Linear Combinations of Partitions of Unity With Restricted Supports

Given a locally finite open covering fC ' : ' 2 Ig of a T 4 -space X we characterize all real-valued functions f 2 C(X) which admit a representation f = P '2I ' ' ' with ' 2 IR and a partition of unity f' ' : ' 2 Ig fulfilling the restriction supp(' ' ) ` C ' . 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 2 C(jPPj) on the underlying set jPPj of a polyhedral complex PP such that, for each polytope P 2 PP, the restriction f j P attains its extrema in vertices of P . Finally, a class of extremal functions on the metric space ([\Gamma1; 1] m ; d1 ) is characterized, which appears in the approximation by so-called controllable partitions of unity. 1 The main theorem and first conclusions Let X be a topological space. A system of continuous real-valued functions f' ' : ' 2 Ig ` C(X) is called a partition of unity on X if ' ' 0 for all ' 2 I, P ..