Rational $\mathbf{\Theta_n}$-Inner Function and its Application in Interpolation Problem

In this paper, we investigate several geometric and function-theoretic properties of the domain $\mathbf{\Theta}_n$. We obtain new characterizations of its distinguished boundary and introduce the notion of a \textit{$\mathbf{\Theta}_n$-inner function}, together with several illustrative examples. We establish connections between $\mathbf{\Theta}n$-inner functions and $\Gamma_n$-inner functions, tetra-inner functions, and $\mathbf{\Theta}_{n+1}$-inner functions. Furthermore, we derive an explicit characterization of rational $\mathbf{\Theta}_n$-inner functions. As an application, for any finite collection of distinct interpolation nodes in $\mathbb{D}$ and prescribed target points in $\mathbf{\Theta}_n$, we obtain an explicit formula for the rational $\mathbf{\Theta}_n$-inner function satisfying the given interpolation data.