Journal of the
Korean Mathematical Society JKMS

An entire function $f \in {\mathcal H}(\mathbb{C})$ is called universal with respect to translations if for any $g \in {\mathcal H}(\mathbb{C}), R > 0,$ and $\epsilon > 0,$ there is $n \in \mathbb{N}$ such that $|f(z+n) - g(z)| < \epsilon$ whenever $|z| \leq \ $R. Similarly, it is universal with respect to differentiation if for any $g,R,$ and $\epsilon,$ there is $n$ such that $|f^{(n)}(z) - g(z)| < \epsilon$ for $|z| \leq$ R. In this note, 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.

Keywords: hypercyclic, analytic functions, convolution operators

MSC numbers: Primary 47A16, 32DXX; Secondary 46B37