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Ergodic averaging sequences

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Abstract

We consider generalizations of the pointwise and mean ergodic theorems to ergodic theorems averaging along different subsequences of the integers or real numbers. The Birkhoff and Von Neumann ergodic theorems give conclusions about convergence of average measurements of systems when the measurements are made at integer times. We consider the case when the measurements are made at timesa(n) or ([a(n)]) where the functiona(x) is taken from a class of functions called a Hardy field, and we also assume that |a(x)| goes to infinity more slowly than some positive power ofx. A special, well-known Hardy field is Hardy’s class of logarithmico-exponential functions.

The main theme of the paper is to point out that for a functiona(x) as described above, a complete characterization of the ergodic averaging behavior of the sequence ([a(n)]) is possible in terms of the distance ofa(x) from (certain) polynomials.

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Authors and Affiliations

  1. Department of Mathematics-MS 136, Rice University, 6100 Main Street, 77005-1892, Houston, TX, USA

    Michael Boshernitzan

  2. Department of Mathematics, California State University, 90032, Los Angeles, CA, USA

    Grigori Kolesnik

  3. Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, 38152-3240, Memphis, TN, USA

    Anthony Quas & Máté Wierdl

Authors
  1. Michael Boshernitzan
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  2. Grigori Kolesnik
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  3. Anthony Quas
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  4. Máté Wierdl
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Corresponding author

Correspondence to Michael Boshernitzan.

Additional information

This research was supported by grants from the NSF.

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Boshernitzan, M., Kolesnik, G., Quas, A. et al. Ergodic averaging sequences. J. Anal. Math. 95, 63–103 (2005). https://doi.org/10.1007/BF02791497

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  • DOI: https://doi.org/10.1007/BF02791497

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