Functional analytic approach to Ces\`aro mean

TL;DR

This paper explores a class of positive linear functionals related to Cesàro means on function spaces, analyzing their structure via semigroup actions and providing bounds for their maximal values.

Contribution

It introduces a functional analytic framework for Cesàro means using semigroup actions, offering new insights into their structure and bounds.

Findings

Characterization of positive linear functionals related to Cesàro means.

Analysis of the role of semigroup actions in the structure of these functionals.

Expressions for the maximal values of these functionals for given functions.

Abstract

We study a certain class $\mathcal{P}$ of positive linear functionals $\varphi$ on $L^{\infty}([1,\infty))$ for which $\varphi(f) = \alpha$ if $\lim_{x \to \infty} \frac{1}{x} \int_1^x f(t)dt = \alpha$. It turns out that translations $f(x) \mapsto f(rx)$ on $L^{\infty}([1, \infty))$, where $r \in [1, \infty)$, which are induced by the action of the multiplicative semigroup $[1, \infty)$ on itself, plays an intrinsic role in the study of $\mathcal{P}$. We also deal with an analogue $\mathcal{K}$ of $\mathcal{P}$ of positive linear functionals on $L^{\infty}([0, \infty))$ partaining to the action of the additive semigroup $[0, \infty)$ on itself. In particular, we give some expressions of maximal possible values of $\mathcal{P}$ and $\mathcal{K}$ for a given function respectively.