On almost convergence on locally compact abelian groups

TL;DR

This paper investigates a summability method called almost convergence for functions on locally compact abelian groups, providing necessary and sufficient conditions and applications to Tauberian theorems, including an analogue of Wiener-Ikehara.

Contribution

It introduces a new framework for almost convergence using topologically invariant means and establishes conditions and applications on groups like integers and reals.

Findings

Characterization of almost convergence via analytic and functional conditions

Necessary and sufficient conditions for almost convergence

Complex Tauberian theorems including Wiener-Ikehara analogue

Abstract

We study a summability method called almost convergence for bounded measurable functions defined on a locally compact abelian group. We define almost convergence using topologically invariant means and exhibit two different kinds of necessary and sufficient conditions, one is analytic and the other is functional analytic, for a given function to be almost convergent. As an application, we show complex Tauberian theorems for almost convergence on the integers and the real numbers. In particular, the latter one can be viewed as an analogue of the Wiener-Ikehara theorem.