Another extension of the disc algebra

TL;DR

This paper extends the classical disc algebra by identifying limits of polynomials on a compactified complex plane, exploring their properties and topological structure, and comparing with previous extensions based on the chordal metric.

Contribution

It introduces a new class of functions, ar{A}(D), as limits of polynomials on a compactification of the complex plane, and analyzes their properties and differences from prior classes.

Findings

The class ar{A}(D) is a richer extension of the disc algebra.

ar{A}(D) differs from the previously studied heck{A}(D) based on the chordal metric.

Properties of ar{A}(D) elements and its topological structure are characterized.

Abstract

We identify the complex plane C with the open unit disc D={z:|z|<1} by the homeomorphism z --> z/(1+|z|). This leads to a compactification $\bar{C}$ of C, homeomorphic to the closed unit disc. The Euclidean metric on the closed unit disc induces a metric d on $\bar{C}$. We identify all uniform limits of polynomials on $\bar{D}$ with respect to the metric d. The class of the above limits is an extension of the disc algebra and it is denoted by $\bar{A}(D)$. We study properties of the elements of $\bar{A}(D)$ and topological properties of the class $\bar{A}(D)$ endowed with its natural topology. The class $\bar{A}(D)$ is different and, from the geometric point of view, richer than the class $\tilde{A}(D)$ introduced in Nestoridis (2010), Arxiv:1009.5364, on the basis of the chordal metric.