Convergence of multipoint Padé approximants of piecewise analytic functions (Q2842988)

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 NEWLINE\[NEWLINE\deg P_n\leq n,\;\deg Q_n\leq n,\;Q_n\not\equiv 0NEWLINE\]NEWLINE and NEWLINE\[NEWLINE(Q_nf_0-P_n)(z)={\mathcal O}(z^{n_1}),\;z\rightarrow 0,\quad (Q_nf_{\infty}-P_n)(z)={\mathcal O}(\textstyle{{1\over z^{n_2-n}}}),\;z\rightarrow \infty.NEWLINE\]NEWLINENEWLINENEWLINE\vskip0.3cm The main results in the paper are analogues of \textit{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 \textit{A. A. Gonchar} and \textit{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.