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 Mat. Sb., 2009, Volume 200, Number 6, Pages 143–160 (Mi msb6811)

On uniform approximation of elliptic functions by Padé approximants

D. V. Khristoforov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Diagonal Padé approximants of elliptic functions are studied. It is known that the absence of uniform convergence of such approximants is related to them having spurious poles that do not correspond to any singularities of the function being approximated. A sequence of piecewise rational functions is proposed, which is constructed from two neighbouring Padé approximants and approximates an elliptic function locally uniformly in the Stahl domain. The proof of the convergence of this sequence is based on deriving strong asymptotic formulae for the remainder function and Padé polynomials and on the analysis of the behaviour of a spurious pole.
Bibliography: 23 titles.

Keywords: Padé approximants, elliptic functions, the Stahl domain, uniform approximations.

DOI: https://doi.org/10.4213/sm6811

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English version:
Sbornik: Mathematics, 2009, 200:6, 923–941

Bibliographic databases:

UDC: 517.538.53
MSC: Primary 41A21; Secondary 41A30, 30E10, 33E05

Citation: D. V. Khristoforov, “On uniform approximation of elliptic functions by Padé approximants”, Mat. Sb., 200:6 (2009), 143–160; Sb. Math., 200:6 (2009), 923–941

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb6811
• https://doi.org/10.4213/sm6811
• http://mi.mathnet.ru/eng/msb/v200/i6/p143

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. V. Khristoforov, “On the Phenomenon of Spurious Interpolation of Elliptic Functions by Diagonal Padé Approximants”, Math. Notes, 87:4 (2010), 564–574
2. 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
3. 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
4. Martínez-Finkelshtein A. Rakhmanov E.A. Suetin S.P., “Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall's Work 25 years later”, advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., 578, ed. Arvesu J. Lagomasino G., Amer. Math. Soc., Providence, RI, 2011, 165–193
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