Unitarity of some barycentric rational approximants

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.