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Mat. Sb. (N.S.), 1984, Volume 125(167), Number 2(10), Pages 231–258 (Mi msb2080)  

This article is cited in 73 scientific papers (total in 74 papers)

Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices

A. I. Aptekarev

Abstract: An expression in the form of a Riemann theta-function is obtained for the asymptotic behavior of polynomials orthogonal on a system of contours. Properties of a limit-periodic discrete Sturm–Liouville operator and the dynamics of a periodic Toda lattice are considered as a consequence of the asymptotic formulas obtained.
Figures: 2.
Bibliography: 21 titles.

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English version:
Mathematics of the USSR-Sbornik, 1986, 53:1, 233–260

Bibliographic databases:

UDC: 517.51+517.9+530.145
MSC: Primary 30E15, 34C25, 42C05; Secondary 14K25, 34A34, 34B25
Received: 16.06.1983

Citation: A. I. Aptekarev, “Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices”, Mat. Sb. (N.S.), 125(167):2(10) (1984), 231–258; Math. USSR-Sb., 53:1 (1986), 233–260

Citation in format AMSBIB
\by A.~I.~Aptekarev
\paper Asymptotic properties of polynomials orthogonal on a~system of contours, and periodic motions of Toda lattices
\jour Mat. Sb. (N.S.)
\yr 1984
\vol 125(167)
\issue 2(10)
\pages 231--258
\jour Math. USSR-Sb.
\yr 1986
\vol 53
\issue 1
\pages 233--260

