English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $L^2(X,\Sigma,\mu,\tau)$ be a measure-preserving system, with $\tau$ a$\mathbb{Z}$-action. In this note, we prove that the ergodic averages alonginteger-valued polynomials, $P(n)$, \[ M_N(f):= \frac{1}{N}\sum_{n \leq N}\tau^{P(n)} f \] converge pointwise for $f \in L^2(X)$. We do so by provingthat, for $r>2$, the $r$-variation, $\mathcal{V}^r(M_N(f))$, extends to abounded operator on $L^2$. We also prove that our result is sharp, in that$\mathcal{V}^2(M_N(f))$ is an unbounded operator on $L^2$.

Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain