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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912901.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912902.png" /> vanishes in an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912903.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912904.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912905.png" />. Using a partition of unity it can be proved that if a generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912906.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912907.png" /> vanishes in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912908.png" /> for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s0912909.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129010.png" /> vanishes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129011.png" />. The union of all neighbourhoods in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129012.png" /> vanishes is called the zero set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129013.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129014.png" />. The support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129015.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129016.png" />, is the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129018.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129019.png" /> is a closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129021.png" /> is a continuous function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129022.png" />, then an equivalent definition of the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129023.png" /> is the following: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129024.png" /> is the closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129025.png" /> of the complement of the set of points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129026.png" /> vanishes (cf. [[Support of a function|Support of a function]]). For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129028.png" />.
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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 singular support (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129029.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129030.png" />-function. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129032.png" />.
 
  
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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  "fx=0"  has no meaning for generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129033.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129034.png" /> in the support of a generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129035.png" /> is called an essential point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091290/s09129036.png" />, cf. [[#References|[a4]]].
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A point $x_0$ in the support of a generalized function $f$ is called an essential point of $f$, cf. [[#References|[a4]]].
  
 
====References====
 
====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>
 
<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>

Revision as of 04:05, 4 January 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 $D^\prime(O)$ 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). 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)
How to Cite This Entry:
Support of a generalized function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_generalized_function&oldid=12815
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article