Pick interpolation and invariant functions

TL;DR

This paper explores the relationship between Pick bodies and invariant functions, introducing new invariant functions that generalize classical concepts and establishing conditions under which they coincide, with applications to the unit disc and polydiscs.

Contribution

It introduces a new invariant function associated with Pick bodies, generalizes the Carathéodory and Lempert functions, and extends some results to polydiscs.

Findings

Invariant functions determine Pick interpolation solvability.

Complete description of the invariant function for the unit disc.

Equality of invariant functions under certain geodesic conditions.

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 Carath\'eodory 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.