Moving averages

S. V. Butler¹ ; J. M. Rosenblatt¹

¹ Department of Mathematics University of Illinois at Urbana-Champaign 273 Altgeld Hall 1409 West Green Street Urbana, IL 61801, U.S.A.

Colloquium Mathematicum, Tome 113 (2008) no. 2, pp. 251-266.

Résumé

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.

Détail

DOI : 10.4064/cm113-2-7

Keywords: ergodic theory certain sequences averages may converge almost everywhere sufficiently rapidly growing subsequence these averages behaved order growth subsequence sufficient often hyperexponential necessarily example averages frac sum subsequence pointwise even infty subsequence pointwise understanding hyperexponential rate growth subsequence required giving simple criteria subject want address here fairly simple description wide class averaging operators which rate growth seen necessary