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

V. I. Buslaev

Sbornik. Mathematics, Tome 193 (2002) no. 6, pp. 811-823

Résumé

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).

Bibliographie

[1] Baker G. A., Jr., Gammel J. L., Wills J. G., “An investigation of the applicability of the Padé approximant method”, J. Math. Anal. Appl., 2 (1961), 405–418 | DOI | MR | Zbl

[2] Baker G. A., Essentials of Padé approximants, Academic Press, New York, 1975 | MR

[3] Wallin H., “The convergence of Padé approximants and the size of the power series coefficients”, Appl. Anal., 4 (1974), 235–251 | DOI | MR | Zbl

[4] Markov A. A., “Dva dokazatelstva skhodimosti nekotorykh nepreryvnykh drobei”, Izbrannye trudy po teorii nepreryvnykh drobei i teorii funktsii, naimenee uklonyayuschikhsya ot nulya, Gostekhizdat, M., 1948, 106–119 | MR

[5] Gonchar A. A., “O skhodimosti approksimatsii Pade dlya nekotorykh klassov meromorfnykh funktsii”, Matem. sb., 97 (139) (1975), 607–629 | MR | Zbl

[6] Rakhmanov E. A., “O skhodimosti diagonalnykh approksimatsii Pade”, Matem. sb., 104 (146) (1977), 271–291 | MR | Zbl

[7] Baker G. A., Jr., Graves-Morris P. R., “Padé approximants. I: Basic theory”, Encyclopedia of Mathematics and its Applications, V. 13, Addison-Wesley Publ. Co., London, 1981, 238–239 | MR

[8] Buslaev V. I., Gonchar A. A., Suetin S. P., “O skhodimosti podposledovatelnostei $m$-i stroki tablitsy Pade”, Matem. sb., 120 (162):4 (1983), 540–545 | MR | Zbl

[9] Stahl H., “Conjectures around the Baker–Gammel–Wills conjecture”, Constr. Approx., 13 (1997), 287–292 | DOI | MR | Zbl

[10] Stahl H., “Ortogonal polynomials with complex valued weight function. I; II”, Constr. Approx., 2 (1986), 225–240 ; 241–251 | DOI | MR | Zbl

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

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

[13] Lubinsky D., “Rogers–Ramanujan and the Baker–Gammel–Wills (Padé) conjecture”, Ann. of Math. (2) (to appear) | MR | Zbl

[14] Buslaev V. I., “Simple counterexample to the Baker–Gammel–Wills conjecture”, East J. Approx., 7:4 (2001), 515–517 | MR | Zbl

[15] Buslaev V. I., “O teoreme Van Fleka dlya pravilnykh $C$-drobei s predelno periodicheskimi koeffitsientami”, Izv. RAN. Ser. matem., 65:4 (2001), 35–48 | MR | Zbl

[16] Buslaev V. I., Remarks on Poincaré and Van Vleck theorems, Industrial Mathematics Institute Preprint Series, No 14, University of South Carolina, 1999, p. –21

[17] Pringsheim A., “Über die Konvergenz unendücher Kettenbruche”, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber., 28 (1899), 295–324

[18] Hirschhorn M. D., “Partitions and Ramanujan's continued fraction”, Duke Math. J., 39 (1972), 789–791 | DOI | MR | Zbl

[19] Poincaré H., “Sur les équations linéaires aux différentielles et aux différences finies”, Amer. J. Math., 7 (1885), 203–258 | DOI | MR

[20] Perron O., “Über einen Satz des Herrn Poincaré”, J. Reine Angew. Math., 136 (1909), 17–37 ; “Über die Poincarésche lineare Differenrengleichung”, J. Reine Angew. Math., 137 (1910), 6–64 | Zbl