Geodesic
On the Phenomenon of Spurious Interpolation of Elliptic Functions by Diagonal Pad\'e Approximants
Matematičeskie zametki, Tome 87 (2010) no. 4, pp. 604-615.

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We study diagonal Padé approximants for elliptic functions. The presence of spurious poles in the approximants not corresponding to the singularities of the original function prevents uniform convergence of the approximants in the Stahl domain. This phenomenon turns out to be closely related to the existence in the Stahl domain of points of spurious interpolation at which the Padé approximants interpolate the other branch of the elliptic function. We also investigate the behavior of diagonal Padé approximants in a neighborhood of points of spurious interpolation.
Keywords: elliptic function, spurious pole, spurious interpolation, Stahl compact set, Riemann surface, complex Green function.
Mots-clés : diagonal Padé approximants, Laurent series 
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D. V. Khristoforov. On the Phenomenon of Spurious Interpolation of Elliptic Functions by Diagonal Pad\'e Approximants. Matematičeskie zametki, Tome 87 (2010) no. 4, pp. 604-615. http://geodesic.mathdoc.fr/item/MZM_2010_87_4_a11/

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

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

[3] H. Stahl, “Spurious poles in Padé approximation”, J. Comput. Appl. Math., 99:1–2 (1998), 511–527 | DOI | MR | Zbl

[4] S. P. Suetin, “O skhodimosti chebyshevskikh nepreryvnykh drobei dlya ellipticheskikh funktsii”, Matem. sb., 194:12 (2003), 63–92 | MR | Zbl

[5] S. P. Suetin, “O ravnomernoi skhodimosti diagonalnykh approksimatsii Pade dlya giperellipticheskikh funktsii”, Matem. sb., 191:9 (2000), 81–114 | MR | Zbl

[6] S. P. Suetin, “Ob interpolyatsionnykh svoistvakh diagonalnykh approksimatsii Pade ellipticheskikh funktsii”, UMN, 59:4 (2004), 201–202 | MR | Zbl

[7] Dzh. Beiker ml., P. Greivs-Morris, Approksimatsii Pade. Ch. 1. Osnovy teorii. Ch. 2. Obobscheniya i prilozheniya, Mir, M., 1986 | MR | Zbl

[8] J. Gammel, W. Marshall, L. Morgan, “An application of Padé approximants to Heisenberg ferromagnetism and antiferromagnetism”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 275:1361 (1963), 257–270 | DOI

[9] V. A. Ilina, P. K. Silaev, Chislennye metody dlya fizikov-teoretikov, Ch. 1, IKI, M, Izhevsk, 2003

[10] S. Dumas, Sur le développement des fonctions elliptiques en fractions continues, Thesis, Zürich, 1908, 59 pp. | Zbl

[11] V. I. Buslaev, “O gipoteze Beikera–Gammelya–Uillsa v teorii approksimatsii Pade”, Matem. sb., 193:6 (2002), 25–38 | MR | Zbl

[12] D. V. Khristoforov, “O skhodimosti diagonalnykh approksimatsii Pade dlya ellipticheskikh funktsii”, Matem. sb., 200:6 (2009), 143–160 | MR | Zbl

[13] E. I. Zverovich, “Kraevye zadachi teorii analiticheskikh funktsii v gelderovskikh klassakh na rimanovykh poverkhnostyakh”, UMN, 26:1 (1971), 113–179 | MR | Zbl

[14] O. Forster, Rimanovy poverkhnosti, Mir, M., 1980 | MR | Zbl