Moving averages

S. V. Butler ; J. M. Rosenblatt

Colloquium Mathematicae, Tome 111 (2008), p. 251-266 / Harvested from The Polish Digital Mathematics Library

Résumé

In ergodic theory, certain sequences of averages ${A_k f}$ may not converge almost everywhere for all f ∈ L¹(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 $A_k f(x) = 1/(2^k) ∑_{j=4^k+1}^{4^k+2^k} f(T^jx)$, then the subsequence $A_{k²} f$ will not be pointwise good even on $L^∞$, but the subsequence $A_{2^k} f$ 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.

Détail

Publié le : 2008-01-01

Zbl 1154.28008

EUDML-ID : urn:eudml:doc:283991