The Takagi problem on the disk and bidisk

TL;DR

This paper presents a new proof for solving Pick problems on the disk using rational functions with specific unimodularity and pole constraints, and extends the approach to the bidisk.

Contribution

It introduces a novel proof technique for Pick problems on the disk and applies it to find rational solutions on the bidisk.

Findings

Rational solutions can be constructed with poles constrained by eigenvalues.

The method provides a new perspective on classical interpolation problems.

Extension of the proof technique from disk to bidisk.

Abstract

We give a new proof on the disk that a Pick problem can be solved by a rational function that is unimodular on the unit circle and for which the number of poles inside the disk is no more than the number of non-positive eigenvalues of the Pick matrix. We use this method to find rational solutions to Pick problems on the bidisk.