Convergence Rates of Ergodic Limits and Approximate Solutions Original Research Article

This paper is concerned with the convergence rates of two processes {Aα} and {Bα}, under the assumption that ||Aα|| = O(1) and there is a closed operator A such that BαA ⊂ ABα = I − Aα, ||AAα|| = O(e(α)), and B*αx* = φ(α)x* for x* ∈ R(A)⊥, where e(α) → 0 and |φ(α)| → ∞. It was previously proved that {Aα} converges strongly on N(A) ⊕ R(A) to P, the projection onto N(A) along R(A), and {Bα} converges strongly on A(D(A) ∩ R(A)) to A−11 the inverse operator of A1 = A | R(A). In this paper, the two processes are shown to be saturated with order O(e(α)), and their saturation classes are characterized. The result provides a unified approach to convergence rates for many particular mean ergodic theorems and for various methods of solving the equation Ax = y. We discuss in particular applications to integrated semigroups, cosine operator functions, and tensor product semigroups