Quantified versions of Ingham's theorem

TL;DR

This paper develops quantified versions of Ingham's Tauberian theorem, providing more general and sharper decay estimates for $C_0$-semigroups, and clarifies the role of the 'fudge factor' in contour integral estimates.

Contribution

It introduces a natural modification of Ingham's proof to obtain more general and sharper quantified decay estimates, enhancing understanding of the 'fudge factor' in contour integral methods.

Findings

Quantified decay estimates for $C_0$-semigroups

More general and sharper results than existing literature

Clarification of the 'fudge factor' in contour integral proofs

Abstract

We obtain quantified versions of Ingham's classical Tauberian theorem and some of its variants by means of a natural modification of Ingham's own simple proof. As corollaries of the main general results, we obtain quantified decay estimates for $C_0$-semigroups. The results reproduce those known in the literature but are both more general and, in one case, sharper. They also lead to a better understanding of the previously obscure "fudge factor' appearing in proofs based on estimating contour integrals.