Regularly log-periodic functions and some applications

TL;DR

This paper develops a Tauberian theorem for Laplace--Stieltjes transforms within the context of regularly log-periodic functions and applies it to precisely characterize the tail behavior of fixed points in smoothing transforms.

Contribution

It introduces new Tauberian and Karamata-type theorems for regularly log-periodic functions, expanding the analytical tools available for such functions.

Findings

Established a Tauberian theorem for Laplace--Stieltjes transforms of regularly log-periodic functions.

Derived Karamata-type theorems applicable to this function class.

Determined the exact tail behavior of fixed points in certain smoothing transforms.

Abstract

We prove a Tauberian theorem for the Laplace--Stieltjes transform and Karamata-type theorems in the framework of regularly log-periodic functions. As an application we determine the exact tail behavior of fixed points of certain type smoothing transforms.