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

TL;DR

This paper investigates a special function space with entire Laplace transforms, demonstrating its approximation properties and using these results to establish the optimality of the Ingham-Karamata Tauberian theorem.

Contribution

It introduces a new function space with specific Laplace transform properties and proves its approximation capabilities, leading to an optimality result for a classical Tauberian theorem.

Findings

Established approximation properties of the function space.

Proved the optimality of the Ingham-Karamata theorem.

Connected function space properties with Tauberian theorem optimality.

Abstract

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.