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Non-Optimal Rates of Ergodic Limits and Approximate Solutions

✍ Scribed by Sen-Yen Shaw

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 294 KB
Volume: 94
Category: Article
ISSN: 0021-9045
DOI: 10.1006/jath.1998.3186

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✦ Synopsis

This paper is concerned with non-optimal rates of convergence for two processes [A : ] and [B : ], which satisfy &A : &=O(1), B : A/AB : =I&A : , &AA : &=O(e(:)), where A is a closed operator and e(:) Ä 0. Under suitable conditions, we describe, in terms of K-functionals, those x (resp. y) for which the non-optimal convergence rate of [A : x] (resp. [B : y]) is of the order O( f (:)), where f is a function satisfying e(:) f (:) Ä 0. In case that f (:)Âe(:) Ä , the sharpness of the non-optimal rate of [A : x] is equivalent to that A has 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 equation Ax= y. We discuss in particular applications to :-times integrated semigroups, n-times integrated cosine functions, and tensor product semigroups.

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