A Canonical Characterization of Normal Functions

TL;DR

This paper provides a new characterization of normal families of analytic functions in the unit ball based on their behavior on complex lines and discs, and offers a simple proof of Hartog's theorem on convergence of power series.

Contribution

It introduces a canonical characterization of normal functions through their restrictions to complex lines and discs, extending previous understandings.

Findings

Normal families are characterized by their restrictions to complex lines.

Normal functions can be characterized by their behavior on analytic discs.

A simple proof of Hartog's theorem on power series convergence is provided.

Abstract

We characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. We then generalize this result to a characterization of normal functions according to behavior on analytic discs. A simple proof of an old theorem of Hartog's that a formal power series at 0 in $\Cn$ is convergent if its restriction to each complex line through the origin is convergent are given.