Uniform equivalence of parametric norms in ergodic and approximation theories

Let $\{T(X)\}$ be a uniformly bounded one-parameter strongly continuous group of operators on a Banach space. Suppose its corresponding ergodic projection exists and is zero. Then we can define a certain countable family of parametric norms (norms depending on a positive parameter $t$) whose rate of decay with respect to $t$ at a given element involves the “ergodic” properties of that element. They ate called the ergodic moduli of a given (positive integral) order. This article is basically devoted to explaining the properties of ergodic moduli and to finding one- and two-sided estimates for them in terms of other parametric norms. An interesting analogy can be drawn between the properties of ergodic moduli and those of the smoothness moduli of functions as studied in approximation theory.