Linear combinations of partitions of unity with restricted supports

Preprint series: 99-30, Reports on Analysis

MSC:
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
49J99 None of the above but in this section
52B70 Polyhedral manifolds
41A30 Approximation by other special function classes

Abstract: Given a locally finite open covering $\{ C_i: i \in I \}$ of
a $T_4$-space $X$ we characterize all real-valued functions
$f \in C(X)$ which admit a representation
$f = \sum_{i \in I} \lambda_i \phi_i$ with $\lambda_i \in R$
and a partition of unity $\{ \phi_i: i \in I \}$ fulfilling
the restriction supp$( \phi_i ) \subseteq C_i$. The crucial
lemma concerns a particular way of extending partitions of
unity in the sense of the Tietze-Urysohn theorem.

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

Keywords: T4-space, linear combinations of partitions of unity, extensions of partitions of unity, polyhedral complex, optimization, approximation