Geodesic
On uniform approximation of elliptic functions by Padé approximants
Sbornik. Mathematics, Tome 200 (2009) no. 6, pp. 923-941 Cet article a éte moissonné depuis la source Math-Net.Ru

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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: elliptic functions, the Stahl domain, uniform approximations.
Mots-clés : Padé approximants 
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D. V. Khristoforov. On uniform approximation of elliptic functions by Padé approximants. Sbornik. Mathematics, Tome 200 (2009) no. 6, pp. 923-941. http://geodesic.mathdoc.fr/item/SM_2009_200_6_a5/

[1] G. Frobenius, “Ueber Relationen zwischen den Näherungsbrüchen von Potenzreihen”, J. Reine Angew. Math., 90 (1880), 1–17 | Zbl

[2] H. Padé, “Sur la représentation approchée d'une fonction par des fractions rationnel”, Ann. Sci. École Norm. Sup. (3), 9 (1892), 3–93 | MR | Zbl

[3] A. A. Gončar, “On convergence of Padé approximants for some classes of meromorphic functions”, Math. USSR-Sb., 26:4 (1975), 555–575 | DOI | MR | Zbl

[4] G. A. Baker, jr.; P. Graves-Morris, Padé approximants. Part I, II, Encyclopedia Math. Appl., 13–14, Addison-Wesley, Reading, MA, 1981 | MR | MR | Zbl | Zbl

[5] E. M. Nikishin, V. N. Sorokin, Rational approximations and orthogonality, Transl. Math. Monogr., 92, Amer. Math. Soc., Providence, RI, 1991 | MR | MR | Zbl | Zbl

[6] S. P. Suetin, “Padé approximants and efficient analytic continuation of a power series”, Russian Math. Surveys, 57:1 (2002), 43–141 | DOI | MR | Zbl

[7] A. A. Gonchar, “Ratsionalnye approksimatsii analiticheskikh funktsii”, Sovr. probl. matem., 2003, no. 1, 83–106 | DOI | MR

[8] J. Gammel, W. Marshall, L. Morgan, “An application of Padé approximants to heisenberg ferromagnetism and antiferromagnetism”, Proc. Roy. Soc. London Ser. A, 275:1361 (1963), 257–270 | DOI

[9] R. March, P. Barone, “Reconstruction of a piecewise constant function from noisy Fourier coefficients by Padé method”, SIAM J. Appl. Math., 60:4 (2000), 1137–1156 | DOI | MR | Zbl

[10] V. Ilina, P. Silaev, Chislennye metody dlya fizikov-teoretikov, In-t komp. issledovanii, M.–Izhevsk, 2003

[11] H. N. A. Ismail, K. R. Raslan, G. S. E. Salem, A. A. Abd Rabboh, “Comparison study between restrictive Taylor, restrictive Padé approximations and Adomian decomposition method for the solitary wave solution of the General KdV equation”, Appl. Math. Comput., 167:2 (2005), 849–869 | DOI | MR | Zbl

[12] M. El-Shahed, “Application of differential transform method to non-linear oscillatory systems”, Commun. Nonlinear Sci. Numer. Simul., 13:8 (2008), 1714–1720 | DOI

[13] S. B. Gashkov, I. B. Gashkov, “Berlekamp–Massey algorithm, continued fractions, Padé approximations, and orthogonal polynomials”, Math. Notes, 79:1–2 (2006), 41–54 | DOI | MR | Zbl

[14] I. V. Andrianov, “Approksimatsii Pade i kontinualizatsiya dlya odnomernoi tsepochki mass”, Matem. modelirovanie, 18:1 (2006), 43–58 | MR | Zbl

[15] H. Stahl, “The convergence of Padé approximants to functions with branch points”, J. Approx. Theory, 91:2 (1997), 139–204 | DOI | MR | Zbl

[16] S. P. Suetin, “Convergence of Chebyshëv continued fractions for elliptic functions”, Sb. Math., 194:12 (2003), 1807–1835 | DOI | MR | Zbl

[17] S. P. Suetin, “Uniform convergence of Padé diagonal approximants for hyperelliptic functions”, Sb. Math., 191:9 (2000), 1339–1373 | DOI | MR | Zbl

[18] S. P. Suetin, “On interpolation properties of diagonal Padé approximants of elliptic functions”, Russian Math. Surveys, 59:4 (2004), 800–802 | DOI | MR | Zbl

[19] H. Stahl, “Diagonal Padé approximants to hyperelliptic functions”, Ann. Fac. Sci. Toulouse Math. (6), 1996, 121–193 | MR | Zbl

[20] S. Dumas, Sur le développement des fonctions elliptiques en fractions continues, Thesis, Zurich, 1908 | Zbl

[21] B. Beckermann, V. Kaliaguine, “The Diagonal of the Padé table and the approximation of the Weyl function of second-order difference operators”, Constr. Approx., 13:4 (1997), 481–510 | DOI | MR | Zbl

[22] E. I. Zverovich, “Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces”, Russian Math. Surveys, 26:1 (1971), 117–192 | DOI | MR | Zbl | Zbl

[23] O. Forster, Riemannsche Flächen, Heidelberger Taschenbücher, 184, Springer-Verlag, Berlin–Heidelberg–New York, 1977 | MR | MR | Zbl