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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.


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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
\by S.~P.~Suetin
\paper Uniform convergence of Pad\'e diagonal approximants for hyperelliptic functions
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 9
\pages 81--114
\jour Sb. Math.
\yr 2000
\vol 191
\issue 9
\pages 1339--1373

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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
    3. V. N. Sorokin, “Cyclic graphs and Apéry's theorem”, Russian Math. Surveys, 57:3 (2002), 535–571  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. S. P. Suetin, “On Dumas' theorem in the theory of continued fractions”, Russian Math. Surveys, 57:5 (2002), 1010–1012  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. A. I. Aptekarev, “Sharp constants for rational approximations of analytic functions”, Sb. Math., 193:1 (2002), 1–72  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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
    9. V. I. Buslaev, S. F. Buslaeva, “On the Rogers–Ramanujan Periodic Continued Fraction”, Math. Notes, 74:6 (2003), 783–793  mathnet  crossref  crossref  mathscinet  zmath  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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    14. A. A. Gonchar, S. P. Suetin, “On Padé Approximants of Meromorphic Functions of Markov Type”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S58–S95  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. S. P. Suetin, “On interpolation properties of diagonal Padé approximants of elliptic functions”, Russian Math. Surveys, 59:4 (2004), 800–802  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. Aptekarev A.I., Van Assche W., “Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight”, J. Approx. Theory, 129:2 (2004), 129–166  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
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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
    26. D. V. Khristoforov, “On the Phenomenon of Spurious Interpolation of Elliptic Functions by Diagonal Padé Approximants”, Math. Notes, 87:4 (2010), 564–574  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    27. Baratchart L., Yattselev M., “Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights”, International Mathematics Research Notices, 2010, no. 22, 4211–4275  mathscinet  zmath  isi  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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    33. A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, “Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets”, Russian Math. Surveys, 66:6 (2011), 1015–1048  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    41. Yattselev M.L., “Nuttall's Theorem With Analytic Weights on Algebraic S-Contours”, J. Approx. Theory, 190:SI (2015), 73–90  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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    45. Yattselev M.L., “Symmetric Contours and Convergent Interpolation”, J. Approx. Theory, 225 (2018), 76–105  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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