Support of a generalized function

The set of those (and only those) points such that in any neighbourhood of them the generalized function does not vanish. A generalized function in vanishes in an open set if for all . Using a partition of unity it can be proved that if a generalized function in vanishes in some neighbourhood for each point , then vanishes in . The union of all neighbourhoods in which vanishes is called the zero set of and is denoted by . The support of , denoted by , is the complement of in , that is, is a closed set in . If is a continuous function in , then an equivalent definition of the support of is the following: is the closure in of the complement of the set of points at which vanishes (cf. Support of a function). For example, , .

The singular support () of a generalized function is the set of those (and only those) points such that in any neighbourhood of them the generalized function is not equal to a -function. For example, , .

Comments

The notion of a zero set as used above is somewhat unusual and does not agree with the zero set of an ordinary function (not a generalized function) as the set of points where that function assumes the value zero. Of course, the statement "fx=0" has no meaning for generalized functions .

A point in the support of a generalized function is called an essential point of , cf. [a4].

References

[a1]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1966)
[a2]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) pp. §7.7
[a3]	V.S. Vladimirov, Yu.N. Drozzinov, B.I. Zavialov, "Tauberian theory for generalized functions" , Kluwer (1988) (Translated from Russian)
[a4]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1. Properties and operations , Acad. Press (1964) pp. 5 (Translated from Russian)