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12
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
, 2013
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Ultraproducts and metastability
 NEW YORK J. MATH. 19 (2013) 713–727.
, 2013
"... Given a convergence theorem in analysis, under very general conditions a modeltheoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory. ..."
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Given a convergence theorem in analysis, under very general conditions a modeltheoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory.
Effective results on nonlinear ergodic averages in CAT(κ) spaces
, 2013
"... In this paper we apply proof mining techniques to compute, in the setting of CAT(κ) spaces (with κ> 0), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a un ..."
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In this paper we apply proof mining techniques to compute, in the setting of CAT(κ) spaces (with κ> 0), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.
Quantitative results on Fejér monotone sequences
, 2015
"... We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejér monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of soca ..."
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We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejér monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of socalled metastability in the sense of T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (xn) for firmly nonexpansive, asymptotically nonexpansive, strictly pseudocontractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (socalled Whyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)spaces due to Gromov.
Norm convergence of nilpotent ergodic averages along Følner nets. Preprint, available online at arXiv.org
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