Partition of unity

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In mathematics, a partition of unity of a topological space X is a set of continuous functions, \{\rho_i\}_{i\in I}, from X to the unit interval [0,1] such that for every point, x\in X,

  • there is a neighbourhood of x where all but a finite number of the functions are 0, and
  • the sum of all the function values at x is 1, i.e., \;\sum_{i\in I} \rho_i(x) = 1.

Sometimes, the requirement not as strict: the sum of all the function values at a particular point is only required to be positive rather than a fixed number for all points in the space

Partitions of unity are useful because they often allow one to extend local constructions to the whole space.

The existence of partitions of unity assumes two distinct forms:

Thus one chooses either to have the supports indexed by the open cover, or the supports compact. If the space is compact, then there exist partitions satisfying both requirements.

Paracompactness of the space is a necessary condition to guarantee the existence of a partition of unity. 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 the continuous and smooth manifold categories, but not the analytic category. Thus analytic partitions of unity do not exist. See analytic continuation.

Applications

A partition of unity can be used to integrate a function defined over a manifold. It is also used in the proofs to demonstrate integration over a manifold. One does this by demonstrating the integral of a function whose support is contained in a single coordinate patch of the manifold, and then using a partition of unity admissible to a cover of the manifold to extend the result to the entire manifold.

See also

External links


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