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    1. Nevai P., “Freud,Geza, Orthogonal Polynomials and Christoffel Functions - a Case-Study”, J. Approx. Theory, 48:1 (1986), 3–167  crossref  mathscinet  zmath  isi
    2. Lubinsky D., “A Survey of General Orthogonal Polynomials for Weights on Finite and Infinite Intervals”, Acta Appl. Math., 10:3 (1987), 237–296  mathscinet  zmath  isi
    3. Franz Peherstorfer, “On Bernstein-Szegö orthogonal polynomials on several intervals. II. Orthogonal polynomials with periodic recurrence coefficients”, Journal of Approximation Theory, 64:2 (1991), 123  crossref  mathscinet  zmath
    4. J.S Geronimo, W Van Assche, “Approximating the weight function for orthogonal polynomials on several intervals”, Journal of Approximation Theory, 65:3 (1991), 341  crossref  mathscinet  zmath
    5. Franz Peherstorfer, “Orthogonal and extremal polynomials on several intervals”, Journal of Computational and Applied Mathematics, 48:1-2 (1993), 187  crossref  mathscinet  zmath
    6. John Zhang, “Relative growth of linear iterations and orthogonal polynomials on several intervals”, Linear Algebra and its Applications, 186 (1993), 97  crossref  mathscinet  zmath
    7. Vanassche W., “Christoffel Functions and Turan Determinants on Several Intervals”, J. Comput. Appl. Math., 48:1-2 (1993), 207–223  crossref  mathscinet  isi
    8. A. I. Aptekarev, J. S. Dehesa, R. J. Yáñez, “Spatial entropy of central potentials and strong asymptotics of orthogonal polynomials”, J Math Phys (N Y ), 35:9 (1994), 4423  crossref  zmath  adsnasa
    9. D. Barrios, G. L. Lopes, E. Torrano, “Polynomials generated by a three-term recurrence relation with asymptotically periodic complex coefficients”, Sb. Math., 186:5 (1995), 629–659  mathnet  crossref  mathscinet  zmath  isi
    10. Alphonse P. Magnus, “Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials”, Journal of Computational and Applied Mathematics, 57:1-2 (1995), 215  crossref  mathscinet  zmath
    11. Aptekarev A., Kaliaguine V., Vanassche W., “Criterion for the Resolvent Set of Nonsymmetric Tridiagonal Operators”, Proc. Amer. Math. Soc., 123:8 (1995), 2423–2430  crossref  mathscinet  zmath  isi
    12. A.I. Aptekarev, A. Branquinho, F. Marcellán, “Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation”, Journal of Computational and Applied Mathematics, 78:1 (1997), 139  crossref  mathscinet  zmath
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    14. A. L. Lukashov, “Rational interpolation processes on two intervals”, Russian Math. (Iz. VUZ), 42:5 (1998), 33–40  mathnet  mathscinet  elib
    15. A.L. Lukashov, “On Chebyshev–Markov Rational Functions over Several Intervals”, Journal of Approximation Theory, 95:3 (1998), 333  crossref  mathscinet  zmath
    16. S. P. Suetin, “Uniform convergence of Padé diagonal approximants for hyperelliptic functions”, Sb. Math., 191:9 (2000), 1339–1373  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. A.Almendral Vázquez, “The Spectrum of a Periodic Complex Jacobi Matrix Revisited”, Journal of Approximation Theory, 105:2 (2000), 344  crossref  mathscinet  zmath
    18. Franz Peherstorfer, Robert Steinbauer, “Orthogonal Polynomials on the Circumference and Arcs of the Circumference”, Journal of Approximation Theory, 102:1 (2000), 96  crossref  mathscinet  zmath
    19. Lebedev V., “On the Solution of Inverse Problems and Trigonometric Forms for the Geronimus Polynomials. Application to the Theory of Iterative Methods”, Russ. J. Numer. Anal. Math. Model, 15:1 (2000), 73–93  crossref  mathscinet  zmath  isi
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    21. Franz Peherstorfer, “Deformation of Minimal Polynomials and Approximation of Several Intervals by an Inverse Polynomial Mapping”, Journal of Approximation Theory, 111:2 (2001), 180  crossref  mathscinet  zmath
    22. Mori M., Sugihara M., “The Double-Exponential Transformation in Numerical Analysis”, J. Comput. Appl. Math., 127:1-2, SI (2001), 287–296  crossref  mathscinet  zmath  adsnasa  isi
    23. Totik V., “Polynomial Inverse Images and Polynomial Inequalities”, Acta Math., 187:1 (2001), 139–160  crossref  mathscinet  zmath  isi
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    26. S. P. Suetin, “Approximation properties of the poles of diagonal Padé approximants for certain generalizations of Markov functions”, Sb. Math., 193:12 (2002), 1837–1866  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    27. S. P. Suetin, “Padé approximants and efficient analytic continuation of a power series”, Russian Math. Surveys, 57:1 (2002), 43–141  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. Lukashov, AL, “Automorphic orthogonal and extremal polynomials”, Canadian Journal of Mathematics-Journal Canadien de Mathematiques, 55:3 (2003), 576  crossref  mathscinet  zmath  isi  elib
    29. Peherstorfer, F, “Zeros of polynomials orthogonal on several intervals”, International Mathematics Research Notices, 2003, no. 7, 361  crossref  mathscinet  zmath  isi  elib
    30. A. L. Lukashov, “Inequalities for derivatives of rational functions on several intervals”, Izv. Math., 68:3 (2004), 543–565  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    33. Lukashov, AL, “Zeros of polynomials orthogonal on two arcs of the unit circle”, Journal of Approximation Theory, 132:1 (2005), 42  crossref  mathscinet  zmath  isi  elib
    34. Totik V., “Equilibrium measures and polynomials”, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, 501–514  mathscinet  zmath  isi
