Inverse theorems on generalized Pad\'e approximants

S. P. Suetin

Sbornik. Mathematics, Tome 37 (1980) no. 4, pp. 581-597

Résumé

In this paper the following theorem is proved. Theorem. {\it For $m>0$ and all sufficiently large $n$, let the Padé approximants $R_{n,m}$ of the series $$ f(z)=\sum_{\nu=0}^\infty A_\nu F_\nu(z),\qquad A_\nu=(f,F_\nu)=\int_{-1}^1f(x)F_\nu(x)\,d\alpha(x), $$ have exactly $m$ finite poles, and let there exist a polynomial $\omega_m(z)=\prod_{j=1}^m(z-z_j)$ such that $$ \varlimsup_{n\to\infty}\|q_{n,m}-\omega_m\|^{1/n}\leqslant\delta1. $$ Then $$ \rho_m(f)\geqslant\frac1\delta\max_{1\leqslant j\leqslant m}|\varphi(z_j)| $$ and in the region $D_m(f)=D_{\rho_m}$ the function $f$ has exactly $m$ poles (at the points $z_1,\dots,z_m$). } Bibliography: 8 titles.