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Concept
Partition of unity
Summary
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 :
there is a neighbourhood of x where all but a finite number of the functions of R are 0, and
the sum of all the function values at x is 1, i.e.,
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:
Given any open cover of a space, there exists a partition indexed over the same set I such that supp Such a partition is said to be subordinate to the open cover
If the space is locally-compact, given any open cover of a space, there exists a partition indexed over a possibly distinct index set J such that each \rho_j has compact support and for each j \in J, supp for some i \in I.
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 to which the space belongs, 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.
If R and T are partitions of unity for spaces X and Y, respectively, then the set of all pairs is a partition of unity for the cartesian product space X \times Y. The tensor product of functions act as
We can construct a partition of unity on by looking at a chart on the complement of a point sending to with center .
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