Entropy and the approximation of bounded functions and operators

Richter, Christian; Stephani, Irmtraud

doi:10.1007/bf01270612

Cited by 8 publications

(4 citation statements)

References 4 publications

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“…Richter and I. Stephani from the University of Jena (cf. [48], [49], [50], [60], [61]). An excellent survey on Shepard's method, probably the most significant application of the general theory presented here, was recently finished by H. Knauf [28].…”

Section: Discussionmentioning

confidence: 99%

The rate of convergence of bounded linear processes on spaces of continuous functions

Gonska

2023

J. Numer. Anal. Approx. Theory

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Quantitative Korovkin-type theorems for approximation by bounded linear operators defined on \(C(X,d)\) are given, where \((X,d)\) is a compact metric space. Special emphasis is on positive linear operators.As is known from previous work of Newman and Shapiro, Jimenez Pozo, Nishishiraho and the author, among others, there are two possible ways to obtain error estimates for bounded linear operator approximation: the so-called direct approach, and the smoothing technique.We give various generalizations and refinements of earlier results which were obtained by using both techniques. Furthermore, it will be shown that, in a certain sense, none of the two methods is superior to the other one.

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Section: Discussionmentioning

confidence: 99%

The rate of convergence of bounded linear processes on spaces of continuous functions

Gonska

2023

J. Numer. Anal. Approx. Theory

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“…. , C n } of X by arbitrary subsets C i is called controllable if, for n = 1, C 1 = X and, for n > 1, ε 1 (C i ) < ε n−1 (X) for 1 ≤ i ≤ n (see [10], [8]). The following two propositions illustrate the close relation between controllability and reduction of coverings.…”

Section: Completely Reduced Coverings and Controllable Coveringsmentioning

confidence: 99%

Tilings, packings, coverings, and the approximation of functions

Hinrichs

Richter

2004

Mathematische Nachrichten

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A packing (resp. covering) F of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of F. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls.Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real-valued functions that rests on so-called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non-compact spaces.

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“…In [15] and [14] other estimates of this type are proved. Then, in order to know the asymptotic behavior of the approximation numbers of a function, it is important to know its modulus of continuity and the asymptotic behavior of the entropy numbers of the underlaying domain.…”

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confidence: 90%

Some remarks on entropy numbers of sets with non-integer dimension

Follo

2007

Ricerche mat.

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We study entropy numbers of sets and generalize some results shown by B. Carl and I. Stephani in [2]. It is well known that if Ω is the closed unit ball in the Euclidean space R N , then for every n ∈ N,We extend estimates of this type to more general subsets of metric spaces. Moreover we study relations between the asymptotic behaviour of the entropy numbers of bounded subsets of R N and their box dimension. Mathematics Subject Classification (2000)28A80 · 28A78

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Entropy and the approximation of bounded functions and operators

Cited by 8 publications

References 4 publications

The rate of convergence of bounded linear processes on spaces of continuous functions

The rate of convergence of bounded linear processes on spaces of continuous functions

Tilings, packings, coverings, and the approximation of functions

Some remarks on entropy numbers of sets with non-integer dimension

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