Discrete sufficient sets for some spaces of entire functions

Abstract:

Let $E$ denote the space of all entire functions $f$ of exponential type (i.e. $|f(z)| = O(\exp (B|z|))$) for some $B > 0$). Let $\mathcal {K}$ denote the space of all positive continuous functions $k$ on the complex plane $C$ with $\exp (B|z|) = O(k(z))$ for each $B > 0$. For $k \in \mathcal {K}$ and $S \subset C$, let $||f|{|_{k,s}} = \sup \{ |f(z)|/k(z):z \in S\}$. We prove that the two families of seminorms ${\{ |||{|_{k,C}}\} _{k \in \mathcal {K}}}$ and ${\{ |||{|_{k,s}}\} _{k \in \mathcal {K}}}$, where \[ S = \{ n + im: - \infty < n,m < + \infty \} \], determine the same topology on $E$.