Harmonic Analogues of G. R. Maclane's Universal Functions

Abstract

Let E denote the space of all entire functions, equipped with the topology of local uniform convergence (the compact-open topology). MacLane [15] constructed an entire function f whose sequence of derivatives (f, f′, f″, …) is dense in E; his construction is succinctly presented in a much later note by Blair and Rubel [2], who unwittingly rederived it (see also [3]). We shall call such a function f a universal entire function. In this note we show that analogous universal functions exist in the space HN of functions harmonic on ℝN, where N⩾2. We also study the permissible growth rates of universal functions in HN and show that the set of all such functions is very large. For purposes of comparison, we first review relevant facts about universal entire functions. The function constructed by MacLane is of exponential type 1. Duyos Ruiz [7] observed that a universal entire function cannot be of exponential type less than 1. G. Herzog [11] refined MacLane's growth estimate by proving the existence of a universal entire function f such that ∣f(z)∣=O(rer) as ∣z∣=r→∞. Finally, Grosse–Erdmann [10] proved the following sharp result.