Sampling and interpolation for analytic selfmappings of the disc

TL;DR

This paper characterizes sampling sequences and explores an interpolation problem for analytic self-mappings of the disc, extending classical results with new characterizations and a generalized Schwarz-Pick lemma.

Contribution

It provides a new characterization of sampling sequences and formulates an interpolation problem based on a generalized Schwarz-Pick lemma for analytic self-mappings of the disc.

Findings

Characterization of sampling sequences for analytic self-mappings.

A version of Schwarz-Pick Lemma for multiple points.

Formulation of an interpolation problem related to Nevanlinna-Pick.

Abstract

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