Hölder Continuity for the Inverse of Mañé′s Projection

Ben-Artzi, Asher; Eden, A.; Foiaş, Ciprian; Nicolaenko, B.

doi:10.1006/jmaa.1993.1288

Cited by 39 publications

(54 citation statements)

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“…In this paper we prove several results about Bouligand dimension and its relation to the Mañé type projection theorems of [16], [3] and [10]. The use of Bouligand dimension in studying projections was initiated by Movahedi-Lankarani in [18] where he constructs a set A with finite fractal dimension for which there are no finite rank projections P with P | A having Lipschitz continuous inverse.…”

Section: Introductionmentioning

confidence: 86%

Bouligand dimension and almost Lipschitz embeddings

Olson¹

2002

Pacific J. Math.

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In this paper we present some new properties of the metric dimension defined by Bouligand in 1928 and prove the following new projection theorem: Let dim b (A − A) denote the Bouligand dimension of the set A − A of differences between elements of A. Given any compact set A ⊆ R N such that dim b (A − A) < m, then almost every orthogonal projection P of A of rank m is injective on A and P | A has Lipschitz continuous inverse except for a logarithmic correction term.

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Section: Introductionmentioning

confidence: 86%

Bouligand dimension and almost Lipschitz embeddings

Olson¹

2002

Pacific J. Math.

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“…Mañé [33] demonstrates the existence of a dense set of injective projections for a compact subset of a Banach space. Ben-Artzi, Eden, Foias, and Nicolaenko [5] study the Hölder continuity of the inverse and establish sharp bounds on the Hölder exponent in the case X ⊂ R n . Foias and Olson [16] give the first proof that the inverse is Hölder continuous when X is a subset of an infinite-dimensional space.…”

Section: Conjecture 84 ([43]mentioning

confidence: 99%

Untitled

2005

Bull. Amer. Math. Soc.

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Abstract. Many problems in mathematics and science require the use of infinite-dimensional spaces. Consequently, there is need for an analogue of the finite-dimensional notions of 'Lebesgue almost every' and 'Lebesgue measure zero' in the infinite-dimensional setting. The theory of prevalence addresses this need and provides a powerful framework for describing generic behavior in a probabilistic way. We survey the theory and applications of prevalence.

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“…In this section we formulate three lemmas which will be used in the proof of Theorem 1. note that the definitions and the propositions of this section are also mentioned in [1] and intensively studied in [24].…”

Section: Statement Of Auxiliary Lemmasmentioning

confidence: 99%

“…We can further assume that S X 1 and δ ∈ (0, 1 2 ). In the proof of the following lemma we use a method which was developed in [1].…”

Section: Proof Of Main Theoremmentioning

confidence: 99%

“…Therefore, as was proved in [22], any orthogonal projection in the underlying Hilbert space H of sufficiently high finite rank can be perturbed so that its restriction to the global attractor is injective. It means that in the case of these systems the global attractor is a subset of the graph of a continuous mapping defined on a finite dimensional subspace of H. It was proved in [1] that the mapping is Hölder continuous as long as the underlying Hilbert space H is finite dimensional. It is natural to ask whether the same theorem also holds for the infinite dimensional space H. In this paper we show the following weaker result:…”

Section: Introductionmentioning

confidence: 99%

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