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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Polynomially moving ergodic averages


Author: Mark Schwartz
Journal: Proc. Amer. Math. Soc. 103 (1988), 252-254
MSC: Primary 28D05
DOI: https://doi.org/10.1090/S0002-9939-1988-0938678-0
MathSciNet review: 938678
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Abstract: Given an increasing sequence of positive integers $ \left\{ {{m_n}} \right\}$, a non-decreasing sequence of positive integers $ \left\{ {{b_n}} \right\}$, and a measurable, measure-preserving ergodic transformation $ \tau $ on a probability space $ \left( {\Omega ,\mathcal{F},\mu } \right)$, the a.s. convergence of the moving averages $ {T_n}\left( f \right) = b_n^{ - 1}\sum\nolimits_{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left( {{\tau ^k}} \right)} $ is considered, for $ f \in {L_p}\left( \Omega \right)$. A counterexample is constructed in the case of polynomial-like $ \left\{ {{m_n}} \right\}$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0938678-0
Article copyright: © Copyright 1988 American Mathematical Society

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