 34L05 47B80 60H25 82B44)}, MRNUMBER = {98m:34173}, MRREVIEWER = {Nariyuki Minami}, DOI = {10.1016/S0960-0779(97)00042-8}, ZBLNUMBER={0936.34074}, }
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@article {Kr89, MRKEY = {990871}, AUTHOR = {Krall, Allan M.}, TITLE = {{$M(\lambda)$} theory for singular {H}amiltonian systems with one singular point}, JOURNAL = {{\rm SIAM} J. Math. Anal.}, FJOURNAL = {SIAM Journal on Mathematical Analysis}, VOLUME = {20}, YEAR = {1989}, NUMBER = {3}, PAGES = {664--700}, ISSN = {0036-1410}, CODEN = {SJMAAH}, MRCLASS = {34B20 (34B05 34B27 34L10 47E05)}, MRNUMBER = {91c:34036}, DOI = {10.1137/0520047}, ZBLNUMBER = {0683.34008}, }
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@article{KO78, author={Krein, M. G. and Ov{\v{c}}arenko, I. E.}, TITLE={Inverse problems for $Q$-functions and resolvent matrices of positive Hermitian operators}, JOURNAL={Soviet Math. Dokl.}, VOLUME={19}, YEAR={1978}, PAGES={1131--1134}, }
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@article {Kup2003, MRKEY = {1981479}, AUTHOR = {Kupin, Stanislav}, TITLE = {On sum rules of special form for {J}acobi matrices}, JOURNAL = {C. R. Math. Acad. Sci. Paris}, FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris}, VOLUME = {336}, YEAR = {2003}, NUMBER = {7}, PAGES = {611--614}, ISSN = {1631-073X}, MRCLASS = {39A70 (47B36)}, MRNUMBER = {2004f:39052}, DOI = {10.1016/S1631-073X(03)00132-8}, ZBLNUMBER = {1057.47037}, }
[LNS2003] 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.

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@article {LNS2003, MRKEY = {2013753}, AUTHOR = {Laptev, A. and Naboko, S. and Safronov, O.}, TITLE = {On new relations between spectral properties of {J}acobi matrices and their coefficients}, JOURNAL = {Comm. Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}, VOLUME = {241}, YEAR = {2003}, NUMBER = {1}, PAGES = {91--110}, ISSN = {0010-3616}, CODEN = {CMPHAY}, MRCLASS = {47B36 (39A70 47A10)}, MRNUMBER = {2004g:47041}, MRREVIEWER = {Denis Krutikov}, ZBLNUMBER = {1135.47303}, }
[LastS] $Go to document$ 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.

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@article {LastS, MRKEY = {1666767}, AUTHOR = {Last, Yoram and Simon, Barry}, TITLE = {Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional {S}chrödinger operators}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {135}, YEAR = {1999}, NUMBER = {2}, PAGES = {329--367}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {47B39 (34L05 34L40 47E05 81Q10)}, MRNUMBER = {2000f:47060}, MRREVIEWER = {G{ü}nter Stolz}, DOI = {10.1007/s002220050288}, ZBLNUMBER={0931.34066}, }
[LS_jdam] Y. Last and B. Simon, "The essential spectrum of Schrödinger, Jacobi, and CMV operators," J. Anal. Math., vol. 98, pp. 183-220, 2006.

