TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, a partition of unity of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which ''closeness'' is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of ...
''X'' is a set ''R'' of
continuous function In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to suff ...
s from ''X'' to the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter I). In addition to its role in real analysis, ...
,1such that for every point, $x\in X$, * there is a
neighbourhood A neighbourhood (British English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area. Neighbourhoods ar ...
of ''x'' where all but a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
number of the functions of ''R'' are 0, and * the sum of all the function values at ''x'' is 1, i.e., $\;\sum_ \rho\left(x\right) = 1$. Image:Partition of unity illustration.svg, center, 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 functions.

# Existence

The existence of partitions of unity assumes two distinct forms: # Given any
open cover In mathematics, particularly topology, a cover of a set X is a collection of sets whose union includes X as a subset. Formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of sets U_\alpha, then C is a cover of X if :X \subs ...
''i''∈''I'' of a space, there exists a partition ''i''∈''I'' indexed ''over the same set I'' such that supp ρ''i''⊆''U''''i''. Such a partition is said to be subordinate to the open cover ''i''. # If the space is locally-compact, given any open cover ''i''∈''I'' of a space, there exists a partition ''j''∈''J'' indexed over a possibly distinct index set ''J'' such that each ρ''j'' has
compact support In mathematics, the support of a real-valued function ''f'' is the subset of the domain containing the elements which are not mapped to zero. If the domain of ''f'' is a topological space, the support of ''f'' is instead defined as the smallest c ...
and for each ''j''∈''J'', supp ρ''j''⊆''U''''i'' for some ''i''∈''I''. Thus one chooses either to have the supports indexed by the open cover, or compact supports. If the space is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British N ...
, 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. 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, 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''×''Y''. The tensor product of functions act as $\left(\rho \otimes \tau \right)\left(x,y\right) = \rho\left(x\right)\tau\left(y\right)$.

# Example

We can construct a partition of unity on $S^1$ by looking at a chart on the complement of a point $p \in S^1$ sending $S^1 -\$ to $\mathbb$ with center $q \in S^1$. Now, let $\Phi$ be a bump function on $\mathbb$ defined by
$\Phi\left(x\right) = \begin \exp\left\left(\frac\right\right) & x \in \left(-1,1\right) \\ 0 & \text \end$
then, both this function and $1 - \Phi$ can be extended uniquely onto $S^1$ by setting $\Phi\left(p\right) = 0$. Then, the set $\$ forms a partition of unity over $S^1$.

# 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 $\_^\infty$ one can obtain a partition of unity in the strict sense by dividing by the sum; the partition becomes $\_^\infty$ where $\sigma\left(x\right) := \sum_^\infty \psi_i\left(x\right)$, 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 $\sum_^\infty \psi_i\left(x\right) < \infty$ for all $x$.

# 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 polynomials of a fixed degree ''m'' are a family of ''m''+1 linearly independent polynomials that are a partition of unity for the unit interval $\left[0,1\right]$. Partition of unity is used to establish global smooth approximations for Sobolev space, Sobolev functions in bounded domains.

* *Gluing axiom *Fine sheaf

# References

* , see chapter 13