Interpolation and sampling for analytic selfmappings of the disc

TL;DR

This paper explores interpolation and sampling problems for analytic self-maps of the disc, extending classical results and providing new characterizations for these function classes.

Contribution

It introduces a specialized interpolation problem based on a Schwarz-Pick lemma variant and characterizes sampling sequences for analytic self-maps of the disc.

Findings

A version of Schwarz-Pick lemma for multiple points

A new interpolation problem for disc self-maps

Characterization of sampling sequences for these functions

Abstract

Two different problems are considered here. First, a version of Schwarz-Pick Lemma for $n$ points leads to an interpolation problem for analytic functions from the disc into itself, which may be considered as a particular case of the classical Nevanlinna-Pick interpolation problem. Second, a characterization of sampling sequences for this class of functions is given.