Pointwise Convergence for Subsequences of Weighted Averages

Details Pointwise Convergence for Subsequences of Weighted Averages Pointwise Convergence for Subsequences of Weighted Averages

2009-11-20

arxiv.org/abs/0911.3927v3

Mathematics

Classical Analysis and ODEs

LaTeX, 11 pages; corrected from previous version (which included an erroneous minor result)

Scientific paper

We prove that if $\mu_n$ are probability measures on $Z$ 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.

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