In
mathematics, the support of a
real-valued
In mathematics, value may refer to several, strongly related notions.
In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is the
subset of the function
domain containing the elements which are not mapped to zero. If the domain of
is 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 po ...
, then the support of
is instead defined as the smallest
closed set containing all points not mapped to zero. This concept is used very widely in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
.
Formulation
Suppose that
is a real-valued function whose
domain is an arbitrary set
The of
written
is the set of points in
where
is non-zero:
The support of
is the smallest subset of
with the property that
is zero on the subset's complement. If
for all but a finite number of points
then
is said to have .
If the set
has an additional structure (for example, a topology), then the support of
is defined in an analogous way as the smallest subset of
of an appropriate type such that
vanishes in an appropriate sense on its complement. The notion of support also extends in a natural way to functions taking values in more general sets than
and to other objects, such as
measures
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Measu ...
or
distributions.
Closed support
The most common situation occurs when
is 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 po ...
(such as the
real line or
-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
) and
is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
real (or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
)-valued function. In this case, the of
,
, or the of
, is defined topologically as the
closure (taken in
) of the subset of
where
is non-zero
that is,
Since the intersection of closed sets is closed,
is the intersection of all closed sets that contain the set-theoretic support of
For example, if
is the function defined by
then
, the support of
, or the closed support of
, is the closed interval
since
is non-zero on the open interval
and the
closure of this set is
The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require that
(or
) be continuous.
Compact support
Functions with on a topological space
are those whose closed support is a
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 ...
subset of
If
is the real line, or
-dimensional Euclidean space, then a function has compact support if and only if it has , since a subset of
is compact if and only if it is closed and bounded.
For example, the function
defined above is a continuous function with compact support
If
is a smooth function then because
is identically
on the open subset
all of
's partial derivatives of all orders are also identically
on
The condition of compact support is stronger than the condition of
vanishing at infinity. For example, the function
defined by
vanishes at infinity, since
as
but its support
is not compact.
Real-valued compactly supported
smooth functions on a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
are called
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 bum ...
s.
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 are an important special case of bump functions as they can be used in
distribution theory to create
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s of smooth functions approximating nonsmooth (generalized) functions, via
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
.
In
good cases, functions with compact support are
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. As an intuition for more complex examples, and in the language of
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
, for any
any function
on the real line
that vanishes at infinity can be approximated by choosing an appropriate compact subset
of
such that
for all
where
is the
indicator function of
Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.
Essential support
If
is a topological
measure space with a
Borel measure (such as
or a
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
subset of
equipped with Lebesgue measure), then one typically identifies functions that are equal
-almost everywhere. In that case, the of a measurable function
written
is defined to be the smallest closed subset
of
such that
-almost everywhere outside
Equivalently,
is the complement of the largest
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
on which
-almost everywhere
The essential support of a function
depends on the
measure as well as on
and it may be strictly smaller than the closed support. For example, if
is the
Dirichlet function
In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number).
\mathbf 1_\Q(x) = \begin
1 & ...
that is
on irrational numbers and
on rational numbers, and