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@article {LS_jdam, MRKEY = {2254485}, AUTHOR = {Last, Yoram and Simon, Barry}, TITLE = {The essential spectrum of {S}chrödinger, {J}acobi, and {CMV} operators}, JOURNAL = {J. Anal. Math.}, FJOURNAL = {Journal d'Analyse Mathématique}, VOLUME = {98}, YEAR = {2006}, PAGES = {183--220}, ISSN = {0021-7670}, CODEN = {JOAMAV}, MRCLASS = {47A10 (34L40 35P05 47B36 47E05 47F05 81Q10)}, MRNUMBER = {2008a:47008}, MRREVIEWER = {Jaouad Sahbani}, ZBLNUMBER={1145.34052}, }
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@article {Lev, MRKEY = {0019257}, AUTHOR = {Levinson, Norman}, TITLE = {The {W}iener {RMS} (root mean square) error criterion in filter design and prediction}, JOURNAL = {J. Math. Phys. Mass. Inst. Tech.}, FJOURNAL = {Journal of Mathematics and Physics}, VOLUME = {25}, YEAR = {1947}, PAGES = {261--278}, ISSN = {0097-1421}, MRCLASS = {60.0X}, MRNUMBER = {8,391e}, MRREVIEWER = {J. L. Doob}, }
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@book {MagWin, MRKEY = {0197830}, AUTHOR = {Magnus, Wilhelm and Winkler, Stanley}, TITLE = {Hill's Equation}, SERIES = {Interscience Tracts in Pure and Applied Mathematics}, NUMBER={20}, PUBLISHER = {Interscience Publishers, John Wiley \& Sons}, ADDRESS={New York}, YEAR = {1966}, PAGES = {viii+127}, MRCLASS = {34.30}, MRNUMBER = {33 \#5991}, MRREVIEWER = {H. Hochstadt}, ZBLNUMBER = {0158.09604}, }
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@article {M1, MRKEY = {1925913}, AUTHOR = {M{\u{a}}ntoiu, Marius}, TITLE = {{$C\sp \ast$}-algebras, dynamical systems at infinity and the essential spectrum of generalized {S}chrödinger operators}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik}, VOLUME = {550}, YEAR = {2002}, PAGES = {211--229}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {81R15 (35P20 35Q40 37B10 46L05 47F05 47L65 81Q10)}, MRNUMBER = {2003i:81109}, MRREVIEWER = {Vladimir Georgescu}, DOI = {10.1515/crll.2002.073}, ZBLNUMBER={1036.46052}, }
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@article {MRG89, MRKEY = {1011740}, AUTHOR = {Marcell{á}n, Francisco and Rodr{\'ı}guez Gonz{á}lez, Isabel}, TITLE = {A class of matrix orthogonal polynomials on the unit circle}, NOTE = {Linear algebra and applications (Valencia, 1987)}, JOURNAL = {Linear Algebra Appl.}, FJOURNAL = {Linear Algebra and its Applications}, VOLUME = {121}, YEAR = {1989}, PAGES = {233--241}, ISSN = {0024-3795}, CODEN = {LAAPAW}, MRCLASS = {30C10 (33A65 42C05)}, MRNUMBER = {90i:30009}, MRREVIEWER = {Joaquin Bustoz}, DOI = {10.1016/0024-3795(89)90704-0}, ZBLNUMBER = {0681.33013}, }
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@article {MS93, MRKEY = {1204344}, AUTHOR = {Marcell{á}n, Francisco and Sansigre, Gabriela}, TITLE = {On a class of matrix orthogonal polynomials on the real line}, JOURNAL = {Linear Algebra Appl.}, FJOURNAL = {Linear Algebra and its Applications}, VOLUME = {181}, YEAR = {1993}, PAGES = {97--109}, ISSN = {0024-3795}, CODEN = {LAAPAW}, MRCLASS = {42C05}, MRNUMBER = {94b:42012}, MRREVIEWER = {Gongli Zhang}, DOI = {10.1016/0024-3795(93)90026-K}, ZBLNUMBER = {0769.15010}, }
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@article {MY03, MRKEY = {1929058}, AUTHOR = {Marcell{á}n, Francisco and Yakhlef, H. O.}, TITLE = {Recent trends on analytic properties of matrix orthonormal polynomials}, NOTE = {Orthogonal polynomials, approximation theory, and harmonic analysis (Inzel, 2000)}, JOURNAL = {Electron. Trans. Numer. Anal.}, FJOURNAL = {Electronic Transactions on Numerical Analysis}, VOLUME = {14}, YEAR = {2002}, PAGES = {127--141}, ISSN = {1068-9613}, MRCLASS = {42C05 (15A15 15A23 33C99)}, MRNUMBER = {2003f:42041}, MRREVIEWER = {D. Waterman}, ZBLNUMBER = {1033.42026}, }
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@book {Marc, MRKEY = {897106}, AUTHOR = {Marchenko, Vladimir A.}, TITLE = {Sturm-{L}iouville Operators and Applications}, SERIES = {Operator Theory: Advances and Applications}, NUMBER = {22}, PUBLISHER = {Birkhäuser}, ADDRESS = {Basel}, YEAR = {1986}, PAGES = {xii+367}, ISBN = {3-7643-1794-9}, MRCLASS = {34B25 (35Q20 47B99 47E05)}, MRNUMBER = {88f:34034}, MRREVIEWER = {John Adam}, ZBLNUMBER = {0592.34011}, }
[MNT85a] $Go to document$ 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.

