Generalization of the Wiener-Ikehara theorem

TL;DR

This paper explores the Wiener-Ikehara theorem's generalization under specific conditions and clarifies the relationship between two hypotheses on the Laplace transform in its exact form.

Contribution

It extends the Wiener-Ikehara theorem to log-linearly slowly decreasing functions and clarifies the connection between key hypotheses on the Laplace transform.

Findings

Extended Wiener-Ikehara theorem under new conditions

Clarified the relationship between hypotheses on the Laplace transform

Provided insights into the exact form of the theorem

Abstract

We study the Wiener-Ikehara theorem under the so-called log-linearly slowly decreasing condition. Moreover, we clarify the connection between two different hypotheses on the Laplace transform occurring in exact forms of the Wiener-Ikehara theorem, that is, in "if and only if" versions of this theorem.