On a space of functions with entire Laplace transforms and its
connection with the optimality of the Ingham-Karamata theorem
On a space of functions with entire Laplace transforms and its
connection with the optimality of the Ingham-Karamata theorem
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 …