Linear combinations of partitions of unity with restrictedsupportsChristian

Given a locally nite 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 fulllling 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(jP Pj) on the underlying set jP Pj of a polyhedral complex P P such that, for each polytope P 2 P P, the restriction fj P attains its extrema in vertices of P. Finally, a class of extremal functions on the metric space ((?1; 1] m ; d 1) is characterized, which appears in the approximation by so-called controllable partitions of unity. 1 The main theorem and rst 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 2I ' i = 1, and the covering f supp(') : 2 Ig formed by the supports of the partition functions is locally nite, i.e., for all x 2 X, there exists a neighbourhood of x intersecting at most nitely many sets of the covering. In the context of the present paper the support of a function f 2 C(X) is meant to be the open set supp(f) = fx 2 X : f(x) 6 = 0g in contrast with the usual deenition, which refers to the closure of the above set. Given a locally nite open covering C = fC : 2 Ig of X a partition of unity f' : 2 Ig is said to be subordinated to C if supp(') C for all 2 I (cf. Que], p. 80). For a xed locally nite open covering C = fC : 2 Ig of X we want to characterize all functions f 2 C(X) which admit a representation f = P 2I ' with real coeecients and a partition of unity f' : 2 Ig subordinated to C. The following theorem solves this question