en>Hymath: /* Non-Hausdorff examples */ adding a phrase "containing the particular point" at the end of a sentence "because it has no nonempty closed compact subspaces."

2013-12-02T10:10:20Z

Non-Hausdorff examples: adding a phrase "containing the particular point" at the end of a sentence "because it has no nonempty closed compact subspaces."

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In [[mathematics]], a '''partition of unity''' of a [[topological space]] ''X'' is a set ''R'' of [[continuous function (topology)|continuous function]]s from ''X'' to the [[unit interval]] [0,1] such that for every point, <math>x\in X</math>,
* there is a [[neighbourhood (mathematics)|neighbourhood]] of ''x'' where all but a [[finite set|finite]] number of the functions of ''R'' are 0, and
* the sum of all the function values at ''x'' is 1, i.e., <math>\;\sum_{\rho\in R} \rho(x) = 1</math>.

[[Image:Partition of unity illustration.svg|center|thumb|500px|A partition of unity of a circle with four functions. The circle is unrolled to a line segment (the bottom solid line) for graphing purposes. The dashed line on top is the sum of the functions in the partition.]]
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 function]]s.

== Existence ==
The existence of partitions of unity assumes two distinct forms:

# Given any [[open cover]] {''U''''i''}''i''∈''I'' of a space, there exists a partition {ρ''i''}''i''∈''I'' indexed ''over the same set I'' such that [[Support (mathematics)|supp]] ρ''i''⊆''U''''i''. Such a partition is said to be '''subordinate to the open cover''' {''U''''i''}''i''.
# Given any open cover {''U''''i''}''i''∈''I'' of a space, there exists a partition {ρ''j''}''j''∈''J'' indexed over a possibly distinct index set ''J'' such that each ρ''j'' has [[compact support]] and for each ''j''∈''J'', supp ρ''j''⊆''U''''i'' for some ''i''∈''I''.

Thus one chooses either to have the [[support (mathematics)|supports]] indexed by the open cover, or compact supports. If the space is [[compact space|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.<ref>{{cite book|last=Rudin|first=Walter|title=Real and complex analysis|year=1987|publisher=McGraw-Hill|location=New York|isbn=0-07-054234-1|pages=40|edition=3rd ed.}}</ref>
[[Paracompact space|Paracompactness]] of the space is a necessary condition to guarantee the existence of a partition of unity [[paracompact space|subordinate to any open cover]]. Depending on the [[category (mathematics)|category]] which the space belongs to, it may also be a sufficient condition.<ref>{{cite book|last=Border|first=Charalambos D. Aliprantis, Kim C.|title=Infinite dimensional analysis : a hitchhiker's guide|year=2007|publisher=Springer|location=Berlin|isbn=978-3-540-32696-0|pages=66|edition=3rd ed.}}</ref> The construction uses [[mollifier]]s (bump functions), which exist in continuous and [[smooth manifolds]], but not in [[analytic manifold]]s. 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 ''S'' are partitions of unity for spaces ''X'' and ''Y'', respectively, then the set of all pairwise products <math>\{\; \rho\sigma \;:\; \rho\in R \wedge \sigma \in S\;\}</math> is a partition of unity for the [[cartesian product]] space ''X''×''Y''.

==Variant definitions==
Sometimes a less restrictive definition is used: the sum of all the function values at a particular point is only required to be positive, rather than 1, for each point in the space. However, given such a set of functions, one can obtain a partition of unity in the strict sense by dividing every function by the sum of all functions (which is defined, since at any point it has only a finite number of terms).

==Applications==
A partition of unity can be used to define the integral (with respect to a [[volume form]]) of a function defined over a manifold: One first defines the integral of a function whose support is contained in a single coordinate patch of the manifold; then one uses a partition of unity to define the integral of an arbitrary function; finally one shows that the definition is independent of the chosen partition of unity.

A partition of unity can be used to show the existence of a [[Riemannian metric]] on an arbitrary manifold.

[[Method_of_steepest_descent#The_case_of_multiple_non-degenerate_saddle_points|Method of steepest descent]] employs a partition of unity to construct asymptotics of integrals.

[[Linkwitz–Riley filter]] is an example of practical implementation of partition of unity to separate input signal into two output signals containing only high- or low-frequency components.

The [[Bernstein polynomial]]s of a fixed degree ''m'' are a family of ''m''+1 linearly independent polynomials that are a partition of unity for the unit interval <math>[0,1]</math>.

==See also==
*[[gluing axiom]]
*[[fine sheaf]]

==References==
{{Reflist}}
* {{Citation | last1=Tu | first1=Loring W. | title=An introduction to manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Universitext | isbn=978-1-4419-7399-3 | doi=10.1007/978-1-4419-7400-6 | year=2011}}, see chapter 13

==External links==
*[http://mathworld.wolfram.com/PartitionofUnity.html General information on partition of unity] at [Mathworld]
*[http://planetmath.org/encyclopedia/PartitionOfUnity.html Applications of a partition of unity] at [Planet Math]

{{DEFAULTSORT:Partition Of Unity}}
[[Category:Differential topology]]
[[Category:Topology]]

Locally compact space - Revision history

en>Hymath: /* Non-Hausdorff examples */ adding a phrase "containing the particular point" at the end of a sentence "because it has no nonempty closed compact subspaces."