Difference between revisions of "Support of a generalized function"
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| − | 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 $D^\prime(O)$ vanishes in an open set $O^\prime \subset O$ if $(f,\phi) = 0$ for all $\phi \in D(O^\prime)$. Using a partition of unity it can be proved that if a generalized function $f$ in $D^\prime(O)$ 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 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 $ | + | 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==== | ====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 $ | + | 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]]]. | A point $x_0$ in the support of a generalized function $f$ is called an essential point of $f$, cf. [[#References|[a4]]]. | ||
Latest revision as of 09:18, 12 August 2022
The set of those (and only those) points such that in any neighbourhood of them the 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). 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. [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) |
Support of a generalized function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_generalized_function&oldid=38872