Let [Escr ] 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 [Escr ]; 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
[Hscr ]N of functions harmonic on ℝN,
where
N[ges ]2. We also study the permissible growth
rates of universal functions in [Hscr ]N 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
[mid ]f(z)[mid ]=O(rer)
as [mid ]z[mid ]=r→∞. Finally,
Grosse–Erdmann [10] proved the following sharp
result.