Linear combinations of partitions of unity with restricted supports

Christian Richter¹

¹ Equipe d'Analyse Université Paris VI Case 186 4, Place Jussieu 75252 Paris Cedex 05, France

Studia Mathematica, Tome 153 (2002) no. 1, pp. 1-11

Résumé

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.

Détail

DOI : 10.4064/sm153-1-1

Keywords: given locally finite covering cal normal space hausdorff topological vector space characterize continuous functions rightarrow which admit representation sum cal varphi partition unity varphi cal subordinate cal application determine class functions boldsymbol mathcal underlying space boldsymbol mathcal euclidean complex boldsymbol mathcal each polytope boldsymbol mathcal restriction attains its extrema vertices finally class extremal functions metric space infty characterized which appears approximation so called controllable partitions unity