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15
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 45 (26 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
On the computational content of the Krasnoselski and Ishikawa fixed point theorems
, 2000
"... This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general ..."
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Cited by 13 (10 self)
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This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general socalled KrasnoselskiMann iterations. These iterations converge to fixed points of f under certain compactness conditions. But, as we show, already for uniformly convex spaces in general no bound on the rate of convergence can be computed uniformly in f . This is related to the nonuniqueness of fixed points. However, the iterations yield even without any compactness assumption and for arbitrary normed spaces approximate fixed points of arbitrary quality for bounded C (asymptotic regularity, Ishikawa 1976). We apply proof theoretic techniques (developed in previous papers of us) to noneffective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...
Rates of Asymptotic Regularity for Halpern Iterations of Nonexpansive Mappings
, 2008
"... In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, prooftheoretic techniques. We give effective rates of asymptotic regularity for the Halpern iterations of nonexpansive selfmappings of nonemp ..."
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Cited by 7 (3 self)
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In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, prooftheoretic techniques. We give effective rates of asymptotic regularity for the Halpern iterations of nonexpansive selfmappings of nonempty convex sets in normed spaces. The paper presents another case study in the project of proof mining, which is concerned with the extraction of effective uniform bounds from (primafacie) ineffective proofs.
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.
A metastable dominated convergence theorem
, 2012
"... The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then ( fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: ..."
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Cited by 2 (1 self)
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The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then ( fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence (fn) and the underlying space. We prove a slight strengthening of Tao’s theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorov’s theorem, and introduce a new mode of convergence related to these notions.
Alternative iterative methods for nonexpansive mappings, rates of convergence and applications
, 2009
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EFFECTIVE ASYMPTOTIC REGULARITY FOR ONEPARAMETER NONEXPANSIVE SEMIGROUPS
"... Abstract. We give explicit bounds on the computation of approximate common fixed points of oneparameter strongly continuous semigroups of nonexpansive mappings on a subset C of a general Banach space. Moreover, we provide the first explicit and highly uniform rate of convergence for an iterative ..."
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Abstract. We give explicit bounds on the computation of approximate common fixed points of oneparameter strongly continuous semigroups of nonexpansive mappings on a subset C of a general Banach space. Moreover, we provide the first explicit and highly uniform rate of convergence for an iterative procedure to compute such points for convex C. Our results are obtained by a logical analysis of the proof (proof mining) of a theorem by T. Suzuki.
interpretations
, 2003
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS