Moving averages

Abstract

In ergodic theory, certain sequences of averages {Akf} may not converge almost everywhere for all f ∈ L1(X), but a sufficiently rapidly growing subsequence {Amkf} of these averages will be well behaved for all f. The order of growth of this sub- sequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are (Formula Presented) then the subsequence {Ak2f} will not be pointwise good even on L∞, but the subsequence {A2kf} will be pointwise good on L1. Understanding when the hyperexponential rate of growth of the subsequence is required, and giving simple criteria for this, is the subject that we want to address here.We give a fairly simple description of a wide class of averaging operators for which this rate of growth can be seen to be necessary.