Entropy and the approximation of continuous functions

Stephani, Irmtraud

doi:10.1007/bf01292929

Cited by 10 publications

(12 citation statements)

References 3 publications

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“…Lemma 5.1 yields lim r→0 a(f, r) = 0, provided that there exist partitions of unity subordinate to completely reduced coverings of X by balls of arbitrarily small radius r > 0. On compact spaces (X, d) this new approximation theory coincides essentially with that one from [10]. The relation between the numbers a(f, r) and a k (f ) is based on Kolmogoroff's entropy function N r (X) of X defined by N r (X) = min{k ≥ 1 : X can be covered by k closed balls of radius r} for r > 0 (see [6], p. 4).…”

Section: A Related Approximation Theorymentioning

confidence: 81%

“…. , 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%

“…The last named concept comprises a class of finite coverings of X by arbitrary subsets that satisfy a geometric uniformity condition in terms of entropy numbers. It has been introduced in [10] in the framework of the approximation of real-valued functions on X. In the last section we embed this kind of approximation in a more general theory that acts on metric spaces possessing completely reduced open coverings.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: A Related Approximation Theorymentioning

confidence: 81%

Section: Completely Reduced Coverings and Controllable Coveringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tilings, packings, coverings, and the approximation of functions

Hinrichs

Richter

2004

Mathematische Nachrichten

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“…In this section we shall sketch the approximation by so-called controllable partitions of unity (see [11], [8], [7]). This method of approximating real-valued continuous functions on compact metric spaces gave rise to the fairly general considerations of the present paper.…”

Section: An Approximation Class Onmentioning

confidence: 99%

Linear combinations of partitions of unity with restricted supports

Richter

2002

Studia Math.

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Abstract. Given a locally finite open covering C of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f : X → E which admit a representation f = C∈C a C ϕ C with a C ∈ E and a partition of unity {ϕ C : C ∈ C} subordinate to C.As an application, we determine the class of all functions f ∈ C(|P|) on the underlying space |P| of a Euclidean complex P such that, for each polytope P ∈ P, the restriction f | P attains its extrema at vertices of P . Finally, a class of extremal functions on the metric space (is characterized, which appears in approximation by so-called controllable partitions of unity. [10], p. 352). For a fixed locally finite open covering C of X and for a Hausdorff topological vector space E, we want to characterize all continuous functions f : X → E which admit a representation f = C∈C a C ϕ C with vectors a C ∈ E and a partition of unity {ϕ C : C ∈ C} subordinate to C. The following theorem solves this question for normal spaces X. We shall see that the claim of this very general theorem can even be used for characterizing normal spaces among T 1 -spaces. Besides that, we shall present an application concerning continuous real-valued functions on polytopes and on Euclidean complexes in R m as they appear in optimization theory (see Section 2).

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“…Survey over the main concepts and results. The present paper refers to the approximation theory for continuous functions on an arbitrary compact metric space (X. d) developed in [7]. In that paper linear combinations ~p = ~ 2j g0j j=l of so called controllable partitions of unity ~01, ~p> ... ; q),on X were used as approximating functions for bounded and, in particular, for continuous functions f or~ X.…”

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confidence: 99%