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Linear combinations of partitions of unity with restricted supports

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Richter C.

Studia Mathematica

2002

tom 153

nr 1

1-11

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 $([-1,1]^{m},d_{∞})$ is characterized, which appears in approximation by so-called controllable partitions of unity.

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1 Studia Mathematica

1 Richter C.

1 2002

Wyniki wyszukiwania

Linear combinations of partitions of unity with restricted supports