Support (mathematics)

Summary

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.

FormulationEdit

Suppose that   is a real-valued function whose domain is an arbitrary set   The set-theoretic support 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 finite support.

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 supportEdit

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 support of  ,  , or the closed support of  , is defined topologically as the closure (taken in  ) of the subset of   where   is non-zero[1][2][3] 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.[4]

Compact supportEdit

Functions with compact support 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 bounded support, 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 supportEdit

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 essential support 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[5]

 

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.[5][6]

GeneralizationEdit

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.[7]

In probability and measure theoryEdit

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).[8]

Note that the word support can refer to the logarithm of the likelihood of a probability density function.[9]

Support of a distributionEdit

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 support of   as the complement of the largest open set on which   vanishes. For example, the support of the Dirac delta is  

Singular supportEdit

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   (a function) except 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 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 supportsEdit

An abstract notion of family of supports 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 family of supports, if it is down-closed and closed under finite union. Its extent is the union over   A paracompactifying 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 alsoEdit

CitationsEdit

  1. ^ Folland, Gerald B. (1999). Real Analysis, 2nd ed. New York: John Wiley. p. 132.
  2. ^ Hörmander, Lars (1990). Linear Partial Differential Equations I, 2nd ed. Berlin: Springer-Verlag. p. 14.
  3. ^ Pascucci, Andrea (2011). PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series. Berlin: Springer-Verlag. p. 678. doi:10.1007/978-88-470-1781-8. ISBN 978-88-470-1780-1.
  4. ^ Rudin, Walter (1987). Real and Complex Analysis, 3rd ed. New York: McGraw-Hill. p. 38.
  5. ^ a b Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. Vol. 14 (2nd ed.). American Mathematical Society. p. 13. ISBN 978-0821827833.
  6. ^ In a similar way, one uses the essential supremum of a measurable function instead of its supremum.
  7. ^ Tomasz, Kaczynski (2004). Computational homology. Mischaikow, Konstantin Michael,, Mrozek, Marian. New York: Springer. p. 445. ISBN 9780387215976. OCLC 55897585.
  8. ^ Taboga, Marco. "Support of a random variable". statlect.com. Retrieved 29 November 2017.
  9. ^ Edwards, A. W. F. (1992). Likelihood (Expanded ed.). Baltimore: Johns Hopkins University Press. pp. 31–34. ISBN 0-8018-4443-6.

ReferencesEdit