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Inverse theorems for multi-point Padé approximations. (Russian) Zbl 0712.30038
Let $$f\in H(B)$$ where B is the closed unit disk and let $$D_ m=[| z| <R_ m]$$ denote the largest disk with center at $$z=0$$ where f admits a holomorphic extension with no more than m poles. Assume that the sequence of polynomials $$\{\omega_ n\}$$, $$n\in {\mathbb{N}}$$ whose zeros lie in B, satisfies the condition $\frac{\omega_ n(z)}{z^ n}\rightrightarrows_{n}\psi (z)\in H({\bar {\mathbb{C}}}\setminus B)$ uniformly on each compact subset of $${\bar {\mathbb{C}}}\setminus B$$. Let $$\pi_ n=\frac{P_ n}{q_ n}$$ be the n-th element of the m-th row of the multipoint Padé table obtained interpolating f at the zeros of $$\omega_{n+m+1}$$. That is: $i)\quad \deg p_ n\leq n,\quad \deg q_ n\leq m,\quad q_ n\not\equiv 0,$
$ii)\quad \frac{q_ nf-p_ n}{\omega_{n+m+1}}\in H(B).$ The author investigates inverse type theorems for the sequences $$\{\pi_ n\}$$, $$n\in {\mathbb{N}}$$, (m fixed). That is, in terms of the asymptotic behavior of the poles of $$\pi_ n$$ a description of the meromorhic extendibility of f and the location of its poles is given. In particular, he proves: Theorem. Assume that for all large n, $$\pi_ n$$ has exactly m finite poles $$\zeta_{n,1},...,\zeta_{n,m}$$ and $$\zeta_{nj}\to a_ j\in {\mathbb{C}}$$, $$| a_ j| >1$$, $$j=1,...,m$$ (1). Then $$R_ m\geq \max_{i}| a_ j|$$ and all the points $$a_ j$$ are singular points of f. Moreover, if $$| a_ j| <R_ m$$ then $$a_ j$$ is a pole of f of order equal to the number of times in which this point appears as limit in (1).

##### MSC:
 30E10 Approximation in the complex plane 41A21 Padé approximation
##### Keywords:
Padé approximation; meromorphic extension
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