Convergence and divergence of averages along subsequences in certain Orlicz spaces
Abstract
Description
The classical theorem of Birkhoff states that the $T^N f(x) = (1/N)\sum_{k=0}^{N-1} f(σ^k x)$ converges almost everywhere for $x\in X$ and $f\in L^{1}(X)$, where $σ$ is a measure preserving transformation of a probability measure space $X$. It was shown that there are operators of the form $T^N f(x)=(1/N)\sum_{k=0}^{N-1}f(σ^{n_k}x)$ for a subsequence $\{n_k\}$ of the positive integers that converge in some $L^p$ spaces while diverging in others. The topic of this talk will examine this phenomenon in the class of Orlicz spaces $\{L{Log}^βL:β>0\}$.