A general quantified Ingham-Karamata Tauberian theorem

TL;DR

This paper introduces a generalized quantified Ingham-Karamata Tauberian theorem with flexible conditions, improving previous results by removing growth restrictions and establishing optimal rates in most cases.

Contribution

It extends the Ingham-Karamata Tauberian theorem with a more flexible one-sided condition and removes growth restrictions, achieving optimal rates.

Findings

Improved theorem by removing growth restrictions.

Established optimality of the quantified rate.

Enhanced applicability under various boundary behaviors.

Abstract

We provide a general quantified Ingham-Karamata Tauberian theorem with a flexible one-sided Tauberian condition under several types of boundary behavior for the Laplace transform. Our results in particular improve a theorem by Stahn, removing a vexing restriction on the growth of the Laplace transform. Improving existing optimality results, we also show that the obtained quantified rate is optimal in almost all cases.