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@article {MNT85a, MRKEY = {766095}, AUTHOR = {M{á}t{é}, Attila and Nevai, Paul and Totik, Vilmos}, TITLE = {Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle}, JOURNAL = {Constr. Approx.}, FJOURNAL = {Constructive Approximation. An International Journal for Approximations and Expansions}, VOLUME = {1}, YEAR = {1985}, NUMBER = {1}, PAGES = {63--69}, ISSN = {0176-4276}, MRCLASS = {42C05 (33A65)}, MRNUMBER = {85j:42045}, MRREVIEWER = {G. Gasper}, DOI = {10.1007/BF01890022}, ZBLNUMBER = {0582.42012}, }
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@article {McKT, MRKEY = {0427731}, AUTHOR = {McKean, H. P. and Trubowitz, E.}, TITLE = {Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points}, JOURNAL = {Comm. Pure Appl. Math.}, FJOURNAL = {Communications on Pure and Applied Mathematics}, VOLUME = {29}, YEAR = {1976}, NUMBER = {2}, PAGES = {143--226}, ISSN = {0010-3640}, MRCLASS = {34B25 (58F99 34B30)}, MRNUMBER = {55 \#761}, MRREVIEWER = {Walter Eberhard}, DOI = {10.1002/cpa.3160290203}, ZBLNUMBER = {0339.34024}, }
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@article {Nai, MRKEY = {0142558}, AUTHOR = {Na{\u\i}man, P. B.}, TITLE = {On the theory of periodic and limit-periodic {J}acobian matrices}, JOURNAL = {Dokl. Akad. Nauk SSSR}, FJOURNAL = {Doklady Akademii Nauk SSSR}, VOLUME = {143}, YEAR = {1962}, PAGES = {277--279}, ISSN = {0002-3264}, MRCLASS = {15.35}, MRNUMBER = {26 \#127}, MRREVIEWER = {S. Kurepa}, ZBLNUMBER = {0118.11003}, NOTE = {in Russian; translated in {Soviet Math. Dokl.} {\bf 3} (1962), 383--385.}, }
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@article {Nai2, MRKEY = {0218915}, AUTHOR = {Na{\u\i}man, P. B.}, TITLE = {On the spectral theory of non-symmetric periodic {J}acobi matrices}, JOURNAL = {Zap. Meh.-Mat. Fak. Har'kov. Gos. Univ. i Har'kov. Mat. Obšč.}, VOLUME = {30}, YEAR = {1964}, PAGES = {138--151}, MRCLASS = {47.30}, MRNUMBER = {36 \#1999}, MRREVIEWER = {S. Kurepa}, }
[NPVY] $Go to document$ 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.

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@article {NPVY, MRKEY = {2130241}, AUTHOR = {Nazarov, F. and Peherstorfer, F. and Volberg, A. and Yuditskii, P.}, TITLE = {On generalized sum rules for {J}acobi matrices}, JOURNAL = {Int. Math. Res. Not.}, FJOURNAL = {International Mathematics Research Notices}, YEAR = {2005}, NUMBER = {3}, PAGES = {155--186}, ISSN = {1073-7928}, MRCLASS = {47B36 (47A10 47A55)}, MRNUMBER = {2006h:47054}, MRREVIEWER = {David Damanik}, DOI = {10.1155/IMRN.2005.155}, ZBLNUMBER = {1089.47025}, }
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@article {Nev79, MRKEY = {519926}, AUTHOR = {Nevai, Paul G.}, TITLE = {Orthogonal polynomials}, JOURNAL = {Mem. Amer. Math. Soc.}, FJOURNAL = {Memoirs of the American Mathematical Society}, VOLUME = {18}, YEAR = {1979}, NUMBER = {213}, PAGES = {v+185}, ISSN = {0065-9266}, CODEN = {MAMCAU}, MRCLASS = {42C05}, MRNUMBER = {80k:42025}, MRREVIEWER = {T. S. Chihara}, }
[Nev92] P. G. Nevai, "Orthogonal polynomials, recurrences, Jacobi matrices, and measures," in Progress in Approximation Theory, New York: Springer-Verlag, 1992, pp. 79-104.

