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Mat. Sb., 2003, Volume 194, Number 12, Pages 63–92 (Mi msb787)  

This article is cited in 13 scientific papers (total in 13 papers)

Convergence of Chebyshëv continued fractions for elliptic functions

S. P. Suetin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Dumas's classical theorem on the behaviour of the Chebyshëv continued fraction corresponding to an elliptic function $f(z)=\sqrt{(z-e_1)\dotsb(z-e_4)}-z^2+z{(e_1+\dotsb+e_4)}/2$ holomorphic at $z=\infty$ is extended to a fairly general class of elliptic functions. The behaviour of the Chebyshëv continued fractions corresponding to functions in that class is characterized in terms relating to the mutual position of the branch points $e_1,…,e_4$. The proof is based on the investigation of the properties of the solution of a certain Riemann boundary-value problem on an elliptic Riemann surface.

DOI: https://doi.org/10.4213/sm787

Full text: PDF file (440 kB)
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English version:
Sbornik: Mathematics, 2003, 194:12, 1807–1835

Bibliographic databases:

Document Type: Article
UDC: 517.53
MSC: Primary 40A15, 4121; Secondary 14K20, 30B70, 34M50
Received: 12.03.2003

Citation: S. P. Suetin, “Convergence of Chebyshëv continued fractions for elliptic functions”, Mat. Sb., 194:12 (2003), 63–92; Sb. Math., 194:12 (2003), 1807–1835

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. 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
    2. S. P. Suetin, “On polynomials orthogonal on several segments with indefinite weight”, Russian Math. Surveys, 60:5 (2005), 991–993  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. 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
    4. 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
    5. 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
    6. 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
    7. 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
    8. Aleksandr Vladimirovich Komlov, A.V.ladimirovich Komlov, Sergei Pavlovich Suetin, “Asimptoticheskaya formula dlya dvukhtochechnogo analoga polinomov Yakobi”, Uspekhi matematicheskikh nauk, 68:4 (2013), 183  mathnet  crossref  mathscinet
    9. 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
    10. R. K. Kovacheva, S. P. Suetin, “Distribution of zeros of the Hermite–Padé polynomials for a system of three functions, and the Nuttall condenser”, Proc. Steklov Inst. Math., 284 (2014), 168–191  mathnet  crossref  crossref  isi
    11. 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
    12. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Aptekarev A.I. Yattselev M.L., “Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials”, Acta Math., 215:2 (2015), 217–280  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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