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Convergence of multipoint Padé approximants of piecewise analytic functions. (English. Russian original) Zbl 1276.41011

Sb. Math. 204, No. 2, 190-222 (2013); translation from Mat. Sb. 204, No. 2, 39-72 (2013).
The author studies two-point Padé approximants defined as usual, i.e., \(R_n=P_n/Q_n\) is a two-point Padé approximant of order \((n_1,n_2)\), with \(n_1+n_2=2n+1\), to holomorphic functions \(f_0\) and \(f_{\infty}\) defined in a neighbourhood of \(z=0\) and \(z=\infty\) if \[ \deg P_n\leq n,\;\deg Q_n\leq n,\;Q_n\not\equiv 0 \] and \[ (Q_nf_0-P_n)(z)={\mathcal O}(z^{n_1}),\;z\rightarrow 0,\quad (Q_nf_{\infty}-P_n)(z)={\mathcal O}(\textstyle{{1\over z^{n_2-n}}}),\;z\rightarrow \infty. \]
The main results in the paper are analogues of H. Stahl’s famous first and second theorem (cf. [Constructive Approximation 2, 225–240 (1986; Zbl 0592.42016); Complex Variables, Theory Appl. 4, 311–324 (1985; Zbl 0542.30027)]) and of a general result by A. A. Gonchar and E. A. Rakhmanov [Mat. Sb., N. Ser. 134(176), No. 3(11), 306–352 (1987; Zbl 0645.30026)], all three given for ordinary Padé approximation.

MSC:

41A21 Padé approximation
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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