Pick interpolation and invariant functions
Anindya Biswas
TL;DR
The paper addresses multi-point Pick interpolation on Carathéodory hyperbolic domains by linking Pick bodies to two invariant distances, $d^{\Omega}_{(\underline{z},\underline{\alpha})}$ and $\delta^{\Omega}_{(\underline{z},\underline{\alpha})}$, thereby generalizing the Carathéodory pseudodistance and connecting to a Lempert-type framework. It introduces the Pick body $\mathscr{D}_\Omega$ and analyzes its Minkowski functional $\mu_{\mathscr{D}_n}$, establishing a solvability criterion for interpolation and deriving explicit expressions in the disc, including the two-point reduction to $c^*_\mathbb{D}$. The study provides explicit evaluations of $d^{\mathbb{D}}_{\underline{\alpha}}$ for special $\underline{\alpha}$ and describes the boundary $\partial\mathscr{D}_n$ in terms of Blaschke extremals, linking to Schwarz–Pick theory. A key result is that for $n=3$ in the bidisc and tridisc settings, the two invariant functions coincide ($d=\delta$), offering a three-point Lempert-type equivalence, while for $n\ge 4$ the equality fails in general, clarifying the limits of this generalization.
Abstract
In this article, we establish a connection between Pick bodies and invariant functions. We demonstrate that an invariant function can be associated with any Pick body, which determines the solvability of a given Pick interpolation problem and serves as a generalization of the Carathéodory pseudodistance. A complete description of this invariant function is provided for the open unit disc, and it is shown that it leads to another invariant function that can be regarded as a generalized Lempert function. It is also proved that these two invariant functions are equal if certain geodesics can be found. Lastly, we show that, in a very special case, a result analogous to Lempert's theorem holds for the bidisc and the tridisc.