Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

Abstract

A holomorphic function on a planar domain \(\Omega \) is said to possess a universal Taylor series about a point \(\zeta \) of \(\Omega \) if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in \(\mathbb {C}\backslash \Omega \) that have connected complement. In the case where \(\Omega \) is simply connected, such functions are known to be unbounded and to form a collection that is independent of the choice of \(\zeta \). This paper uses tools from potential theory to show that, even for domains \(\Omega \) of arbitrary connectivity, such functions are unbounded whenever they exist. In the doubly connected case, a further analysis of boundary behaviour reveals that the collection of such functions can depend on the choice of \(\zeta \). This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.