Distributional versions of Littlewood's Tauberian theorem

TL;DR

This paper extends Littlewood's Tauberian theorem to Laplace transforms of Schwartz distributions, providing new versions applicable to power series and offering a simplified proof of a classical result.

Contribution

It introduces several generalized versions of Littlewood's Tauberian theorem for Schwartz distributions and applies them to power series and classical theorems.

Findings

New Tauberian theorems for Laplace transforms of Schwartz distributions

Cesàro summability follows from Abel summability for certain power series

Simplified proof of Littlewood's classical one-sided Tauberian theorem

Abstract

We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series where Ces\`{a}ro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.