# Symmetric Contours and Convergent Interpolation

## TL;DR

This paper constructs symmetric contours using hyperelliptic Riemann surfaces to analyze the convergence of multipoint Padé approximants, providing strong asymptotic error formulas under mild conditions.

## Contribution

It introduces a method to construct symmetric contours via hyperelliptic Riemann surfaces for analyzing rational interpolants with free poles.

## Key findings

- Constructed symmetric contours using hyperelliptic Riemann surfaces.
- Derived strong asymptotics for interpolation error.
- Applied ar-Riemann-Hilbert techniques for analysis.

## Abstract

The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as applied to the multipoint Pad\'e approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ, if there exists a contour that is "symmetric" with respect to the interpolation scheme, does not separate the plane, and in the complement of which the germ has a single-valued continuation with non-identically zero jump across the contour, then the interpolants converge to that continuation in logarithmic capacity in the complement of the contour. The existence of such a contour is not guaranteed. In this work we do construct a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author. We consider rational interpolants with free poles of Cauchy transforms of non-vanishing complex densities on such contours under mild smoothness assumptions on the density. We utilize \( \bar\partial \)-extension of the Riemann-Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation.