$L^2$ and $L^\infty$ rational approximation

TL;DR

This paper presents a computational study comparing best $L^2$ and $L^ Infty$ rational approximations of analytic functions on the unit disk, utilizing new algorithms and formulations.

Contribution

It introduces a novel computational approach for $L^2$ rational approximation using a new barycentric TF-IRKA formulation, enabling the first detailed comparison with $L^ Infty$ approximations.

Findings

Comparison of $L^2$ and $L^ Infty$ rational approximations.

Implementation of a new barycentric TF-IRKA method.

First computational study of these approximation problems.

Abstract

Using recently developed algorithms, we compute and compare best $L^2$ and $L^\infty$ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first computational study. To compute the $L^2$ best approximations, we employ a new formulation of TF-IRKA in barycentric form.