Geodesic
On the convergence of Padé approximants in classes of holomorphic functions
Sbornik. Mathematics, Tome 40 (1981) no. 2, pp. 149-155 
E. A. Rakhmanov. On the convergence of Padé approximants in classes of holomorphic functions. Sbornik. Mathematics, Tome 40 (1981) no. 2, pp. 149-155. http://geodesic.mathdoc.fr/item/SM_1981_40_2_a1/
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     author = {E. A. Rakhmanov},
     title = {On the convergence of {Pad\'e} approximants in classes of holomorphic functions},
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     year = {1981},
     volume = {40},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_40_2_a1/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper it is proved that if, for any function $f$ holomorphic in a domain $D\subset\overline{\mathbf C}$ ($\infty\in D$), the sequence $\{\pi_n(f)\}_{n\in\mathbf N}$ of its diagonal Padé approximants (corresponding to the point $z=\infty$) converges to $f$ in measure in $D$, then $\operatorname{cap}(\overline{\mathbf C}\setminus D)=0$. Bibliography: 8 titles.

[1] Ch. Pommerenke, “Pade approximants and convergence in capacity”, J. Math. Anal. Appl., 41 (1973), 147–153 | DOI | MR

[2] Ch. Pommerenke, “Problems in complex theory”, Bull. London Math. Soc., 4:3 (1972), 354–366 | DOI | MR | Zbl

[3] A. A. Gonchar, “O skhodimosti approksimatsii Pade”, Matem. sb., 92(134) (1973), 152–164 | Zbl

[4] A. A. Gonchar, “Lokalnoe uslovie odnoznachnosti analiticheskikh funktsii neskolkikh peremennykh”, Matem. sb., 93(135) (1974), 296–313 | Zbl

[5] H. S. Wall, Analytic theory of continued fractions, Van Nostrand, New York, 1948 | MR | Zbl

[6] G. A. Baker, Jr., Essentials of Pade Approximants, Academic Press, New York, 1975 | MR

[7] Dzh. Uolsh, Interpolyatsiya i approksimatsiya ratsionalnymi funktsiyami v kompleksnoi oblasti, IL, Moskva, 1961 | MR

[8] A. A. Gonchar, K. N. Lungu, “Polyusy approksimatsii Pade i analiticheskoe prodolzhenie funktsii”, Matem. sb., 111(153):2 (1980), 279–292 | MR | Zbl