On Almost Convergence on Locally Compact Abelian Groups

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. These results are closely related to some of the classical Tauberian theorems like the Ingham–Karamata and Katznelson–Tzafriri theorems.