Analytic Continuation of Concrete Realizations and the Mccarthy Champagne Conjecture

Publication Date

2023

Description

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 that 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 �-variable perspective functions.

Journal

Int. Math. Res. Not. IMRN

Volume

2023

Issue

9

First Page

7845

Last Page

7882

Department

Mathematics

DOI

https://doi.org/10.1093/imrn/rnac050

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