View source for Support of a generalized function

{{TEX|done}}
The set of those (and only those) points such that in any neighbourhood of them the [[Generalized function|generalized function]] does not vanish. A generalized function $f$ in $\mathcal D^\prime(O)$ vanishes in an open set $O^\prime \subset O$ if $(f,\phi) = 0$ for all $\phi \in \mathcal D(O^\prime)$. Using a partition of unity it can be proved that if a generalized function $f$ in $\mathcal D^\prime(O)$ vanishes in some neighbourhood  $U_y \subset O$ for each point $y \in O$, then $f$ vanishes in $O$. The union of all neighbourhoods in which $f$ vanishes is called the zero set of $f$ and is denoted by $O_f$. The support of $f$, denoted by $\mathop{\mathrm{supp}} f$, is the complement of $O_f$ in $O$, that is, $\mathop{\mathrm{supp}} f = O \setminus O_f$ is a closed set in $O$. If $f$ is a continuous function in  $O$, then an equivalent definition of the support of $f$ is the following: $\mathop{\mathrm{supp}} f$ is the closure in $O$ of the complement of the set of points at which $f$ vanishes (cf. [[Support of a function|Support of a function]]). For example, $\mathop{\mathrm{supp}} x = \mathbf{R}^1$, $\mathop{\mathrm{supp}} \delta = \{0\}$.

The singular support ($\mathop{\mathrm{sing}} \mathop{\mathrm{supp}}$) 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 $C^\infty$-function. For example, $\mathop{\mathrm{sing}} \mathop{\mathrm{supp}} x = \emptyset$, $\mathop{\mathrm{sing}} \mathop{\mathrm{supp}} \delta = \{0\}$.


====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  "$f(x)=0$"  has no meaning for generalized functions $f$.

A point $x_0$ in the support of a generalized function $f$ is called an essential point of $f$, cf. [[#References|[a4]]].

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