Pointwise convergence for subsequences of weighted averages
Colloquium Mathematicum, Tome 124 (2011) no. 2, pp. 157-168.

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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.
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

Patrick LaVictoire 1

1 Department of Mathematics University of Wisconsin at Madison Madison, WI 53706-1388, U.S.A.
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Patrick LaVictoire. Pointwise convergence for subsequences of weighted averages. Colloquium Mathematicum, Tome 124 (2011) no. 2, pp. 157-168. doi : 10.4064/cm124-2-2. http://geodesic.mathdoc.fr/articles/10.4064/cm124-2-2/

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