Compactly supported

In mathematics, the support of a function is the set of points where the function is not zero-valued, or the closure of that set.[1]:678 This concept is used very widely in mathematical analysis. In the form of functions with support that is bounded, it also plays a major part in various types of mathematical duality theories.

Formulation

A function supported in Y must vanish in X \ Y. For instance, f with domain X is said to have finite support if f(x) = 0 for all but a finite number of x in X. Since any superset of a support is also a support, attention is given to properties of subsets of X that admit at least one support for f. When the support of f (written supp(f)) is mentioned, it may be the intersection of all supports, {x in X:  f(x) ≠ 0} (the set-theoretic support), or the smallest support with some property of interest.

Closed supports

The most common situation occurs when X is a topological space (such as the real line) and f : XR is a continuous function. In this case, only closed supports of X are considered. So a (topological) support of f  is a closed subset of X outside of which f  vanishes, i.e.

\operatorname{supp}(f) := \overline{\{x \in X \,|\, f(x) \neq 0 \}}.

In this sense, supp(f ) is the intersection of all closed supports, since the intersection of closed sets is closed. The topological supp(f ) is the topological closure of the set-theoretic supp(f ).

Generalization

If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : XM. M may also be 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 ZN of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily { f  in ZN :f  has finite support } is the countable set of all integer sequences that have only finitely many nonzero entries.

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.

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

Compact support

Functions with compact support in X are those with support that is a compact subset of X. For example, if X is the real line, they are functions of bounded support and therefore vanish at infinity (and negative infinity).

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 ε > 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

|f(x) - I_C(x)f(x)| < \varepsilon

for all xX, where I_C is the indicator function of C. Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.

Support of a distribution

It is possible also to talk about the support of a distribution, such as the Dirac delta function δ(x) on the real line. In that example, we can consider test functions F, which are smooth functions with support not including the point 0. Since δ(F) (the distribution δ applied as linear functional to F) is 0 for such functions, we can say that the support of δ is {0} 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 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_{\alpha} of open sets, then for any test function \phi supported in \bigcup U_{\alpha}, 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 support of f as the complement of the largest open set on which f vanishes. For example, the support of the Dirac delta is \{0\}.

Singular support

In Fourier analysis in particular, it is interesting to study the singular support of a distribution. This has the intuitive interpretation as the set of points at which a distribution fails to be a smooth function.

For example, the Fourier transform of the Heaviside step function can, up to constant factors, be considered to be 1/x (a function) except 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 {0}: it cannot accurately be expressed as a function in relation to test functions with support including 0. It can be expressed as an application of a Cauchy principal value improper integral.

For distributions in several variables, singular supports allow one to define wave front sets 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 family of supports on a topological space X, 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 Φ of closed subsets of X is a family of supports, if it is down-closed and closed under finite union. Its extent is the union over Φ. A paracompactifying family of supports satisfies further than any Y in Φ is, with the subspace topology, a paracompact space; and has some Z in Φ which is a neighbourhood. If X is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.

See also

References

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