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Pointwise convergence for subsequences of weighted averages

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LaVictoire P.

Colloquium Mathematicum

2011

tom 124

nr 2

157-168

We prove that if μₙ are probability measures on ℤ such that μ̂ₙ 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 $μ_{n_{k}}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions, the rate of growth of ρ'(x) determines the existence of a "good" subsequence of these averages.

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1 Colloquium Mathematicum

1 LaVictoire P.

1 2011

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Pointwise convergence for subsequences of weighted averages