An extension of the disc Algebra

TL;DR

This paper characterizes all uniform limits of polynomials on the closed unit disc under the chordal metric, revealing a class of functions including holomorphic functions and the point at infinity, and explores their properties.

Contribution

It extends the classical polynomial approximation theory by identifying the limit functions in the chordal metric and analyzing their topological and functional properties.

Findings

Limits include holomorphic functions and the point at infinity.

The class A(D) is characterized by boundary limits existing in C or at infinity.

Open questions and new research directions are proposed.

Abstract

We identify all uniform limits of polynomials on the closed unit disc with respect to the chordal metric \c{hi} . One such limit is f=oo. The other limits are holomorphic functions f:-->C so that for every {\zeta} in the boundary of unit disc D the limf(z) while z-->{\zeta} exists in C U {oo}. The class of the above functions is denoted by A(D)~. We study properties of the members of A(D)~, as well as, some topological properties of A(D)~ endowed with its natural metric topology. There are several open questions and new directions of investigation.