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@incollection {Nev92, MRKEY = {1240779}, AUTHOR = {Nevai, Paul G.}, TITLE = {Orthogonal polynomials, recurrences, {J}acobi matrices, and measures}, BOOKTITLE = {Progress in Approximation Theory}, VENUE={{T}ampa, {FL}, 1990}, SERIES = {Springer Ser. Comput. Math.}, NUMBER = {19}, PAGES = {79--104}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1992}, MRCLASS = {42C05 (41A10)}, MRNUMBER = {94g:42037}, ZBLNUMBER = {0789.42016}, }
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@article {NT89, MRKEY = {1018677}, 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}, CODEN = {ASMCBP}, MRCLASS = {33A65 (30C10 42C05)}, MRNUMBER = {90i:33021}, MRREVIEWER = {N. K. Govil}, ZBLNUMBER = {0691.42020}, }
[NTppt] $Go to document$ P. G. Nevai and V. Totik, "Denisov’s theorem on recurrence coefficients," J. Approx. Theory, vol. 127, iss. 2, pp. 240-245, 2004.

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@article {NTppt, MRKEY = {2058161}, AUTHOR = {Nevai, Paul G. and Totik, Vilmos}, TITLE = {Denisov's theorem on recurrence coefficients}, JOURNAL = {J. Approx. Theory}, FJOURNAL = {Journal of Approximation Theory}, VOLUME = {127}, YEAR = {2004}, NUMBER = {2}, PAGES = {240--245}, ISSN = {0021-9045}, CODEN = {JAXTAZ}, MRCLASS = {42C05}, MRNUMBER = {2005e:42076}, MRREVIEWER = {K. Pan}, DOI = {10.1016/j.jat.2004.03.006}, ZBLNUMBER = {1061.42014}, }
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@book {PF, MRKEY = {1223779}, AUTHOR = {Pastur, Leonid and Figotin, Alexander}, TITLE = {Spectra of Random and Almost-Periodic Operators}, SERIES = {Grundl. Math. Wissen.}, NUMBER = {297}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1992}, PAGES = {viii+587}, ISBN = {3-540-50622-5}, MRCLASS = {47B80 (34L40 35P25 47E05 47F05 60H25 82B10 82C10)}, MRNUMBER = {94h:47068}, MRREVIEWER = {Russell A. Johnson}, ZBLNUMBER = {0752.47002}, }
[PY] F. Peherstorfer and P. Yuditskii, "Asymptotic behavior of polynomials orthonormal on a homogeneous set," J. Anal. Math., vol. 89, pp. 113-154, 2003.

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@article {PY, MRKEY = {1981915}, AUTHOR = {Peherstorfer, Franz and Yuditskii, Peter}, TITLE = {Asymptotic behavior of polynomials orthonormal on a homogeneous set}, JOURNAL = {J. Anal. Math.}, FJOURNAL = {Journal d'Analyse Mathématique}, VOLUME = {89}, YEAR = {2003}, PAGES = {113--154}, ISSN = {0021-7670}, CODEN = {JOAMAV}, MRCLASS = {42C05 (33C99 47B36)}, MRNUMBER = {2004j:42022}, MRREVIEWER = {D. S. Lubinsky}, ZBLNUMBER = {1032.42028}, }
[Rab5] $Go to document$ V. S. Rabinovich, "Operator-valued discrete convolutions and some of their applications," Mat. Zametki, vol. 51, iss. 5, pp. 90-101, 158, 1992.

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@article {Rab5, MRKEY = {1186537}, AUTHOR = {Rabinovich, V. S.}, TITLE = {Operator-valued discrete convolutions and some of their applications}, 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}, MRREVIEWER = {A. B{ö}ttcher}, DOI = {10.1007/BF01262183}, }
[Rakh77] E. A. Rakhmanov, "On the asymptotics of the ratio of orthogonal polynomials," Math. USSR Sb., vol. 32, pp. 199-213, 1977.

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@article{Rakh77, author={Rakhmanov, E. A.}, TITLE={On the asymptotics of the ratio of orthogonal polynomials}, JOURNAL={Math. {\rm USSR} Sb.}, VOLUME={32}, YEAR={1977}, PAGES={199--213}, ZBLNUMBER={0401.30033}, }
[Rakh83] E. A. Rakhmanov, "On the asymptotics of the ratio of orthogonal polynomials, II," Math. USSR Sb., vol. 46, pp. 105-117, 1983.

