Optimal approximation of multivariate periodic Sobolev functions in the sup-norm

Cobos, Fernando; Kühn, Thomas; Sickel, Winfried

doi:10.1016/j.jfa.2016.03.018

Cited by 39 publications

(56 citation statements)

References 18 publications

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“…Let us recall a result of Temlyakov [53], see also [14]. [53] and [14] the authors deal with approximation numbers a n (id : By using specific properties of Weyl numbers we will extent Prop.…”

Section: The Extreme Case P 2 = ∞mentioning

confidence: 99%

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Weyl numbers of embeddings of tensor product Besov spaces

Sickel

2015

Journal of Approximation Theory

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Section: The Extreme Case P 2 = ∞mentioning

confidence: 99%

“…[53] and [14] the authors deal with approximation numbers a n (id : By using specific properties of Weyl numbers we will extent Prop. 3.2 to the following result.…”

Section: The Extreme Case P 2 = ∞mentioning

confidence: 99%

Weyl numbers of embeddings of tensor product Besov spaces

Sickel

2015

Journal of Approximation Theory

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“…Now we turn to extreme cases given by either p 2 = ∞ or p 2 = 1. Let us recall a result of Temlyakov [36], see also [6].…”

Section: The Main Resultsmentioning

confidence: 99%

“…In the literature many times the notation H t mix (Ω) and M W t 2 (Ω) are used instead of S t 2,2 B(Ω). In [36,6] the authors deal with approximation numbers. However, for Banach spaces Y and Hilbert spaces H we always have …”

Section: The Main Resultsmentioning

confidence: 99%

Gelfand numbers of embeddings of mixed Besov spaces

Nguyen

2017

Journal of Complexity

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Gelfand numbers represent a measure for the information complexity which is given by the number of information needed to approximate functions in a subset of a normed space with an error less than ε. More precisely, Gelfand numbers coincide up to the factor 2 with the minimal error e wor (n, Λ all ) which describes the error of the optimal (non-linear) algorithm that is based on n arbitrary linear functionals. This explains the crucial role of Gelfand numbers in the study of approximation problems. Let S t p1,p1 B((0, 1) d ) be the Besov spaces with dominating mixed smoothness on (0, 1)d . In this paper we consider the problem App : 0, 1) d ) and investigate the asymptotic behaviour of Gelfand numbers of this embedding. We shall give the correct order of convergence of Gelfand numbers in almost all cases. In addition we shall compare these results with the known behaviour of approximation numbers which coincide with e wor−lin (n, Λ all ) when we only allow linear algorithms.

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“…like the cube, see [6,20,23,24]. Quite remarkable are recent results of Mieth [26], to be discussed later, since they hold for general domains D d .…”

Section: Introductionmentioning

confidence: 84%

Algorithms and complexity for functions on general domains

Novak

2020

Journal of Complexity

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Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on D d ⊂ R d and only assume that D d is a bounded Lipschitz domain. Some results are even more general. We study three different concepts to measure the complexity: order of convergence, asymptotic constant, and explicit uniform bounds, i.e., bounds that hold for all n (number of pieces of information) and all (normalized) domains.It is known for many problems that the order of convergence of optimal algorithms does not depend on the domain D d ⊂ R d . We present examples for which the following statements are true:1. Also the asymptotic constant does not depend on the shape of D d or the imposed boundary values, it only depends on the volume of the domain.2. There are explicit and uniform lower (or upper, respectively) bounds for the error that are only slightly smaller (or larger, respectively) than the asymptotic error bound. *Much is known about the order of convergence of the numbers e n , in particular for simple domains D d , such as the cube or the torus. General bounded Lipschitz domains are studied in [7,10,11,12,26,27,29,38,39,40,42,43]. In many cases, the optimal order of the e n is of the form e n ≍ n −α (log n) β , where α and β do not depend on the domain D d . It is interesting to know also the exact asymptotic constant C := lim n→∞ e n n α (log n) −β , if it exists. The value of C is known only in rare cases (unless d = 1, we do not discuss the univariate case in detail), and usually only for very special domains,

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Optimal approximation of multivariate periodic Sobolev functions in the sup-norm

Abstract: Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for L 2 -approximation of Sobolev functions into estimates for L ∞ -approximation, with precise control of all involved constants. As

Cited by 39 publications

References 18 publications

Weyl numbers of embeddings of tensor product Besov spaces

Weyl numbers of embeddings of tensor product Besov spaces

Gelfand numbers of embeddings of mixed Besov spaces

Algorithms and complexity for functions on general domains

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