Inverse theorems on generalized Padé approximants

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
$$\varlimsup_{n\to\infty}\|q_{n,m}-\omega_m\|^{1/n}\leqslant\delta<1.$$
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 $z_1,\dots,z_m$). }
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