Analytic continuation of concrete realizations and the McCarthy Champagne conjecture

TL;DR

This paper develops formulas for transforming multivariate Schur, Herglotz, and Pick functions between different realizations, and proves a positive solution to McCarthy's Champagne conjecture for certain classes of functions.

Contribution

It introduces new formulas for realization transformations and establishes the Champagne conjecture for two-variable quasi-rational and d-variable perspective functions.

Findings

Formulas for moving between transfer function realizations.

Concrete realizations that analytically continue through the boundary.

Positive solution to McCarthy's Champagne conjecture for specific function classes.

Abstract

In this paper, we give formulas that allow one to move between transfer function type realizations of multi-variate Schur, Herglotz and Pick functions, without adding additional singularities except perhaps poles coming from the conformal transformation itself. In the two-variable commutative case, we use a canonical de Branges-Rovnyak model theory to obtain concrete realizations that analytically continue through the boundary for inner functions which are rational in one of the variables (so-called quasi-rational functions). We then establish a positive solution to McCarthy's Champagne conjecture for local to global matrix monotonicity in the settings of both two-variable quasi-rational functions and $d$-variable perspective functions.