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@article{Rakh83, author={Rakhmanov, E. A.}, TITLE={On the asymptotics of the ratio of orthogonal polynomials, {\rm II}}, JOURNAL={Math. {\rm USSR} Sb.}, VOLUME={46}, YEAR={1983}, PAGES={105--117}, ZBLNUMBER={0515.30030}, }
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@article {RT, MRKEY = {967643}, AUTHOR = {Ralston, James and Trubowitz, Eugene}, TITLE = {Isospectral sets for boundary value problems on the unit interval}, JOURNAL = {Ergodic Theory Dynam. Systems}, FJOURNAL = {Ergodic Theory and Dynamical Systems}, VOLUME = {8$\sp *$}, YEAR = {1988}, NUMBER = {Charles Conley Memorial Issue}, PAGES = {301--358}, ISSN = {0143-3857}, MRCLASS = {34B25 (34A55 58F19 58G25)}, MRNUMBER = {89m:34035}, MRREVIEWER = {K. S. Harinath}, ZBLNUMBER={0678.34025}, }
[RS4] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, New York: Academic Press, 1978.

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@book {RS4, MRKEY = {0493421}, AUTHOR = {Reed, Michael and Simon, Barry}, TITLE = {Methods of Modern Mathematical Physics. {\rm IV}. {A}nalysis of Operators}, PUBLISHER = {Academic Press}, ADDRESS = {New York}, YEAR = {1978}, PAGES = {xv+396}, ISBN = {0-12-585004-2}, MRCLASS = {47-02 (81.47)}, MRNUMBER = {58 \#12429c}, MRREVIEWER = {P. R. Chernoff}, ZBLNUMBER = {0401.47001}, }
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@misc{R07, author={Remling, C.}, TITLE={The absolutely continuous spectrum of Jacobi matrices}, ARXIV={0706.1101}, }
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@incollection {Rod90, MRKEY = {1100301}, AUTHOR = {Rodman, Leiba}, TITLE = {Orthogonal matrix polynomials}, BOOKTITLE = {Orthogonal Polynomials ({C}olumbus, {OH}, 1989)}, SERIES = {NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.}, NUMBER = {294}, PAGES = {345--362}, PUBLISHER = {Kluwer Acad. Publ.}, ADDRESS = {Dordrecht}, YEAR = {1990}, MRCLASS = {42C05}, MRNUMBER = {93b:42039}, ZBLNUMBER = {0703.42022}, }
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@article {Ryck, MRKEY = {2291796}, AUTHOR = {Ryckman, E.}, TITLE = {A strong {S}zegő theorem for {J}acobi matrices}, JOURNAL = {Comm. Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}, VOLUME = {271}, YEAR = {2007}, NUMBER = {3}, PAGES = {791--820}, ISSN = {0010-3616}, CODEN = {CMPHAY}, MRCLASS = {47B36 (39A70 42C05 47A10)}, MRNUMBER = {2008e:47081a}, MRREVIEWER = {Boris Petrovich Osilenker}, ZBLNUMBER={1133.47025}, }
[Sak92] A. L. Sakhnovich, "Spectral functions of a second-order canonical system," Mat. Sb., vol. 181, iss. 11, pp. 1510-1524, 1990.

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@article {Sak92, MRKEY = {1090913}, AUTHOR = {Sakhnovich, A. L.}, TITLE = {Spectral functions of a second-order canonical system}, JOURNAL = {Mat. Sb.}, FJOURNAL = {MatematicheskiĭSbornik}, VOLUME = {181}, YEAR = {1990}, NUMBER = {11}, PAGES = {1510--1524}, ISSN = {0368-8666}, MRCLASS = {47E05 (34L05 47A70)}, MRNUMBER = {92c:47056}, MRREVIEWER = {Bj{ö}rn Textorius}, NOTE = {in Russian; translated in {\it Math. {\rm USSR} Sb.} {\bf 71} (1992), 355--369.}, }
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@article {SF, MRKEY = {2013964}, AUTHOR = {Sebbar, Ahmed and Falliero, Th{é}r{è}se}, TITLE = {Capacities and {J}acobi matrices}, JOURNAL = {Proc. Edinb. Math. Soc.}, FJOURNAL = {Proceedings of the Edinburgh Mathematical Society. Series II}, VOLUME = {46}, YEAR = {2003}, NUMBER = {3}, PAGES = {719--745}, ISSN = {0013-0915}, MRCLASS = {31B15 (30C20 39A70 47B36)}, MRNUMBER = {2005e:31007}, MRREVIEWER = {Dimitrios Betsakos}, DOI = {10.1017/S0013091502001141}, ZBLNUMBER = {1041.31003}, }
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@article{Sh, author={Shohat, J. A.}, TITLE={Théorie générale des polinomes orthogonaux de Tchebichef}, JOURNAL={Mémorial des Sciences Mathématiques}, VOLUME={66}, PAGES={1--69}, YEAR={1934}, }
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@article {S168, MRKEY = {709464}, AUTHOR = {Simon, Barry}, TITLE = {Kotani theory for one-dimensional stochastic {J}acobi matrices}, JOURNAL = {Comm. Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}, VOLUME = {89}, YEAR = {1983}, NUMBER = {2}, PAGES = {227--234}, ISSN = {0010-3616}, CODEN = {CMPHAY}, MRCLASS = {60H25 (35P99 47B99 58F19 82A57)}, MRNUMBER = {85d:60122}, MRREVIEWER = {Ren{é} Carmona}, URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103922682}, ZBLNUMBER = {0534.60057}, }
[Sim288] $Go to document$ 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.

