Cesaro averaging and extension of functionals on Loo(0, oo)

On the space of essentially bounded functions L & INFIN;(0, oo) we consider the Cesaro f x averaging operator Jf (x) := 1 0 f (t) dt. We then extend the concept of integer x iterates of Cesaro averaging Jn, to an operator of the form Jrf (x), where r is any positive real number and f E L & INFIN;(0, oo). Our definition of fractional powers of Cesaro averaging is such that (Jr)r>0 has the semigroup property. Our paper contains the following result: [ For any f E L & INFIN;(0, oo), 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 ? on L & INFIN;(0, oo) that are invariant under our continuous generalization of Cesaro iterates Jr. We also construct an example of a Banach limit & psi; on L & INFIN;(0, oo) that preserves Cesaro convergence, but is not Cesaro invariant.& COPY; 2023 Elsevier Inc. All rights reserved.