    35. S. P. Suetin, “Spectral properties of a class of discrete Sturm–Liouville operators”, Russian Math. Surveys, 61:2 (2006), 365–367  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    36. S. P. Suetin, “Comparative Asymptotic Behavior of Solutions and Trace Formulas for a Class of Difference Equations”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S96–S137  mathnet  crossref  crossref  mathscinet  zmath  isi
    37. Pastur, L, “From random matrices to quasi-periodic Jacobi matrices via orthogonal polynomials”, Journal of Approximation Theory, 139:1–2 (2006), 269  crossref  mathscinet  zmath  isi
    38. S. P. Suetin, “Trace formulae for a class of Jacobi operators”, Sb. Math., 198:6 (2007), 857–885  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    39. A. L. Lukashov, “Estimates for derivatives of rational functions and the fourth Zolotarev problem”, St. Petersburg Math. J., 19:2 (2008), 253–259  mathnet  crossref  mathscinet  zmath  isi
    40. Golinskii L., Totik V., “Orthogonal polynomials: from Jacobi to Simon”, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon's 60th birthday, Proc. Sympos. Pure Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007, 821–874  crossref  mathscinet  zmath  isi
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    42. Pastur L., “Orthogonal polynomials, Jacobi matrices and random matrices”, Integrable Systems and Random Matrices: in Honor of Percy Deift, Contemporary Mathematics Series, 458, 2008, 249–263  crossref  mathscinet  zmath  isi
    43. Hundertmark D., Simon B., “Eigenvalue Bounds in the Gaps of Schrodinger Operators and Jacobi Matrices”, J. Math. Anal. Appl., 340:2 (2008), 892–900  crossref  mathscinet  zmath  isi
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    49. Totik, V, “Chebyshev constants and the inheritance problem”, Journal of Approximation Theory, 160:1–2 (2009), 187  crossref  mathscinet  zmath  isi
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    52. Mina-Diaz E., “An Expansion for Polynomials Orthogonal Over an Analytic Jordan Curve”, Commun. Math. Phys., 285:3 (2009), 1109–1128  crossref  mathscinet  zmath  adsnasa  isi  elib
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    56. Simon B., “Fine Structure of the Zeros of Orthogonal Polynomials: a Progress Report”, Recent Trends in Orthogonal Polynomials and Approximation Theory, Contemporary Mathematics, 507, eds. Arvesu J., Marcellan F., MartinezFinkelshtein A., Amer Mathematical Soc, 2010, 241–254  crossref  mathscinet  zmath  isi
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    59. Eremenko A., Yuditskii P., “Comb Functions”, Recent Advances in Orthogonal Polynomials, Special Functions, and their Applications, Contemporary Mathematics, 578, eds. Arvesu J., Lagomasino G., Amer Mathematical Soc, 2011, 99–118  crossref  mathscinet  isi
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    61. A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight”, Trans. Moscow Math. Soc., 73 (2012), 139–159  mathnet  crossref  mathscinet  zmath  elib
    62. Mirta M. Castro, F. Alberto Grünbaum, “On a Seminal Paper by Karlin and McGregor”, SIGMA, 9 (2013), 020, 11 pp.  mathnet  crossref  mathscinet
    63. I. E. Egorova, L. A. Pastur, “On asymptotic properties of polynomials orthogonal with respect to varying weights and related topics of spectral theory”, St. Petersburg Math. J., 25:2 (2014), 223–240  mathnet  crossref  mathscinet  zmath  isi  elib
    64. Helge Krüger, Barry Simon, “Cantor polynomials and some related classes of OPRL”, Journal of Approximation Theory, 2014  crossref  mathscinet
    65. Moale I., Yuditskii P., “Spectral Sets of Periodic Matrices Related To the Strong Moment Problem”, J. Spectr. Theory, 4:1 (2014), 23–52  crossref  mathscinet  zmath  isi
    66. Alexey Lukashov, Dmitri Prokhorov, “Approximation of sgn
      ( x ) on Two Symmetric Intervals by Rational Functions with Fixed Poles”, Comput. Methods Funct. Theory, 2015  crossref  mathscinet
    67. Journal of Computational and Applied Mathematics, 2015  crossref
    68. V. V. Borzov, E. V. Damaskinsky, “The discrete spectrum of Jacobi matrix related to recurrence relations with periodic coefficients”, J. Math. Sci. (N. Y.), 213:5 (2016), 694–705  mathnet  crossref  mathscinet
    69. V. M. Buchstaber, V. N. Dubinin, V. A. Kaliaguine, B. S. Kashin, V. N. Sorokin, S. P. Suetin, D. N. Tulyakov, B. N. Chetverushkin, E. M. Chirka, A. A. Shkalikov, “Alexander Ivanovich Aptekarev (on his 60th birthday)”, Russian Math. Surveys, 70:5 (2015), 965–973  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    70. Benjamin Eichinger, “Periodic GMP Matrices”, SIGMA, 12 (2016), 066, 19 pp.  mathnet  crossref
    71. A. B. Bogatyrev, O. A. Grigoriev, “Closed Formula for the Capacity of Several Aligned Segments”, Proc. Steklov Inst. Math., 298 (2017), 60–67  mathnet  crossref  crossref  isi  elib
    72. B. Eichinger, P. Yuditskii, “Ahlfors problem for polynomials”, Sb. Math., 209:3 (2018), 320–351  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    73. A. A. Kononova, “On measures generating orthogonal polynomials with similar asymptotic behavior of the ratio at infinity”, Ufa Math. J., 10:1 (2018), 64–75  mathnet  crossref  isi  elib
    74. R. Ch. Kulaev, A. B. Shabat, “Conservation laws for Volterra chain with initial step-like condition”, Ufa Math. J., 11:1 (2019), 63–69  mathnet  crossref  isi
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