Meromorphic continuation of functions and arbitrary distribution of interpolation points

TL;DR

This paper characterizes the meromorphic continuation region of an analytic function using the convergence rate of multi-point rational interpolants with arbitrary point distribution and bounded poles.

Contribution

It introduces a new characterization of meromorphic continuation based on convergence rates of rational interpolants with arbitrary interpolation points.

Findings

Provides a geometric criterion for meromorphic continuation

Handles arbitrary distribution of interpolation points

Establishes convergence properties of rational approximants

Abstract

We characterize the region of meromorphic continuation of an analytic function $f$ in terms of the geometric rate of convergence on a compact set of sequences of multi-point rational interpolants of $f$. The rational approximants have a bounded number of poles and the distribution of interpolation points is arbitrary.