A functional analytic approach to Cesàro mean

We study the class P of positive linear functionals φ on L^∞([1,∞)) for which φ(f)=α if [Formula presented]. The semigroup of translations f(x)↦f(rx) on L^∞([1,∞)), where r∈[1,∞), plays a crucial role in the study of P. In particular, we give three different expressions of their extremal values, which can be considered main results of this paper. We also study linear functionals on l_∞, the set of all real-valued bounded functions on natural numbers N, which extend Cesàro mean and give similar results about their extremal values, including a functional analytic proof of the classical result of Pólya.