Linear combinations of partitions of unity with restricted supports
Tom 153 / 2002
Streszczenie
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.
