Abel universal functions: boundary behaviour and Taylor polynomials

TL;DR

This paper explores the boundary behavior and Taylor polynomial convergence of Abel universal functions, which are dense in the space of holomorphic functions on the unit disk, revealing their complex approximation properties.

Contribution

It advances the theory of Abel universal functions by analyzing their boundary behavior and Taylor polynomial convergence outside the disk.

Findings

Boundary behavior includes local growth and Picard points.

Taylor polynomials of these functions exhibit specific convergence properties outside the disk.

Universal radial approximation is characterized by dense dilates in function spaces.

Abstract

A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A(\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the space of continuous functions on $K$, for any proper compact subset $K$ of the unit circle. It has been recently shown that $\mathcal{U}_A(\mathbb{D})$ is a dense $G_{\delta}$ subset of the space of holomorphic functions on $\mathbb{D}$ endowed with the topology of local uniform convergence. In this paper, we develop further the theory of universal radial approximation by investigating the boundary behaviour of functions in $\mathcal{U}_A(\mathbb{D})$ (local growth, existence of Picard points and asymptotic values) and the convergence properties of their Taylor polynomials outside $\mathbb{D}$.