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Mat. Sb., 2000, Volume 191, Number 9, Pages 81–114 (Mi msb508)  

This article is cited in 45 scientific papers (total in 45 papers)

Uniform convergence of Padé diagonal approximants for hyperelliptic functions

S. P. Suetin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The uniform convergence of Padé diagonal approximants is studied for functions in some class that is a natural generalization of hyperelliptic functions. The study is based on Nuttall's approach, which consists in the analysis of a certain Riemann boundary-value problem on the corresponding hyperelliptic Riemann surface. In terms of the solution of this problem, a strong asymptotic formula is obtained for non-Hermitian orthogonal polynomials that are the denominators of the Padé approximants. Under some fairly general assumptions, which are formulated in terms of the periods of the complex Green's function corresponding to the problem and which hold in “general position”, a version of the Baker–Gammel–Willes conjecture is proved.

DOI: https://doi.org/10.4213/sm508

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English version:
Sbornik: Mathematics, 2000, 191:9, 1339–1373

Bibliographic databases:

UDC: 517.53
MSC: Primary 41A21, 41A25, 41A27; Secondary 30F35
Received: 28.10.1999 and 14.06.2000

Citation: S. P. Suetin, “Uniform convergence of Padé diagonal approximants for hyperelliptic functions”, Mat. Sb., 191:9 (2000), 81–114; Sb. Math., 191:9 (2000), 1339–1373

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. 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
    2. S. P. Suetin, “On the dynamics of “wandering” zeros of polynomials that are orthogonal on certain intervals”, Russian Math. Surveys, 57:2 (2002), 425–427  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    6. 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
    7. V. I. Buslaev, “On the Baker–Gammel–Wills conjecture in the theory of Padé approximants”, Sb. Math., 193:6 (2002), 811–823  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. S. P. Suetin, “The asymptotic behaviour of diagonal Padé approximants for hyperelliptic functions of genus $g=2$”, Russian Math. Surveys, 58:4 (2003), 802–804  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    11. Lukashov A.L., Peherstorfer F., “Automorphic orthogonal and extremal polynomials”, Canad. J. Math., 55:3 (2003), 576–608  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    12. Peherstorfer F., “Zeros of polynomials orthogonal on several intervals”, Int. Math. Res. Not., 2003, no. 7, 361–385  crossref  mathscinet  zmath  isi
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    19. Aptekarev A., Cachafeiro A., Marcellán F., “A scalar Riemann boundary value problem approach to orthogonal polynomials on the circle”, J. Approx. Theory, 141:2 (2006), 174–181  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    20. 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 - ERGODIC SCHRODINGER OPERATORS, SINGULAR SPECTRUM, ORTHOGONAL POLYNOMIALS, AND INVERSE SPECTRAL THEORY, Proceedings of Symposia in Pure Mathematics, 76, no. 2, 2007, 821–874  crossref  mathscinet  zmath  adsnasa  isi
    21. A. I. Aptekarev, “Matrix Riemann–Hilbert analysis for the case of higher genus — asymptotics of polynomials orthogonal on a system of intervals”, Preprinty IPM im. M. V. Keldysha, 2008, 028, 23 pp.  mathnet
    22. L. A. Knizhnerman, “Gauss–Arnoldi quadrature for $\langle(zI-A)^{-1}\varphi,\varphi\rangle$ and rational Padé-type approximation for Markov-type functions”, Sb. Math., 199:2 (2008), 185–206  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    23. Baratchart L., Yattselev M., “Convergent interpolation to cauchy integrals over analytic arcs”, Found. Comput. Math., 9:6 (2009), 675–715  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    24. Baratchart L., Yattselev M., “Multipoint Padé approximants to complex Cauchy transforms with polar singularities”, J. Approx. Theory, 156:2 (2009), 187–211  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    25. D. V. Khristoforov, “On uniform approximation of elliptic functions by Padé approximants”, Sb. Math., 200:6 (2009), 923–941  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    30. Martinez-Finkelshtein A. Rakhmanov E.A. Suetin S.P., “Heine, Hilbert, Pade, Riemann, and Stieltjes: John Nuttall's Work 25 Years Later”, Recent Advances in Orthogonal Polynomials, Special Functions, and their Applications, Contemporary Mathematics, 578, ed. Arvesu J. Lagomasino G., Amer Mathematical Soc, 2011, 165–193  crossref  mathscinet  isi
    31. Yu. A. Labych, A. P. Starovoitov, “Priblizhenie nepreryvnykh funktsii ratsionalnymi drobyami Pade–Chebysheva”, PFMT, 2011, no. 1(6), 69–78  mathnet
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