by **
C. Richter**

**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*

**Upload:** 1999-06-01

**Update:** 1999-06-01

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