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In mathematics, the support of a real-valued function f is the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of the function domain containing the elements which are not mapped to zero. If the domain of f 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 f is instead defined as the smallest
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric spac ...
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 f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f 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 \R and to other objects, such as measures or distributions.


Closed support

The most common situation occurs when X 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 In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
or n-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 sp ...
) and f : X \to \R is a continuous real (or complex)-valued function. In this case, the of f, \operatorname(f), or the of f, is defined topologically as the closure (taken in X) of the subset of X where f is non-zero that is, \operatorname(f) := \operatorname_X\left(\\right) = \overline. Since the intersection of closed sets is closed, \operatorname(f) is the intersection of all closed sets that contain the set-theoretic support of f. For example, if f : \R \to \R is the function defined by f(x) = \begin 1 - x^2 & \text , x, < 1 \\ 0 & \text , x, \geq 1 \end then \operatorname(f), the support of f, or the closed support of f, is the closed interval 1, 1 since f is non-zero on the open interval (-1, 1) and the closure of this set is 1, 1 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 f : X \to \R (or f : X \to \Complex) be continuous.


Compact support

Functions with on a topological space X are those whose closed support is a compact subset of X. If X is the real line, or n-dimensional Euclidean space, then a function has compact support if and only if it has , since a subset of \R^n is compact if and only if it is closed and bounded. For example, the function f : \R \to \R defined above is a continuous function with compact support 1, 1 If f : \R^n \to \R is a smooth function then because f is identically 0 on the open subset \R^n \smallsetminus \operatorname(f), all of f's partial derivatives of all orders are also identically 0 on \R^n \smallsetminus \operatorname(f). The condition of compact support is stronger than the condition of vanishing at infinity. For example, the function f : \R \to \R defined by f(x) = \frac vanishes at infinity, since f(x) \to 0 as , x, \to \infty, but its support \R is not compact. Real-valued compactly supported
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s 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 sp ...
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 ...
s. Mollifiers 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 called ...
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 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, for any \varepsilon > 0, any function f on the real line \R that vanishes at infinity can be approximated by choosing an appropriate compact subset C of \R such that \left, f(x) - I_C(x) f(x)\ < \varepsilon for all x \in X, where I_C is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
of C. 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 X is a topological
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
with a Borel measure \mu (such as \R^n, or a Lebesgue measurable subset of \R^n, equipped with Lebesgue measure), then one typically identifies functions that are equal \mu-almost everywhere. In that case, the of a measurable function f : X \to \R written \operatorname(f), is defined to be the smallest closed subset F of X such that f = 0 \mu-almost everywhere outside F. Equivalently, \operatorname(f) 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 a ...
on which f = 0 \mu-almost everywhere \operatorname(f) := X \setminus \bigcup \left\. The essential support of a function f depends on the measure \mu as well as on f, and it may be strictly smaller than the closed support. For example, if f : , 1\to \R is the Dirichlet function that is 0 on irrational numbers and 1 on rational numbers, and , 1/math> is equipped with Lebesgue measure, then the support of f is the entire interval , 1 but the essential support of f is empty, since f is equal almost everywhere to the zero function. In analysis one nearly always wants to use the essential support of a function, rather than its closed support, when the two sets are different, so \operatorname(f) is often written simply as \operatorname(f) and referred to as the support.


Generalization

If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : X \to M. Support may also be defined for any
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
with identity (such as a group,
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
, or composition algebra), in which the identity element assumes the role of zero. For instance, the family \Z^ of functions from the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
to the
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
is the uncountable set of integer sequences. The subfamily \left\ is the countable set of all integer sequences that have only finitely many nonzero entries. Functions of finite support are used in defining algebraic structures such as group rings and free abelian groups.


In probability and measure theory

In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the support of a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
can be loosely thought of as the closure of the set of possible values of a random variable having that distribution. There are, however, some subtleties to consider when dealing with general distributions defined on a sigma algebra, rather than on a topological space. More formally, if X : \Omega \to \R is a random variable on (\Omega, \mathcal, P) then the support of X is the smallest closed set R_X \subseteq \R such that P\left(X \in R_X\right) = 1. In practice however, the support of a
discrete random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
X is often defined as the set R_X = \ and the support of a continuous random variable X is defined as the set R_X = \ where f_X(x) is a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
of X (the set-theoretic support). Note that the word can refer to the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
of the
likelihood The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...
of a probability density function.


Support of a distribution

It is possible also to talk about the support of a
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...
, such as the Dirac delta function \delta(x) on the real line. In that example, we can consider test functions F, which are
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s with support not including the point 0. Since \delta(F) (the distribution \delta applied as
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , th ...
to F) is 0 for such functions, we can say that the support of \delta is \ only. Since measures (including
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
s) on the real line are special cases of distributions, we can also speak of the support of a measure in the same way. Suppose that f is a distribution, and that U is an open set in Euclidean space such that, for all test functions \phi such that the support of \phi is contained in U, f(\phi) = 0. Then f is said to vanish on U. Now, if f vanishes on an arbitrary family U_ of open sets, then for any test function \phi supported in \bigcup U_, a simple argument based on the compactness of the support of \phi and a partition of unity shows that f(\phi) = 0 as well. Hence we can define the of f as the complement of the largest open set on which f vanishes. For example, the support of the Dirac delta is \.


Singular support

In Fourier analysis in particular, it is interesting to study the of a distribution. This has the intuitive interpretation as the set of points at which a distribution . For example, the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
can, up to constant factors, be considered to be 1/x (a function) at x = 0. While x = 0 is clearly a special point, it is more precise to say that the transform of the distribution has singular support \: it cannot accurately be expressed as a function in relation to test functions with support including 0. It be expressed as an application of a Cauchy principal value integral. For distributions in several variables, singular supports allow one to define and understand Huygens' principle in terms of
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 ...
. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).


Family of supports

An abstract notion of on 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 ...
X, suitable for
sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
, was defined by Henri Cartan. In extending Poincaré duality to
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for example Alexander–Spanier cohomology. Bredon, ''Sheaf Theory'' (2nd edition, 1997) gives these definitions. A family \Phi of closed subsets of X is a , if it is down-closed and closed under
finite union In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of ze ...
. Its is the union over \Phi. A family of supports that satisfies further that any Y in \Phi is, with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
, a paracompact space; and has some Z in \Phi which is a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
. If X is a locally compact space, assumed Hausdorff the family of all
compact subset In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s satisfies the further conditions, making it paracompactifying.


See also

* * * *


Citations


References

* * Set theory Real analysis Topology Topology of function spaces