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@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.}, FJOURNAL = {Journal of Functional Analysis}, VOLUME = {214}, YEAR = {2004}, NUMBER = {2}, PAGES = {396--409}, ISSN = {0022-1236}, CODEN = {JFUAAW}, MRCLASS = {47B36 (30D30 30D50 39A70)}, MRNUMBER = {2005j:47033}, MRREVIEWER = {Jan Janas}, DOI = {10.1016/j.jfa.2003.11.006}, ZBLNUMBER = {1064.30030}, }
[Sim303] $Go to document$ B. Simon, "Meromorphic Szegő functions and asymptotic series for Verblunsky coefficients," Acta Math., vol. 195, pp. 267-285, 2005.

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@article {Sim303, MRKEY = {2233692}, AUTHOR = {Simon, Barry}, TITLE = {Meromorphic {S}zegő functions and asymptotic series for {V}erblunsky coefficients}, JOURNAL = {Acta Math.}, FJOURNAL = {Acta Mathematica}, VOLUME = {195}, YEAR = {2005}, PAGES = {267--285}, ISSN = {0001-5962}, CODEN = {ACMAA8}, MRCLASS = {30C10}, MRNUMBER = {2007c:30007}, MRREVIEWER = {D. H. Hamilton}, DOI = {10.1007/BF02588083}, ZBLNUMBER={1117.42005}, }
[OPUC1] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1; Classical Theory, Providence, RI: Amer. Math. Soc., 2005.

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@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)}, MRNUMBER = {2006a:42002a}, MRREVIEWER = {P. L. Duren}, ZBLNUMBER = {1082.42020}, }
[OPUC2] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2: Spectral theory, Providence, RI: Amer. Math. Soc., 2005.

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@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}, MRCLASS = {42-02 (30C85 33C45 42C05 47B36 47N50)}, MRNUMBER = {2006a:42002b}, MRREVIEWER = {P. L. Duren}, ZBLNUMBER = {1082.42021}, }
[Sim_Sturm] B. Simon, "Sturm oscillation and comparison theorems," in Sturm-Liouville Theory, Basel: Birkhäuser, 2005, pp. 29-43.

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@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] $Go to document$ 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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@article {Jost3, MRKEY = {2345042}, AUTHOR = {Simon, Barry}, TITLE = {Meromorphic {J}ost functions and asymptotic expansions for {J}acobi parameters}, 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}, MRREVIEWER = {Stanislav Kupin}, DOI = {10.1007/s10688-007-0013-z}, ZBLNUMBER={pre05206865}, }
[SimonNew] B. Simon, Szegő’s Theorem and Its Descendants : Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials, Princeton, NJ: Princeton Univ. Press, 2010.

