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On the m-th row of the table of multipoint Padé approximations. (Russian) Zbl 0623.30048

Let \(\alpha =(\alpha_{n,i})\), \(i=1,2,...,n\), \(n\in {\mathbb{N}}\), be a table of points contained in \(E_ 1=[| z| \leq 1]\) such that \[ z^{- n}\quad \prod^{n}_{i=1}(z-\alpha_{n,i})\quad \rightrightarrows \quad \phi (z) \] uniformly on each compact subset of \({\bar {\mathbb{C}}}\setminus E_ 1\), where \(\phi\in H({\bar {\mathbb{C}}}\setminus E_ 1).\)
Suppose \(f\in H(E_ 1)\) then, in terms of the asymptotic behavior of the poles of the m-th row multipoint Padé approximants of f with respect to \(\alpha\) information is supplied on the largest disk \(D_ m=[| z| \leq R_ m]\) inside of which f has no more than m poles and the location of the poles of f inside \(D_ m\). If \(D_ m={\mathbb{C}}\) it is shown that a subsequence of the m-th row converges to f on each compact subset of \({\mathbb{C}}\setminus {\mathcal P}\), where \({\mathcal P}\) is the set of poles of f.

MSC:

30E10 Approximation in the complex plane
41A21 Padé approximation
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