Non-Optimal Rates of Ergodic Limits and Approximate Solutions Original Research Article

This paper is concerned with non-optimal rates of convergence for two processes {Aα} and {Bα}, which satisfy ‖Aα‖=O(1),BαA⊂ABα=I−Aα, ‖AAα‖=O(e(α)), whereAis a closed operator ande(α)→0. Under suitable conditions, we describe, in terms ofK-functionals, thosex(resp.y) for which the non-optimal convergence rate of {Aαx} (resp. {Bαy}) is of the orderO(f(α)), wherefis a function satisfyinge(α)⩽f(α)→0. In case thatf(α)/e(α)→∞, the sharpness of the non-optimal rate of {Aαx} is equivalent to thatAhas non-closed range. The result provides a unified approach to dealing with non-optimal rates for many particular mean ergodic theorems and for various methods of solving the equationAx=y. We discuss in particular applications toα-times integrated semigroups,n-times integrated cosine functions, and tensor product semigroups.