Rational ${\mathbf{\Theta }}_{\mathbf{n}}$-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 ${\mathbf{\Theta }}_n\mathit{\text{-inner}}\mathit{\text{function}}$, together with several illustrative examples. We establish connections between $\mathbf{\Theta }n$-inner functions and ${\mathrm{\Gamma }}_n$-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.