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

Abstract

We study approximation properties of the Fréchet space of all continuously differentiable functions $\tau$ such that ${\tau }^{\prime }(x)=o(1)$ 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.

arXiv comment11 pages, 2 figures

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