有理$\mathbf{\Theta_n}$-内函数及其在插值问题中的应用
Rational $\mathbf{Θ_n}$-Inner Function and its Application in Interpolation Problem
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中文总结 AI 辅助
研究区域$\mathbf{\Theta}_n$的几何与函数理论性质，引入$\mathbf{\Theta}_n$-内函数概念并建立其与其他函数的联系，推导有理$\mathbf{\Theta}_n$-内函数的刻画，还给出满足特定插值数据的该函数的显式公式。
中文摘要 AI 辅助
本文研究了区域$\mathbf{\Theta}_n$的若干几何和函数理论性质。得到了其特征边界的新刻画，引入了“$\mathbf{\Theta}_n$-内函数”的概念并给出示例。建立了$\mathbf{\Theta}n$-内函数与$\Gamma_n$-内函数、四面体-内函数以及$\mathbf{\Theta}_{n + 1}$-内函数之间的联系。还推导了有理$\mathbf{\Theta}_n$-内函数的显式刻画。作为应用，对于$\mathbb{D}$中任意有限个不同的插值节点和$\mathbf{\Theta}_n$中规定的目标点，得到了满足给定插值数据的有理$\mathbf{\Theta}_n$-内函数的显式公式。
英文摘要
In this paper, we investigate several geometric and function-theoretic properties of the domain $\mathbfΘ_n$. We obtain new characterizations of its distinguished boundary and introduce the notion of a \textit{$\mathbfΘ_n$-inner function}, together with several illustrative examples. We establish connections between $\mathbfΘn$-inner functions and $Γ_n$-inner functions, tetra-inner functions, and $\mathbfΘ_{n+1}$-inner functions. Furthermore, we derive an explicit characterization of rational $\mathbfΘ_n$-inner functions. As an application, for any finite collection of distinct interpolation nodes in $\mathbb{D}$ and prescribed target points in $\mathbfΘ_n$, we obtain an explicit formula for the rational $\mathbfΘ_n$-inner function satisfying the given interpolation data.