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@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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@article {SZ, MRKEY = {2020274}, 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)}, MRNUMBER = {2005f:42057}, MRREVIEWER = {Franz Peherstorfer}, ZBLNUMBER={1046.42017}, }
[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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@article {SY, MRKEY = {1674798}, AUTHOR = {Sodin, Mikhail and Yuditskii, Peter}, TITLE = {Almost periodic {J}acobi matrices with homogeneous spectrum, infinite-dimensional {J}acobi inversion, and {H}ardy spaces of character-automorphic functions}, JOURNAL = {J. Geom. Anal.}, FJOURNAL = {The Journal of Geometric Analysis}, VOLUME = {7}, YEAR = {1997}, NUMBER = {3}, PAGES = {387--435}, ISSN = {1050-6926}, MRCLASS = {47B36 (30F35 37K10 37K20 46E15 46E22 47A10)}, MRNUMBER = {2000k:47033}, MRREVIEWER = {Vladimir S. Matveev}, ZBLNUMBER = {1041.47502}, }
[Sz15] $Go to document$ 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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@article {Sz15, MRKEY = {1511838}, AUTHOR = {Szegő, G.}, TITLE = {Ein {G}renzwertsatz über die {T}oeplitzschen {D}eterminanten einer reellen positiven {F}unktion}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {76}, YEAR = {1915}, NUMBER = {4}, PAGES = {490--503}, ISSN = {0025-5831}, CODEN = {MAANA}, MRCLASS = {Contributed Item}, MRNUMBER = {1511838}, DOI = {10.1007/BF01458220}, JFMNUMBER={45.0518.02}, }
[Sz20] $Go to document$ G. SzegHo, "Beiträge zur Theorie der Toeplitzschen Formen," Math. Z., vol. 6, iss. 3-4, pp. 167-202, 1920.

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@article {Sz20, MRKEY = {1544404}, AUTHOR = {Szegő, G.}, TITLE = {Beiträge zur {T}heorie der {T}oeplitzschen {F}ormen}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {6}, YEAR = {1920}, NUMBER = {3-4}, PAGES = {167--202}, ISSN = {0025-5874}, CODEN = {MAZEAX}, MRCLASS = {Contributed Item}, MRNUMBER = {1544404}, DOI = {10.1007/BF01199955}, JFMNUMBER={48.0376.03}, }
[Teschl] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Providence, RI: Amer. Math. Soc., 2000.

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@book {Teschl, MRKEY = {1711536}, AUTHOR = {Teschl, Gerald}, TITLE = {Jacobi Operators and Completely Integrable Nonlinear Lattices}, SERIES = {Mathematical Surveys and Monographs}, NUMBER = {72}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2000}, PAGES = {xvii+351}, ISBN = {0-8218-1940-2}, MRCLASS = {39A70 (35Q58 37K60 39A12 47B36 82B23)}, MRNUMBER = {2001b:39019}, MRREVIEWER = {Malcolm Adams}, ZBLNUMBER = {1056.39029}, }
[Toda] M. Toda, Theory of Nonlinear Lattices, Second ed., New York: Springer-Verlag, 1989.

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@book {Toda, MRKEY = {971987}, AUTHOR = {Toda, Morikazu}, TITLE = {Theory of Nonlinear Lattices}, SERIES = {Springer Series in Solid-State Sciences}, NUMBER = {20}, EDITION = {Second}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1989}, PAGES = {x+225}, ISBN = {3-540-18327-2}, MRCLASS = {58F07 (70F99 70K99 82A99)}, MRNUMBER = {89h:58082}, ZBLNUMBER = {0694.70001}, }
[newVA] $Go to document$ 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.

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@article {newVA, MRKEY = {2328181}, AUTHOR = {Van Assche, Walter}, TITLE = {Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle}, JOURNAL = {J. Approx. Theory}, FJOURNAL = {Journal of Approximation Theory}, VOLUME = {146}, YEAR = {2007}, NUMBER = {2}, PAGES = {227--242}, ISSN = {0021-9045}, CODEN = {JAXTAZ}, MRCLASS = {33C45 (33C47 42C05)}, MRNUMBER = {2008e:33020}, MRREVIEWER = {Alicia Cachafeiro}, DOI = {10.1016/j.jat.2006.11.002}, ZBLNUMBER = {1136.33005}, }
[vMoer] $Go to document$ P. van Moerbeke, "The spectrum of Jacobi matrices," Invent. Math., vol. 37, iss. 1, pp. 45-81, 1976.

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@article {vMoer, MRKEY = {0650253}, AUTHOR = {van Moerbeke, Pierre}, TITLE = {The spectrum of {J}acobi matrices}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {37}, YEAR = {1976}, NUMBER = {1}, PAGES = {45--81}, ISSN = {0020-9910}, MRCLASS = {58F05 (14K99 34B25 35Q99)}, MRNUMBER = {58 \#31226}, MRREVIEWER = {J. S. Joel}, DOI = {10.1007/BF01418827}, ZBLNUMBER = {0361.15010}, }
[Wat] D. S. Watkins, Fundamentals of Matrix Computations, second ed., New York: Wiley-Interscience, 2002.

