  We study the equation $m(D)f = 0$ in a large class of sub-exponentially
growing functions. Under appropriate restrictions on $m \in C(\mathbb{R}^n)$,
we show that every such solution can be analytically continued to a
sub-exponentially growing entire function on $\mathbb{C}^n$ if and only if
$m(\xi) \not= 0$ for $\xi \not= 0$.
