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17
On the computational content of convergence proofs via Banach limits
"... This paper addresses new developments in the ongoing proof mining program, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method to extract rates of metastability (i ..."
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This paper addresses new developments in the ongoing proof mining program, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method to extract rates of metastability (in the sense of Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all what is known) rely on the axiom of choice. In this paper we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X. Combined with our method of eliminating Banach limits this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.
Effective metastability of Halpern iterates in CAT(0) spaces
"... This paper provides an effective uniform rate of metastability (in the sense of Tao) on the strong convergence of Halpern iterations of nonexpansive mappings in CAT(0) spaces. The extraction of this rate from an ineffective proof due to Saejung is an instance of the general proof mining program whic ..."
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Cited by 5 (2 self)
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This paper provides an effective uniform rate of metastability (in the sense of Tao) on the strong convergence of Halpern iterations of nonexpansive mappings in CAT(0) spaces. The extraction of this rate from an ineffective proof due to Saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs. This methodology is applied here for the first time to a proof that uses Banach limits and hence makes a substantial reference to the axiom of choice.
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
, 2013
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Gödel functional interpretation and weak compactness
, 2011
"... In recent years, proof theoretic transformations (socalled proof interpretations) that are based on extensions of monotone forms of Gödel’s famous functional (‘Dialectica’) interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content c ..."
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Cited by 3 (1 self)
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In recent years, proof theoretic transformations (socalled proof interpretations) that are based on extensions of monotone forms of Gödel’s famous functional (‘Dialectica’) interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for bounded closed and convex subsets of an abstract (not necessarily separable) Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω ∗ definable from bar recursion (in the sense of Spector) of lowest type. While a case study on the analysis of strong convergence results (due to Browder and Wittmann resp.) that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an
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.
METASTABILITY IN THE FURSTENBERGZIMMER TOWER
, 2009
"... According to the FurstenbergZimmer structure theorem, every measurepreserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measurepreserving systems to provide a perspicuous proof of Szemerédi’s theore ..."
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According to the FurstenbergZimmer structure theorem, every measurepreserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measurepreserving systems to provide a perspicuous proof of Szemerédi’s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measurepreserving system can extend arbitrarily far into the countable ordinals. Here we show that the FurstenbergKatznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the ωωω th level.