Bitangential interpolation in generalized Schur classes

TL;DR

This paper studies bitangential interpolation problems within generalized Schur classes, providing linear fractional solution representations for both invertible and singular cases, applicable in the unit disc and right half plane.

Contribution

It introduces new linear fractional representations for solutions to bitangential interpolation problems in generalized Schur classes, including non-holomorphic solutions at interpolation points.

Findings

Solution representations for invertible Hermitian Pick matrices

Solution representations for singular Hermitian Pick matrices

Applicable in both the unit disc and right half plane

Abstract

Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these problems are presented for invertible and singular Hermitian Pick matrices. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in a previous paper.