Let (X, B , P) be a non-atomic probability space and let T be an invertible measure-preserving transformation of ( X, B , P). Fix a sequence (mk ) in Z and let f ∈ Lp( X), 1 ≤ p ≤ infinity. We know that, depending on what the powers are, the averages 1n k=1nfTmk x may or may not converge a.e. x ∈ X, and they may or may not stay bounded a.e. We consider the properties of sequences (Ln) of real numbers and ( wn) of positive integers so that 1Ln k=1wnf Tmkx and 1Lnsup 1≤k≤n1k j=1k fTmjx converge a.e. x ∈ X for any sequence (mk) in Z .
University of Illinois - Urbana-Champaign
Networked Digital Library of Theses and Dissertations
University of Illinois at Urbana-Champaign