Support (mathematics)

Support (mathematics)
In mathematics, the support of a real-valued function is the subset of the function domain containing the elements which are not mapped to zero. If the domain of is a topological space, 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.
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 or distributions.
Closed support
The most common situation occurs when is a topological space (such as the real line or -dimensional Euclidean space) and is a continuous real- (or complex-) 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 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 are 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 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 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 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 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 that is on irrational numbers and on rational numbers, and is equipped with Lebesgue measure, then the support of is the entire interval but the essential support of is empty, since 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 is often written simply as and referred to as the support.
Generalization
If is an arbitrary set containing zero, the concept of support is immediately generalizable to functions Support may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra), in which the identity element assumes the role of zero. For instance, the family of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily 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.
More formally, if is a random variable on then the support of is the smallest closed set such that
In practice however, the support of a discrete random variable is often defined as the set and the support of a continuous random variable is defined as the set where is a probability density function of (the set-theoretic support).
Note that the word can refer to the logarithm of the likelihood of a probability density function.
Support of a distribution
It is possible also to talk about the support of a distribution, such as the Dirac delta function on the real line. In that example, we can consider test functions which are smooth functions with support not including the point Since (the distribution applied as linear functional to ) is for such functions, we can say that the support of is 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.
Suppose that is a distribution, and that is an open set in Euclidean space such that, for all test functions such that the support of is contained in Then is said to vanish on Now, if vanishes on an arbitrary family of open sets, then for any test function supported in a simple argument based on the compactness of the support of and a partition of unity shows that as well. Hence we can define the of as the complement of the largest open set on which 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 of the Heaviside step function can, up to constant factors, be considered to be (a function) at While 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 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 (2nd edition, 1997) gives these definitions. A family of closed subsets of is a , if it is down-closed and closed under finite union. Its is the union over A family of supports that satisfies further that any in is, with the subspace topology, a paracompact space; and has some in which is a neighbourhood. If is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.
See also
Citations
References
Set theory
Real analysis
Topology
Topology of function spaces
Schwartz distributions