Unitarity of some barycentric rational approximants

TL;DR

This paper demonstrates that certain barycentric rational approximants of the exponential function preserve unitarity on the imaginary axis, including those computed via adaptive AAA and Lawson methods, with improved stability and efficiency.

Contribution

It proves unitarity preservation for (k,k)-rational barycentric approximants, including those from linearized error minimization, and introduces a more stable, cost-effective computational procedure.

Findings

Unitarity is conserved by barycentric rational approximants of the exponential on the imaginary axis.

The AAA and AAA-Lawson methods produce approximants that maintain unitarity.

A modified procedure improves numerical stability and reduces computational cost.

Abstract

The exponential function maps the imaginary axis to the unit circle and, for many applications, this unitarity property is also desirable from its approximations. We show that this property is conserved not only by the (k,k)-rational barycentric interpolant of the exponential on the imaginary axis, but also by (k,k)-rational barycentric approximants that minimize a linearized approximation error. These results are a consequence of certain properties of singular vectors of Loewner-type matrices associated to linearized approximation errors. Prominent representatives of this class are rational approximants computed by the adaptive Antoulas--Anderson (AAA) method and the AAA--Lawson method. Our results also lead to a modified procedure with improved numerical stability of the unitarity property and reduced computational cost.