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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 31 (18 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 11 (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...
On the logical analysis of proofs based on nonseparable Hilbert space theory
, 2010
"... Starting in [15] and then continued in [9, 17, 24] and [18], general logical metatheorems were developed that guarantee the extractability of highly uniform effective bounds from proofs of theorems that hold for general classes of structures such as ..."
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Cited by 4 (4 self)
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Starting in [15] and then continued in [9, 17, 24] and [18], general logical metatheorems were developed that guarantee the extractability of highly uniform effective bounds from proofs of theorems that hold for general classes of structures such as
Rates of asymptotic regularity for Halpern
, 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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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. 1
Alternative iterative methods for nonexpansive
, 905
"... mappings, rates of convergence and applications ..."
unknown title
"... 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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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.
2 1 General introduction
"... This paper (and its companion [29]) is another case study in the project of ‘proof mining ’ 1 in analysis by which we mean the logical analysis of mathematical proofs (typically using noneffective analytical tools) with the aim of extracting new numerically relevant information (e.g. effective unif ..."
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This paper (and its companion [29]) is another case study in the project of ‘proof mining ’ 1 in analysis by which we mean the logical analysis of mathematical proofs (typically using noneffective analytical tools) with the aim of extracting new numerically relevant information (e.g. effective uniform bounds or algorithms etc.) hidden in the proofs. 2 Many problems in numerical (functional) analysis can be seen as instances of the following general task: construct a solution x of an equation A(x): ≡ (F(x) = 0), where x is an element of some Polish (i.e. complete separable metric) space (typically with additional structure) and F: X → IR (usually F will depend on certain parameters a which again belong to Polish spaces). Quite often the construction of such a solution is obtained in two steps: 1) One shows how to construct (uniformly in the parameters of A) approximate solutions (sometimes called ‘εsolutions’) xε ∈ X for an εversion of the original equation Aε(x): ≡ ( F(x)  < ε). 2) Exploiting compactness conditions on X one concludes that either (x 1 n)n∈IN itself or some subsequence of it converges to a solution of A(x). The first step usually is constructive. However, the noneffectivity of the second step in many cases prevents one from being able to compute a solution x of A effectively within a prescribed error 1 k, i.e. to compute a function n(k) such that dX(xn(k), x) < 1 k. In many cases X: = K is compact and x is uniquely determined. Then (xn) itself converges to x so that no subsequence needs to be selected. However, the problem of how to get apriori bounds (in particular not depending on x itself) on the rate of convergence of that sequence remains. In numerical analysis, often such rates are not provided (due to the ineffectivity of the proof of the uniqueness of x). 3 In a series of papers we have demonstrated the applicability of proof theoretic techniques to extract socalled uniform moduli of uniqueness (which generalize 1 The term ‘proof mining ’ (instead of G. Kreisel’s ‘unwinding of proofs’) for the activity of extracting additional information hidden in given proofs using proof theoretic tools was suggested to the author by Professor Dana Scott. 2 For a different case study in analysis in the context of best approximation theory see [21],[22]. For other kinds of logical analyses of specific proofs see [33] and [36].
A note on the monotone functional interpretation
, 2011
"... We prove a result relating the author’s monotone functional interpretation to the bounded functional interpretation due to Ferreira and Oliva. More precisely we show that (over model of majorizable functionals) largely a solution for the bounded interpretation also is a solution for monotone functio ..."
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We prove a result relating the author’s monotone functional interpretation to the bounded functional interpretation due to Ferreira and Oliva. More precisely we show that (over model of majorizable functionals) largely a solution for the bounded interpretation also is a solution for monotone functional interpretation although the latter uses the existence of an underlying precise witness. This makes it possible to focus on the extraction of bounds (as in the bounded interpretation) while using the conceptual benefit of having precise realizers at the same time without having to construct them.