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Proceedings of the American Mathematical Society

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

 

 

Discrepancy of behavior of perturbed sequences in $ L\sp p$ spaces


Author: Karin Reinhold-Larsson
Journal: Proc. Amer. Math. Soc. 120 (1994), 865-874
MSC: Primary 28D05; Secondary 47A35
MathSciNet review: 1169889
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Abstract: Given $ p \in [1,\infty )$, examples of sequences $ {\{ {n_k}\} _{k \subset \mathbb{N}}}$ such that for any ergodic dynamical system $ (X,\beta ,m,T)$ the averages

$\displaystyle {A_N}f(x) = \frac{1} {N}\sum\limits_{k = 1}^N {f({T^{{n_k}}}x)} $

converge almost everywhere in all $ {L^q}(X),\,q > p$, but fail to have a finite limit for some function in $ {L^p}(X)$ are shown. Also, sequences such that for all ergodic dynamical systems the averages $ {A_N}f(x)$ do not converge for some function $ f \in {L^p}(X)$ for all $ 1 \leqslant p < \infty $ but do converge for all functions in $ {L^\infty }(X)$ are shown.

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DOI: https://doi.org/10.1090/S0002-9939-1994-1169889-2
Article copyright: © Copyright 1994 American Mathematical Society