# Approximation in the closed unit ball

## TL;DR

This paper reviews classic theorems on the approximation and closure properties of Blaschke products, inner functions, and their quotients in the sup-norm, with applications to spectral theory.

## Contribution

It provides an expository overview of key theorems in complex analysis related to approximation in the unit ball and offers a new proof of a spectral mapping theorem.

## Key findings

- Characterization of closures of sets of Blaschke products and inner functions
- Connections between approximation theorems and spectral theory
- A simplified proof of the Berger-Stampfli spectral mapping theorem

## Abstract

In this expository article, we present a number of classic theorems that serve to identify the closure in the sup-norm of various sets of Blaschke products, inner functions and their quotients, as well as the closure of the convex hulls of these sets. The results presented include theorems of Carath\'eodory, Fisher, Helson-Sarason, Frostman, Adamjan-Arov-Krein, Douglas-Rudin and Marshall. As an application of some of these ideas, we obtain a simple proof of the Berger-Stampfli spectral mapping theorem for the numerical range of an operator.