Abel universal functions: boundary behaviour and Taylor polynomials

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 $\{f_r:0\le r<1\}$ 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 $\mathbb{D}$.