Convergence Rates of Regularized Approximation Processes

Abstract

We define the concept of an A-regularized approximation process and prove for it uniform convergence theorems and strong convergence theorems with optimal and non-optimal rates. The sharpness of non-optimal convergence is also established. The general results provide a unified approach to dealing with convergence rates of various approximation processes, and also of local ergodic limits as well. As applications, approximation theorems, and local Abelian and Cesáro ergodic theorems with rates are deduced for n -times integrated solution families for Volterra integral equations, which include n -times integrated semigroups and cosine functions as special cases. Applications to ( Y )-semigroups and tensor product semigroups are also discussed.