On a Space of Functions with Entire Laplace Transforms and its Connection with the Optimality of the Ingham-Karamata Theorem

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2025年5月24日

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Morgan Callewaert,
Lenny Neyt,
Jasson Vindas

摘要

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