Moving averages
Volume 113 / 2008
Abstract
In ergodic theory, certain sequences of averages $\{A_kf\}$ may not converge almost everywhere for all $f \in L^1(X)$, but a sufficiently rapidly growing subsequence $\{A_{m_k}f\}$ of these averages will be well behaved for all $f$. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are $$ A_kf(x) = \frac 1{2^k} \sum _{j=4^k+1}^{4^k+2^k} f(T^jx), $$ then the subsequence $A_{k^2}f$ will not be pointwise good even on $L^\infty$, but the subsequence $A_{2^k}f$ will be pointwise good on $L^1$. 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.
