Editing Partition of unity

{{short description|Set of functions from a topological space to [0,1] which sum to 1 for any input}}

In [[mathematics]], a '''partition of unity''' on a [[topological space]] {{tmath|X}} is a [[Set (mathematics)|set]] {{tmath|R}} of [[continuous function (topology)|continuous function]]s from {{tmath|X}} to the [[unit interval]] [0,1] such that for every point <math>x\in X</math>:
* there is a [[neighbourhood (mathematics)|neighbourhood]] of {{tmath|x}} where all but a [[finite set|finite]] number of the functions of {{tmath|R}} are non zero<ref>Lee, John M., and John M. Lee. Smooth manifolds. Springer New York, 2003.</ref>, and
* the sum of all the function values at {{tmath|x}} is 1, i.e., <math display="inline">\sum_{\rho\in R} \rho(x) = 1.</math>

[[Image:Partition of unity illustration.svg|center|thumb|500px|A partition of unity on 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]] <math>\{ U_i \}_{i \in I}</math> of a space, there exists a partition <math>\{ \rho_i \}_{i \in I}</math> indexed ''over the same set'' {{tmath|I}} such that [[Support (mathematics)|supp]] <math>\rho_i \subseteq U_i.</math> Such a partition is said to be '''subordinate to the open cover''' <math>\{ U_i \}_i.</math>
# If the space is locally compact, given any open cover <math>\{ U_i \}_{i \in I}</math> of a space, there exists a partition <math>\{ \rho_j \}_{j \in J}</math> indexed over a possibly distinct index set {{tmath|J}} such that each {{tmath|\rho_j}} has [[compact support]] and for each {{tmath|j \in J}}, supp <math>\rho_j \subseteq U_i</math> for some {{tmath|i \in 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 subordinate 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=978-0-07-054234-1|pages=40|edition=3rd}}</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]] to which the space belongs, this may also be a sufficient condition.<ref>{{cite book|first1=Charalambos D.|last1=Aliprantis|first2=Kim C.|last2=Border
|title=Infinite dimensional analysis: a hitchhiker's guide|year=2007|publisher=Springer|location=Berlin|isbn=978-3-540-32696-0| pages=716|edition=3rd}}</ref> In particular, a compact set in the [[Euclidean space]] admits a smooth partition of unity subordinate to any finite open cover. The construction uses [[mollifier]]s (bump functions), which exist in continuous and [[smooth manifolds]], but not necessarily 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 {{tmath|R}} and {{tmath|T}} are partitions of unity for spaces {{tmath|X}} and {{tmath|Y}} respectively, then the set of all pairs <math>\{ \rho\otimes\tau :\ \rho \in R,\ \tau \in T \}</math> is a partition of unity for the [[cartesian product]] space {{tmath|X \times Y}}. The tensor product of functions act as <math>(\rho \otimes \tau )(x,y) = \rho(x)\tau(y).</math>

== Example ==
Let <math>p</math> and <math>q</math> be antipodal points on the circle <math>S^1</math>. We can construct a partition of unity on <math>S^1</math> by looking at a chart on the complement of the point <math>p \in S^1</math> that sends <math>S^1 -\{p\}</math> to <math>\mathbb{R}</math> with center <math>q \in S^1</math>. Now let <math>\Phi</math> be a [[bump function]] on <math>\mathbb{R}</math> defined by <math display="block">\Phi(x) = \begin{cases}
\exp\left(\frac{1}{x^2-1}\right) & x \in (-1,1) \\
0 & \text{otherwise}
\end{cases}</math> then, both this function and <math>1 - \Phi</math> can be extended uniquely onto <math>S^1</math> by setting <math>\Phi(p) = 0</math>. Then, the pair of functions <math>\{ (S^1 - \{p\}, \Phi), (S^1 - \{q\}, 1-\Phi) \}</math> forms a partition of unity over <math>S^1</math>.

