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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
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