Generalization of the effective Wiener-Ikehara theorem

TL;DR

This paper extends the Wiener-Ikehara Tauberian theorem to more general conditions, providing an effective error term and broadening its applicability in asymptotic analysis of Laplace transforms.

Contribution

It introduces a generalized Wiener-Ikehara theorem with effective error bounds under broader conditions, including slow decrease and boundary poles.

Findings

Proves asymptotic evaluation under generalized conditions.

Provides explicit error terms for the theorem.

Adapts lemmas of Ganelius and Tenenbaum for effectiveness.

Abstract

We consider the classical Wiener-Ikehara Tauberian theorem, with a generalized condition of slow decrease and some additional poles on the boundary of convergence of the Laplace transform. In this generality, we prove the otherwise known asymptotic evaluation of the transformed function, when the usual conditions of the Wiener-Ikehara theorem hold. However, our version also provides an effective error term, not known thus far in this generalality. The crux of the proof is a proper, asymptotic variation of the lemmas of Ganelius and Tenenbaum, also constructed for the sake of an effective version of the Wiener-Ikehara Theorem.