Almost-Everywhere Convergence and Polynomials

Details Almost-Everywhere Convergence and Polynomials Almost-Everywhere Convergence and Polynomials

2009-11-10

arxiv.org/abs/0911.1832v1

Journal Of Modern Dynamics, Volume 2, No. 3, 2008, 465-470

Mathematics

Dynamical Systems

6 pages

Scientific paper

Denote by $\Gamma$ the set of pointwise good sequences. Those are sequences of real numbers $(a_k)$ such that for any measure preserving flow $(U_t)_{t\in \mathbb R}$ on a probability space and for any $f\in L^\infty$, the averages $\frac{1}{n} \sum_{k=1}^{n} f(U_{a_k}x) $ converge almost everywhere. We prove the following two results. [1.] If $f: (0,\infty)\to\mathbb R$ is continuous and if $\big(f(ku+v)\big)_{k\geq 1}\in\Gamma$ for all $u, v>0$, then $f$ is a polynomial on some subinterval $J\subset (0,\infty)$ of positive length. [2.] If $f: [0,\infty)\to\mathbb R$ is real analytic and if $\big(f(ku)\big)_{k\geq 1}\in\Gamma$ for all $u>0$, then $f$ is a polynomial on the whole domain $[0,\infty)$. These results can be viewed as converses of Bourgain's polynomial ergodic theorem which claims that every polynomial sequence lies in $\Gamma$.

Affiliated with

Also associated with

No associations

Rating

Almost-Everywhere Convergence and Polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Almost-Everywhere Convergence and Polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Almost-Everywhere Convergence and Polynomials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-50980