A smoother ergodic average

Abstract

We study the pointwise behavior of the smoothed out averages $$P_{n}f(x)= \frac{1}{n} \sum_{k=1}^{n} \frac{1}{\epsilon_{k}} \int_{|t| \lt \epsilon_{k}/2} {f(T_{k+t}x)dt},$$ where $T_{t}$ is a measure preserving flow on a probability space. We show that these are good averages in $L^{P}$, $p \gt 1$, if $\epsilon_{k}$ is a convergent sequence or if they are given by stationary random variables. When $p = 1$ the averages are good if $\lim_{k \rightarrow \infty} \epsilon_{k} = \epsilon \gt 0$