Convergence of Two-point Padé Approximants to Piecewise Holomorphic Functions

Let f0 and f∞ be formal power series at the origin and infinity, and Pn/Qn, with deg(Pn), deg(Qn) 6 n, be a rational function that simultaneously interpolates f0 at the origin with order n and f∞ at infinity with order n + 1. When germs f0, f∞ represent multi-valued functions with finitely many branch points, it was shown by Buslaev [5] that there exists a unique compact set F in the complement of which approximants converge in capacity to the approximated functions. The set Fmight or might not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets F that do separate the plane.