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On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem (2407.15547v2)
Published 22 Jul 2024 in math.CA, math.CV, math.FA, and math.NT
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.
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