On row sequences of Pad\'e and Hermite-Pad\'e approximation

TL;DR

This paper surveys key results and proposes conjectures on the behavior of row sequences in Padé and Hermite-Padé approximation, focusing on inverse results and the role of common denominators in rational approximation.

Contribution

It introduces a conjecture on inverse results for type II Hermite-Padé approximation and discusses inverse results for incomplete Padé approximants, linking these to broader approximation theory.

Findings

Inverse results proved for incomplete Padé approximants

Conjecture posed on inverse results for Hermite-Padé approximation

Discussion on the connection between common denominators and approximation behavior

Abstract

A survey of direct and inverse type results for row sequences of Pad\'e and Hermite-Pad\'e approximation is given. A conjecture is posed on an inverse type result for type II Hermite-Pad\'e approximation when it is known that the sequence of common denominators of the approximating vector rational functions has a limit. Some inverse type results are proved for the so called incomplete Pad\'e approximants which may lead to the proof of the conjecture and the connection is discussed.