Geodesic
The distribution of poles of rational functions of best approximation and related questions
Sbornik. Mathematics, Tome 9 (1969) no. 2, pp. 267-274 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $f(z)\in H_2$ ($|z|<1$), and let $e_n(f)$ and $r_n(f)$ be best approximations of $f$ by means of polynomials and rational functions of degree $\leqslant n$. The fundamental result of this work is the following theorem: if $\varlimsup_{n\to\infty}(e_n(f)-r_n(f))^{1/n}\leqslant\rho<1$, then $f(z)$ is analytic in the disk $|z|<\rho^{1/2}$. In particular, if $\lim_{n\to\infty}(e_n(f)-r_n(f))^{1/n}=0$, then $f(z)$ is an entire function. Bibliography: 4 titles. 
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     title = {The distribution of poles of rational functions of best approximation and related questions},
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A. L. Levin. The distribution of poles of rational functions of best approximation and related questions. Sbornik. Mathematics, Tome 9 (1969) no. 2, pp. 267-274. http://geodesic.mathdoc.fr/item/SM_1969_9_2_a9/

[1] A. A. Gonchar, “Skorost priblizheniya ratsionalnymi drobyami i svoistva funktsii”, Trudy Mezhdunarodnogo kongressa matematikov, Mir, M., 1968, 329–366

[2] G. A. Volkov, “O raspolozhenii polyusov ratsionalnykh funktsii nailuchshego priblizheniya”, Matem. zametki, 7:3 (1970), 289–293 | MR | Zbl

[3] A. L. Levin, V. M. Tikhomirov, “O priblizheniyakh analiticheskikh funktsii ratsionalnymi”, DAN SSSR, 174:2 (1967), 279–282 | MR | Zbl

[4] V. D. Erokhin, “O nailuchshem priblizhenii analiticheskikh funktsii posredstvom ratsionalnykh drobei so svobodnymi polyusami”, DAN SSSR, 128:1 (1959), 29–32 | Zbl