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@book {Wat, MRKEY = {1899577}, AUTHOR = {Watkins, David S.}, TITLE = {Fundamentals of Matrix Computations, {\rm second ed.}}, SERIES = {Pure and Applied Mathematics}, PUBLISHER = {Wiley-Interscience}, ADDRESS = {New York}, YEAR = {2002}, PAGES = {xiv+618}, ISBN = {0-471-21394-2}, MRCLASS = {65-01 (65Fxx)}, MRNUMBER = {2003a:65002}, ZBLNUMBER = {1005.65027}, }
[Wei87] J. Weidmann, Spectral Theory of Ordinary Differential Operators, New York: Springer-Verlag, 1987.

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@book {Wei87, MRKEY = {923320}, AUTHOR = {Weidmann, Joachim}, TITLE = {Spectral Theory of Ordinary Differential Operators}, SERIES = {Lecture Notes in Mathematics}, NUMBER = {1258}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1987}, PAGES = {vi+303}, ISBN = {3-540-17902-X}, MRCLASS = {47E05 (34-02 34B25)}, MRNUMBER = {89b:47070}, MRREVIEWER = {Tuncay Aktosun}, ZBLNUMBER = {0647.47052}, }
[Weyl] H. Weyl, "Über beschränkte quadratische Formen, deren Differenz vollstetig ist," Rend. Circ. Mat. Palermo, vol. 27, pp. 373-392, 1909.

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@article{Weyl, author={Weyl, H.}, TITLE={Über beschränkte quadratische Formen, deren Differenz vollstetig ist}, JOURNAL={Rend. Circ. Mat. Palermo}, VOLUME={27}, YEAR={1909}, PAGES={373--392}, JFMNUMBER={40.0395.01}, }
[YM2001] 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.

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@incollection {YM2001, MRKEY = {1842902}, AUTHOR = {Yakhlef, Hossain O. and Marcell{á}n, Francisco}, TITLE = {Orthogonal matrix polynomials, connection between recurrences on the unit circle and on a finite interval}, BOOKTITLE = {Approximation, Optimization and Mathematical Economics ({P}ointe-à-{P}itre, 1999)}, PAGES = {369--382}, PUBLISHER = {Physica}, ADDRESS = {Heidelberg}, YEAR = {2001}, MRCLASS = {30C10}, MRNUMBER = {2002e:30005}, MRREVIEWER = {Stephen M. Zemyan}, ZBLNUMBER = {0985.15020}, }
[YMP01] $Go to document$ 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 {YMP01, MRKEY = {1840018}, AUTHOR = {Yakhlef, Hossain O. and Marcell{á}n, Francisco and Pi{ñ}ar, Miguel A.}, TITLE = {Relative asymptotics for orthogonal matrix polynomials with convergent recurrence coefficients}, JOURNAL = {J. Approx. Theory}, FJOURNAL = {Journal of Approximation Theory}, VOLUME = {111}, YEAR = {2001}, NUMBER = {1}, PAGES = {1--30}, ISSN = {0021-9045}, CODEN = {JAXTAZ}, MRCLASS = {41A60 (42C05)}, MRNUMBER = {2002d:41044}, MRREVIEWER = {F. M{ó}ricz}, DOI = {10.1006/jath.2001.3557}, ZBLNUMBER = {1005.42014}, }
[YK] $Go to document$ 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.

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@article {YK, MRKEY = {0469461}, AUTHOR = {Youla, Dante C. and Kazanjian, Nerses N.}, TITLE = {Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle}, JOURNAL = {{\rm IEEE} Trans. Circuits and Systems}, FJOURNAL = {Institute of Electrical and Electronics Engineers. Transactions on Circuits and Systems}, VOLUME = {CAS-25}, YEAR = {1978}, NUMBER = {2}, PAGES = {57--69}, ISSN = {0098-4094}, MRCLASS = {93E10 (60G25)}, MRNUMBER = {57 \#9251}, MRREVIEWER = {Bradley W. Dickinson}, DOI = {10.1109/TCS.1978.1084443}, ZBLNUMBER={0417.93018}, }

Perturbations of orthogonal polynomials with periodic recursion coefficients

Abstract

Authors