 
Summary: ON UNIVERSAL FUNCTIONS OCTOBER 29, 2002
RICHARD ARON AND DINESH MARKOSE
Abstract. An entire function f H(C) is called universal with respect to
translations if for any g H(C), R > 0, and > 0, there is n N such that
f(z + n)  g(z) < whenever z < . Similarly, it is universal with respect to
differentiation if for any g, R, and , there is n such that f(n)(z)g(z) < for
z R. In this survey article, we review G. MacLane's proof of the existence
of universal functions with respect to differentiation, and we give a simplified
proof of G. D. Birkhoff's theorem showing the existence of universal functions
with respect to translation. We also discuss Godefroy and Shapiro's extension
of these results to convolution operators as well as some new, related results
and problems.
1. Introduction.
Our interest here will be in what has come to be called hypercyclic oper
ators on the space H(C) of entire functions of one complex variable. This
subject has its origins in 1929 with the paper [2] by G. D. Birkhoff, in
which he proved that there is f H(C) such that the set of all translates
{f(z), f(1 + z), ..., f(n + z), ...} is dense in H(C). About 25 years later, G.
MacLane [9] proved an analogous result for derivatives: There is an entire
function f such that the set of all derivatives {f, f , ..., f(n), ...} is dense in
