In mathematics, the support of a real-valued function is the subset of the function's domain is a topological space is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis.

Formulation

is a real-valued function whose domain The

is said to have .

has an additional structure (for example, a topology and to other objects, such as measures or distributions.

Closed support

is a topological space (such as the real line-dimensional Euclidean space is a continuous real- (or complex-) valued function. In this case, the , or the of , is defined topologically as the closure is non-zero.

and the closure

) be continuous.

Compact support

Functions with are those whose closed support is a compact-dimensional Euclidean space, then a function has compact support if and only if it has is compact if and only if it is closed and bounded.

The condition of compact support is stronger than the condition of vanishing at infinity is not compact.

Real-valued compactly supported smooth functions on a Euclidean space are sometimes called bump functions. Mollifiers are an important special case of bump functions as they can be used in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.

In a locally compact Hausdorff space, continuous functions with compact support are dense in the space of continuous functions that vanish at infinity. As an intuition for more complex examples, and in the language of limits is the indicator function.

In the theory of distributions, it is also true that the smooth compactly supported smooth functions are dense in the space of Schwartz functions. They are also dense in every Lp space. Thus in most good cases, the compactly supported functions of the appropriate class are dense in many different topologies, and so they are fundamental in analysis.

Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.

Essential support

is a topological measure space with a Borel measure or a Lebesgue measurable-almost everywhere. In that case, the written is the complement of the largest open set-almost everywhere

depends on the measure is the Dirichlet function is equal almost everywhere to the zero function.

and referred to as the support.

Generalization

Support may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra of functions from the natural numbers to the integers is the uncountable 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, the support of a probability distribution 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.

In practice however, the support of a discrete random variable and the support of a continuous random variable is a probability density function (the set-theoretic support).

Note that the word can refer to the logarithm of the likelihood of a probability density function.

Support of a distributionSupport (statistics)

It is possible also to talk about the support of a distribution, such as the Dirac delta function which are smooth functions applied as linear functional only. Since measures (including probability measures) on the real line are special cases of distributions, we can also speak of the support of a measure in the same way.

as well. Hence we can define the

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 of the Heaviside step function (a function) 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. 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 suitable for sheaf theory, was defined by Henri Cartan. In extending Poincaré duality to manifolds 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 is a , if it is down-closed and closed under finite union. Its A is, with the subspace topology, a paracompact space which is a neighbourhood is a locally compact space, assumed Hausdorff, the family of all compact subsets satisfies the further conditions, making it paracompactifying.

See also

Citations

References