Convergence of two-point Padé approximants to piecewise holomorphic functions

M. L. Yattselev

Sbornik. Mathematics, Tome 212 (2021) no. 11, pp. 1626-1659

Résumé

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.

Détail

Keywords: non-Hermitian orthogonality, strong asymptotics, matrix Riemann-Hilbert approach.
Mots-clés : two-point Padé approximants, $S$-contours

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