Illinois Journal of Mathematics

A smoother ergodic average

Karin Reinhold

Full-text: Open access

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 $\lim_{k \rightarrow \infty} \epsilon_{k} = \epsilon \gt 0$

Article information

Source
Illinois J. Math., Volume 44, Issue 4 (2000), 843-859.

Dates
First available in Project Euclid: 19 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1255984695

Digital Object Identifier
doi:10.1215/ijm/1255984695

Mathematical Reviews number (MathSciNet)
MR1804316

Zentralblatt MATH identifier
0983.47007

Subjects
Primary: 47A35: Ergodic theory [See also 28Dxx, 37Axx]
Secondary: 28D10: One-parameter continuous families of measure-preserving transformations

Citation

Reinhold, Karin. A smoother ergodic average. Illinois J. Math. 44 (2000), no. 4, 843--859. doi:10.1215/ijm/1255984695. https://projecteuclid.org/euclid.ijm/1255984695


Export citation