Analytic Continuation of Concrete Realizations and the McCarthy Champagne Conjecture
Analytic Continuation of Concrete Realizations and the McCarthy Champagne Conjecture
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具体实现的解析延拓和麦卡锡香槟猜想
DOI：
10.1093/imrn/rnac050
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发表时间：
2022
影响因子：
1
通讯作者：
Tully-Doyle, Ryan
中科院分区：
文献类型：
--
作者：
Bickel, Kelly;Pascoe, J E;Tully-Doyle, Ryan
在本文中，我们给出的公式允许人们在多变量Schur， Herglotz和Pick函数的传递函数类型实现之间移动，而不需要添加额外的奇异点，除了可能来自保角变换本身的极点。在两变量交换的情况下，我们使用典型的de Branges-Rovnyak模型理论来获得在其中一个变量中是有理的内函数（所谓的拟有理函数）的边界上解析连续的具体实现。在两变量拟有理函数和变量透视函数的情况下，我们建立了关于矩阵局部到全局单调性的McCarthy’s Champagne猜想的正解。
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-calledquasi-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.