Cesàro averaging and extension of functionals on L∞(0,∞)

Abstract

On the space of essentially bounded functions L∞(0,∞) we consider the Cesàro averaging operator Jf(x):=1x∫0xf(t)dt. We then extend the concept of integer iterates of Cesàro averaging Jn, to an operator of the form Jrf(x), where r is any positive real number and f∈L∞(0,∞). Our definition of fractional powers of Cesàro averaging is such that (Jr)r>0 has the semigroup property. Our paper contains the following result: [ For any f∈L∞(0,∞), Jrf(x) has a limit at infinity for some r>0, if and only if Jsf(x) has a limit at infinity for any s>0. In this case, the limit values are all the same]. We present a strong quantitative version of the special case where 0<r≤1 and s=1+r. We construct Banach limits Λ on L∞(0,∞) that are invariant under our continuous generalization of Cesàro iterates Jr. We also construct an example of a Banach limit Ψ on L∞(0,∞) that preserves Cesàro convergence, but is not Cesàro invariant.