The distribution of poles of rational functions of best approximation and related questions

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.
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