Pointwise convergence for subsequences of weighted averages

DOI10.4064/CM124-2-2zbMATH Open1228.42024arXiv0911.3927OpenAlexW2964173930MaRDI QIDQ3174464

Patrick La Victoire

Publication date: 14 October 2011

Published in: Colloquium Mathematicum (Search for Journal in Brave)

Abstract: We prove that if $m{u}_n$ are probability measures on $Z$ such that $hatm{u}_n$ 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 $m{u}_{n_k}$ 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 $n^2+lfloorho(n)floor$ for a slowly growing function $ho$. Under some monotonicity assumptions, the rate of growth of $h{o}^{\prime }(x)$ determines the existence of a "good" subsequence of these averages.

Full work available at URL: https://arxiv.org/abs/0911.3927

zbMATH Keywords

ergodic theoremgood sequencessubsequence averagesweighted averages

Mathematics Subject Classification ID

Maximal functions, Littlewood-Paley theory (42B25) Ergodic theorems, spectral theory, Markov operators (37A30)

Related Items (2)

Integer sequences with big gaps and the pointwise ergodic theorem ⋮ Pointwise convergence of nonconventional averages

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