Pointwise convergence for subsequences of weighted averages

Patrick LaVictoire¹

¹ Department of Mathematics University of Wisconsin at Madison Madison, WI 53706-1388, U.S.A.

Colloquium Mathematicum, Tome 124 (2011) no. 2, pp. 157-168

Résumé

We prove that if $\mu_n$ are probability measures on $\mathbb Z$ such that $\hat \mu_n$ converges to 0 uniformly on every compact subset of $(0,1)$, then there exists a subsequence $\{n_k\}$ such that the weighted ergodic averages corresponding to $\mu_{n_k}$ satisfy a pointwise ergodic theorem in $L^1$. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along $n^2+ \lfloor \rho(n)\rfloor$ for a slowly growing function $\rho$. Under some monotonicity assumptions, the rate of growth of $\rho'(x)$ determines the existence of a “good” subsequence of these averages.

Détail

DOI : 10.4064/cm124-2-2

Keywords: prove probability measures mathbb hat converges uniformly every compact subset there exists subsequence weighted ergodic averages corresponding satisfy pointwise ergodic theorem further discuss relationship between fourier decay pointwise ergodic theorems subsequences considering particular averages along lfloor rho rfloor slowly growing function rho under monotonicity assumptions rate growth rho determines existence subsequence these averages