  We study approximation properties of the Fr\'{e}chet space of all
continuously differentiable functions $\tau$ such that $\tau'(x)=o(1)$ and such
that their Laplace transforms admit entire extensions to $\mathbb{C}$. As an
application, these approximation results are combined with the open mapping
theorem to show the optimality theorem for the Ingham-Karamata Tauberian
theorem.
