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Mat. Sb., 2002, Volume 193, Number 6, Pages 25–38 (Mi msb658)  

This article is cited in 27 scientific papers (total in 27 papers)

On the Baker–Gammel–Wills conjecture in the theory of Padé approximants

V. I. Buslaev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The well-known Padé conjecture, which was formulated in 1961 by Baker, Gammel, and Wills states that for each meromorphic function $f$ in the unit disc $D$ there exists a subsequence of its diagonal Padé approximants converging to $f$ uniformly on all compact subsets of $D$ not containing the poles of $f$. In 2001, Lubinsky found a meromorphic function in $D$ disproving Padé's conjecture.
The function presented in this article disproves the holomorphic version of Padé's conjecture and simultaneously disproves Stahl's conjecture (Padé's conjecture for algebraic functions).


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English version:
Sbornik: Mathematics, 2002, 193:6, 811–823

Bibliographic databases:

UDC: 517.524
MSC: 41A21, 30E10
Received: 24.12.2001

Citation: V. I. Buslaev, “On the Baker–Gammel–Wills conjecture in the theory of Padé approximants”, Mat. Sb., 193:6 (2002), 25–38; Sb. Math., 193:6 (2002), 811–823

Citation in format AMSBIB
\by V.~I.~Buslaev
\paper On the Baker--Gammel--Wills conjecture in the~theory of Pad\'e approximants
\jour Mat. Sb.
\yr 2002
\vol 193
\issue 6
\pages 25--38
\jour Sb. Math.
\yr 2002
\vol 193
\issue 6
\pages 811--823

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    This publication is cited in the following articles:
    1. Lubinsky D.S., “On Baker'S Patchwork Conjecture For Diagonal Pade Approximants”, Constr. Approx.  crossref  isi
    2. 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
    3. Baker G.A. (Jr.), “Some structural properties of two counter-examples to the Baker-Gammel-Wills conjecture”, J. Comput. Appl. Math., 161:2 (2003), 371–391  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    4. 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
    5. 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
    6. V. I. Buslaev, “Convergence of the Rogers–Ramanujan continued fraction”, Sb. Math., 194:6 (2003), 833–856  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. S. L. Skorokhodov, “Padé approximation and numerical analysis for the Riemann $\zeta$-function”, Comput. Math. Math. Phys., 43:9 (2003), 1277–1298  mathnet  mathscinet  zmath
    8. 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
    9. S. L. Skorokhodov, “Methods of analytical continuation of the generalized hypergeometric functions $ _pF_{p-1}(a_1,…,a_p;b_1,…,b_{p-1};z)$”, Comput. Math. Math. Phys., 44:7 (2004), 1102–1123  mathnet  mathscinet  zmath
    10. Baker G.A. (Jr.), “Counter-examples to the Baker-Garnmel-Wills conjecture and patchwork convergence”, J. Comput. Appl. Math., 179:1-2 (2005), 1–14  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    11. Borcea J., Bøgvad R., Shapiro B., “On rational approximation of algebraic functions”, Adv. Math., 204:2 (2006), 448–480  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    12. 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
    13. S. P. Suetin, “On the Existence of Nonlinear Padé–Chebyshev Approximations for Analytic Functions”, Math. Notes, 86:2 (2009), 264–275  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. 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
    15. 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
    16. S. P. Suetin, “Numerical Analysis of Some Characteristics of the Limit Cycle of the Free van der Pol Equation”, Proc. Steklov Inst. Math., 278, suppl. 1 (2012), S1–S54  mathnet  crossref  crossref  isi  elib
    17. Derevyagin M., Derkach V., “Convergence of Diagonal Pade Approximants for a Class of Definitizable Functions”, Recent Advances in Operator Theory in Hilbert and Krein Spaces, Operator Theory Advances and Applications, 198, 2010, 97–124  mathscinet  zmath  isi
    18. A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. 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
    20. 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, eds. Arvesu J., Lagomasino G., Amer Mathematical Soc, 2011, 165–193  crossref  mathscinet  isi
    21. Baratchart L., Stahl H., Yattselev M., “Weighted extremal domains and best rational approximation”, Adv Math, 229:1 (2012), 357–407  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    22. Buslaev V.I., “An Estimate of the Capacity of Singular Sets of Functions That Are Defined by Continued Fractions”, Anal. Math., 39:1 (2013), 1–27  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    23. Baratchart L. Yattselev M.L., “Pade Approximants to Certain Elliptic-Type Functions”, J. Anal. Math., 121 (2013), 31–86  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    24. V. I. Buslaev, “On the Van Vleck Theorem for Limit-Periodic Continued Fractions of General Form”, Proc. Steklov Inst. Math., 298 (2017), 68–93  mathnet  crossref  crossref  mathscinet  isi  elib
    25. V. I. Buslaev, “Continued fractions with limit periodic coefficients”, Sb. Math., 209:2 (2018), 187–205  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    26. D. S. Lubinsky, “Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants”, Sb. Math., 209:3 (2018), 432–448  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    27. Lubinsky D.S., “On Uniform Convergence of Diagonal Multipoint Pade Approximants For Entire Functions”, Constr. Approx., 49:1 (2019), 149–174  crossref  mathscinet  zmath  isi  scopus
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