Functional analytic approach to Cesàro mean

Abstract

We study a certain class $\mathcal{P}$ of positive linear functionals $\phi$ on $L^{\infty }([1,\infty ))$ for which $\phi (f)=\alpha$ 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.

14 pages

References in corpus (1)

• Extreme points of the set of density measures