Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior

TL;DR

This paper develops advanced Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior, generalizing classical results and relaxing boundary conditions, with applications to the Katznelson-Tzafriri theorem.

Contribution

It introduces new Tauberian theorems using local pseudofunction boundary behavior, allowing minimal boundary conditions and null boundary singularities.

Findings

Generalized Ingham-Fatou-Riesz and Wiener-Ikehara theorems

Relaxed boundary requirements for Laplace transforms

Refined results for the Katznelson-Tzafriri theorem

Abstract

We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberian hypotheses; in this context, we also establish new boundedness theorems for Laplace transforms with pseudomeasure boundary behavior. As an application, we refine various results related to the Katznelson-Tzafriri theorem for power series.