==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 <math>\{ \psi_i \}_{i=1}^\infty</math> one can obtain a partition of unity in the strict sense by dividing by the sum; the partition becomes <math>\{ \sigma^{-1}\psi_i \}_{i=1}^\infty</math> where <math display="inline">\sigma(x) := \sum_{i=1}^\infty \psi_i(x)</math>, which is well defined since at each point only a finite number of terms are nonzero. Even further, some authors drop the requirement that the supports be locally finite, requiring only that <math display="inline">\sum_{i = 1}^\infty \psi_i(x) < \infty</math> for all <math>x</math>.<ref>{{Cite book| last=Strichartz| first= Robert S.|url=https://www.worldcat.org/oclc/54446554|title=A guide to distribution theory and Fourier transforms |date=2003|publisher=World Scientific Pub. Co|isbn=981-238-421-9|location=Singapore|oclc=54446554}}</ref>

In the field of [[Operator algebra|operator algebras]], a partition of unity is composed of projections<ref>{{cite book |last1=Conway |first1=John B. |title=A Course in Functional Analysis |publisher=Springer |isbn=0-387-97245-5 |page=54 |edition=2nd}}</ref> <math>p_i=p_i^*=p_i^2</math>. In the case of [[C*-algebra|<math>\mathrm{C}^*</math>-algebras]], it can be shown that the entries are pairwise [[Orthogonality|orthogonal]]:<ref>{{cite book |last1=Freslon |first1=Amaury |title=Compact matrix quantum groups and their combinatorics |date=2023 |publisher=Cambridge University Press|bibcode=2023cmqg.book.....F }}</ref>
<math display="block">p_ip_j=\delta_{i,j}p_i\qquad (p_i,\,p_j\in R).</math>
Note it is ''not'' the case that in a general [[*-algebra]] that the entries of a partition of unity are pairwise orthogonal.<ref>{{cite web |last1=Fritz |first1=Tobias |title=Pairwise orthogonality for partitions of unity in a *-algebra|url=https://mathoverflow.net/a/463103/35482 |website=Mathoverflow |access-date=7 February 2024}}</ref>
 
If <math>a</math> is a [[normal element|normal]] element of a unital <math>\mathrm{C}^*</math>-algebra <math>A</math>, and has finite [[Spectrum (functional analysis)|spectrum]] <math>\sigma(a)=\{\lambda_1,\dots,\lambda_N\}</math>, then the projections in the [[Spectral theorem|spectral decomposition]]:
<math display="block">a=\sum_{i=1}^N\lambda_i\,P_i,</math>
form a partition of unity.<ref>{{cite book |last1=Murphy |first1=Gerard J. |title=C*-Algebras and Operator Theory |date=1990 |publisher=Academic Press |isbn=0-12-511360-9 |page=66}}</ref>

In the field of [[Compact quantum group|compact quantum groups]], the rows and columns of the fundamental representation <math>u\in
M_N(C)</math> of a quantum permutation group <math>(C,u)</math> form partitions of unity.<ref>{{cite book |last1=Banica |first1=Teo |title=Introduction to Quantum Groups |date=2023 |publisher=Springer |isbn=978-3-031-23816-1}}</ref>

==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 single-variable polynomials that are a partition of unity for the unit interval <math>[0,1]</math>.

The weak [[Hilbert's Nullstellensatz|Hilbert Nullstellensatz]] asserts that if <math>f_1,\ldots, f_r\in \C[x_1,\ldots,x_n]</math> are polynomials with no common vanishing points in <math>\C^n</math>, then there are polynomials <math>a_1, \ldots, a_r</math> with <math>a_1f_1+\cdots+a_r f_r = 1</math>. That is, <math>\rho_i = a_i f_i</math> form a polynomial partition of unity subordinate to the [[Zariski topology|Zariski-open]] cover <math>U_i = \{x\in \C^n \mid f_i(x)\neq 0\}</math>. 

Partitions of unity are used to establish global smooth approximations for [[Sobolev space|Sobolev]] functions in bounded domains.<ref>{{Citation|last=Evans|first=Lawrence|chapter=Sobolev spaces|date=2010-03-02|pages=253–309|publisher=American Mathematical Society|isbn=9780821849743|doi=10.1090/gsm/019/05|title=Partial Differential Equations|volume=19|series=Graduate Studies in Mathematics}}</ref>

==See also==
*{{section link|Smoothness|Smooth partitions of unity}}
*[[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]

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[[Category:Differential topology]]
[[Category:Topology]]