Universality and Boundary Behaviour of Holomorphic Functions

Universality is a notion with connections to various areas of Analysis. Generally speaking, a mathematical object is called universal if it can approximate, through some specific process, every element of a given space. One of the most well-studied examples of universal objects is the class of universal Taylor series, which consists of holomorphic functions on the unit disc D whose Taylor polynomials can approximate everything plausible outside D. It turns out that the boundary behaviour of such functions (which are many in the sense of Baire category) is extremely chaotic. This thesis will investigate further connections between the boundary behaviour of holomorphic functions and Universality. The first objective is the study of a new class of holomorphic functions on D whose universal approximation property imposes a wild radial behaviour. This is the class of Abel universal functions that was introduced in 2020 by Charpentier to unify several previous results. Each Abel universal function f has dilates which are universal in the following sense: for every proper compact subset K of the unit circle and any continuous function φ on K, there exists a sequence (r_n) in (0,1) such that f(r_nζ) converges to φ(ζ) uniformly for ζ in K as n tendts to infinity. In the first component of this thesis we further develop the theory of Abel universal functions by investigating a wide range of properties: their behaviour on certain boundary regions, their Taylor series and their invariance under composition from the left and the right. We will also make connections with classical topics in Complex Analysis and make comparisons with universal Taylor series. Our second objective is to study the generic boundary behaviour of functions in classical function spaces. In particular, using Baire's Theorem, we will give analogues of Abel universal functions in the Hardy, Bergman and Dirichlet spaces of the unit disc and will also show that one-sided extendability in certain L^∞ (and L^p) spaces of a rectifiable Jordan arc is a rare phenomenon. Finally, we will discuss further developments and avenues of future investigation.