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A quantitative mean ergodic theorem for uniformly convex Banach spaces

  • U. KOHLENBACH (a1) and L. LEUŞTEAN (a1) (a2)

Abstract

We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of Tao of the mean ergodic theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad et al [Local stability of ergodic averages. Trans. Amer. Math. Soc. to appear] and Tao [Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys.28(2) (2008), 657–688].

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[1]Avigad, J., Gerhardy, P. and Towsner, H.. Local stability of ergodic averages. Trans. Amer. Math. Soc. to appear.
[2]Birkhoff, G.. The mean ergodic theorem. Duke Math. J. 5(1) (1939), 1920.
[3]Clarkson, J. A.. Uniformly convex spaces. Trans. Amer. Math. Soc. 40 (1936), 396414.
[4]Gerhardy, P. and Kohlenbach, U.. General logical metatheorems for functional analysis. Trans. Amer. Math. Soc. 360 (2008), 26152660.
[5]Kohlenbach, U.. Uniform asymptotic regularity for Mann iterates. J. Math. Anal. Appl. 279 (2003), 531544.
[6]Kohlenbach, U.. Some logical metatheorems with application in functional analysis. Trans. Amer. Math. Soc. 357 (2005), 89128.
[7]Kohlenbach, U.. Effective uniform bounds from proofs in abstract functional analysis. New Computational Paradigms: Changing Conceptions of What is Computable. Eds. B. Cooper, B. Löwe and A. Sorbi. Springer, Berlin, 2008, pp. 223258.
[8]Kohlenbach, U.. Applied Proof Theory: Proof Interpretations and Their Use in Mathematics (Springer Monographs in Mathematics). Springer, Berlin, 2008.
[9]Kohlenbach, U. and Leuştean, L.. Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. J. Eur. Math. Soc. to appear.
[10]Tao, T.. Soft analysis, hard analysis, and the finite convergence principle. Structures and Randomness: Pages from Year One of a Mathematical Blog. American Mathematical Society, Providence, RI, 2008.
[11]Tao, T.. Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys. 28(2) (2008), 657688.

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A quantitative mean ergodic theorem for uniformly convex Banach spaces

  • U. KOHLENBACH (a1) and L. LEUŞTEAN (a1) (a2)

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