Geodesic
Convergence of two-point Pad\'e approximants to piecewise holomorphic functions
Sbornik. Mathematics, Tome 212 (2021) no. 11, pp. 1626-1659

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Let $f_0$ and $f_\infty$ be formal power series at the origin and infinity, and $P_n/Q_n$, $\deg(P_n),\deg(Q_n)\leq n$, be the rational function that simultaneously interpolates $f_0$ at the origin with order $n$ and $f_\infty$ at infinity with order ${n+1}$. When germs $f_0$ and $f_\infty$ represent multi-valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set $F$ in the complement of which the approximants converge in capacity to the approximated functions. The set $F$ may or may not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets $F$ that do separate the plane. Bibliography: 26 titles.
Keywords: two-point Padé approximants, non-Hermitian orthogonality, strong asymptotics, matrix Riemann-Hilbert approach.
Mots-clés : $S$-contours 
@article{SM_2021_212_11_a6,
     author = {M. L. Yattselev},
     title = {Convergence of two-point {Pad\'e} approximants to piecewise holomorphic functions},
     journal = {Sbornik. Mathematics},
     pages = {1626--1659},
     publisher = {mathdoc},
     volume = {212},
     number = {11},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2021_212_11_a6/}
}
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M. L. Yattselev. Convergence of two-point Pad\'e approximants to piecewise holomorphic functions. Sbornik. Mathematics, Tome 212 (2021) no. 11, pp. 1626-1659. http://geodesic.mathdoc.fr/item/SM_2021_212_11_a6/