ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 41, 47 Shaw, Sen-Yen, and Liu, Hsiang, National Central University, Taiwan Convergence rates of regularized approximation processes Let {T_{α}}, {S_{α}} be two nets of bounded linear operators such that ∥T_{α}∥ = O(α), S_{α} → I strongly, and S_{α}A ⊂ AS_{α} = (e(α))^{−1}(T_{α} − I), where A is a closed linear operator and e(α) → 0. It is shown that {T_{α}} is saturated with the order O(e(α)), its saturation class being the completion of the domain D(A) of A relative to the space X. Under suitable conditions, we further describe, in terms of K-functionals, those x for which the convergence rate of {T_{α}x} is of the order O(f(α)), where f satisfies 0 < e(α) ≤ f(α) → 0. In case A is unbounded and f(α)/e(α) → ∞, the non-optimal rate is seen to be sharp. The general results provide a unified approach to dealing with convergence rates of many particular approximation processes.