Rational ${\mathbf{\Theta }}_{\mathbf{n}}$-Inner Function and its Application in Interpolation Problem

Abstract

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.

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