On Almost convergence on the real line and its application to bounded analytic functions

TL;DR

This paper explores topologically invariant means and almost convergence on the real line, extending Raimi's concept and applying it to analyze the asymptotic behavior of bounded analytic functions in Hardy spaces.

Contribution

It generalizes Raimi's analytic characterization of almost convergence and links it to the asymptotic properties of bounded analytic functions in Hardy spaces.

Findings

Extended Raimi's characterization of almost convergence.

Established relation between asymptotic behavior on imaginary axis and infinity.

Applied results to Hardy space theory.

Abstract

We address the study of topologically invariant means and almost convergence on the real numbers $\mathbb{R}$. Here, the former is a certain class of invariant means on $L^{\infty}(\mathbb{R})$ and the latter is a summability method defined by them. Almost convergence on $\mathbb{R}$ was firstly introduced by Raimi (1957) as a generalization of Lorentz's almost convergence for bounded sequences. We extensively generalize his result of analytic characterization of almost convergence and explore its application to the theory of Hardy space. Specifically, we establish the relation between the asymptotic behavior on the imaginary axis and that at infinity of bounded analytic functions defined on the right half plane.