On multipoint Pad\'e approximants whose poles accumulate on contours that separate the plane

TL;DR

This paper investigates the asymptotic behavior of multipoint Padé approximants to Cauchy integrals with non-vanishing densities on a Jordan arc, especially when poles accumulate on contours separating the plane.

Contribution

It provides new asymptotic analysis of multipoint Padé approximants with poles accumulating on separating contours in the complex plane.

Findings

Asymptotic behavior characterized for approximants with poles on separating contours

Extension of classical results to more complex contour configurations

Insights into the distribution of poles for these approximants

Abstract

In this note we consider asymptotics of the multipoint Pad\'e approximants to Cauchy integrals of analytic non-vanishing densities defined on a Jordan arc connecting $ -1 $ and $ 1 $. We allow for the situation where the (symmetric) contour attracting the poles of the approximants does separate the plane.