Partition of unity
In mathematics, a partition of unity of a topological space X is a set R of continuous functions from X to the unit interval [0,1] such that for every point, ,
Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They are also important in the interpolation of data, in signal processing, and the theory of spline functions.
The existence of partitions of unity assumes two distinct forms:
Thus one chooses either to have the supports indexed by the open cover, or compact supports. If the space is compact, then there exist partitions satisfying both requirements.
A finite open cover always has a continuous partition of unity subordinated to it, provided the space is locally compact and Hausdorff.Paracompactness of the space is a necessary condition to guarantee the existence of a partition of unity subordinate to any open cover. Depending on the category which the space belongs to, it may also be a sufficient condition. The construction uses mollifiers (bump functions), which exist in continuous and smooth manifolds, but not in analytic manifolds. Thus for an open cover of an analytic manifold, an analytic partition of unity subordinate to that open cover generally does not exist. See analytic continuation.
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