In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s from 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 ). In addition to its role in real analysis, ...
,1such that for every point
:
* there is a
neighbourhood
A neighbourhood (British English, Irish 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 are ...
of 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 marke ...
number of the functions of are 0, and
* the sum of all the function values at 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
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, and the theory of
spline function
In mathematics, a spline is a special function defined piecewise by polynomials.
In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree pol ...
s.
Existence
The existence of partitions of unity assumes two distinct forms:
# Given any
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
of a space, there exists a partition
indexed ''over the same set'' 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 such that each has
compact support and for each , supp
for some .
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 ...
, 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
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
of the space is a necessary condition to guarantee the existence of a partition of unity
subordinate to any open cover. Depending on the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
to which the space belongs, it may also be a sufficient condition. The construction uses
mollifier
In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) f ...
s (bump functions), which exist in continuous and
smooth manifolds
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, but not in
analytic manifold
In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic ge ...
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
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
.
If and are partitions of unity for spaces and , respectively, then the set of all pairs
is a partition of unity for the
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
space . The tensor product of functions act as
Example
We can construct a partition of unity on
by looking at a chart on the complement of a point
sending
to
with center
. Now, let
be a
bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
on
defined by
then, both this function and
can be extended uniquely onto
by setting
. Then, the set
forms a partition of unity over
.
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 by the sum; the partition becomes
where
, 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
for all
.
Applications
A partition of unity can be used to define the integral (with respect to a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
) 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
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
on an arbitrary manifold.
Method of steepest descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in r ...
employs a partition of unity to construct asymptotics of integrals.
Linkwitz–Riley filter
A Linkwitz–Riley (L-R) filter is an infinite impulse response filter used in Linkwitz–Riley audio crossovers, named after its inventors Siegfried Linkwitz and Russ Riley. This filter type was originally described in ''Active Crossover Net ...
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
In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein.
A numerically stable way to evaluate pol ...
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