Moving averages

Tom 113 / 2008

S. V. Butler, J. M. Rosenblatt Colloquium Mathematicum 113 (2008), 251-266 MSC: 28D05, 37A05, 37A30. DOI: 10.4064/cm113-2-7

Streszczenie

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.

Autorzy

  • S. V. ButlerDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    273 Altgeld Hall
    1409 West Green Street
    Urbana, IL 61801, U.S.A.
    e-mail
  • J. M. RosenblattDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    273 Altgeld Hall
    1409 West Green Street
    Urbana, IL 61801, U.S.A.
    e-mail

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