Convergence and divergence of averages along subsequences in certain Orlicz spaces

Details Convergence and divergence of averages along subsequences in certain Orlicz spaces Convergence and divergence of averages along subsequences in certain Orlicz spaces

2009-01-07

arxiv.org/abs/0901.0932v1

Mathematics

Dynamical Systems

Scientific paper

The classical theorem of Birkhoff states that the $T^N f(x) = (1/N)\sum_{k=0}^{N-1} f(\sigma^k x)$ converges almost everywhere for $x\in X$ and $f\in L^{1}(X)$, where $\sigma$ is a measure preserving transformation of a probability measure space $X$. It was shown that there are operators of the form $T^N f(x)=(1/N)\sum_{k=0}^{N-1}f(\sigma^{n_k}x)$ 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 $\{L{Log}^\beta L:\beta>0\}$.

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