On a multi-point interpolation problem for generalized Schur functions

TL;DR

This paper extends the understanding of multi-point interpolation problems for generalized Schur functions, providing a linear fractional description of solutions for all admissible negative eigenvalue counts, building on known results for the minimal case.

Contribution

It generalizes the linear fractional parametrization of solutions from the minimal negative eigenvalue case to all cases with higher eigenvalues.

Findings

Provides a linear fractional description for all solutions with 57777777 solutions for 57777777 solutions.

Extends known results from the minimal negative eigenvalue case to arbitrary 57777777 solutions.

Clarifies the structure of the solution set for generalized Schur functions in the context of multi-point interpolation.

Abstract

The nondegenerate Nevanlinna-Pick-Carath\'eodory-Fejer interpolation problem with finitely many interpolation conditions always has infinitely many solutions in a generalized Schur class $\cS_\kappa$ for every $\kappa\ge \kappa_{\rm min}$ where the integer $\kappa_{\rm min}$ equals the number of negative eigenvalues of the Pick matrix associated to the problem and completely determined by interpolation data. A linear fractional description of all $\cS_{\kappa_{\rm min}}$ solutions of the (nondegenerate) problem is well known. In this paper, we present a similar result for an arbitrary $\kappa\ge \kappa_{\rm min}$.