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

A. L. Levin

Sbornik. Mathematics, Tome 9 (1969) no. 2, pp. 267-274

Résumé

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 $\varlimsup_{n\to\infty}(e_n(f)-r_n(f))^{1/n}\leqslant\rho1$, 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.