Linear combinations of partitions of unity with restricted supports

Volume 153 / 2002

Christian Richter Studia Mathematica 153 (2002), 1-11 MSC: 54D15, 49J99, 52B11, 41A30. DOI: 10.4064/sm153-1-1


Given a locally finite open covering $\cal C$ of a normal space $X$ and a Hausdorff topological vector space $E$, we characterize all continuous functions $f: X \rightarrow E$ which admit a representation $f = \sum_{C \in {\cal C}} a_C \varphi_C$ with $a_C \in E$ and a partition of unity $\{\varphi_C: C \in {\cal C} \}$ subordinate to ${\cal C}$.

As an application, we determine the class of all functions $f \in C(|\boldsymbol{\mathcal P}|)$ on the underlying space $|\boldsymbol{\mathcal P}|$ of a Euclidean complex $\boldsymbol{\mathcal P}$ such that, for each polytope $P \in \boldsymbol{\mathcal P}$, the restriction $f|_P$ attains its extrema at vertices of $P$. Finally, a class of extremal functions on the metric space $([-1,1]^m,d_\infty)$ is characterized, which appears in approximation by so-called controllable partitions of unity.


  • Christian RichterEquipe d'Analyse
    Université Paris VI
    Case 186
    4, Place Jussieu
    75252 Paris Cedex 05, France

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