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A function defined in some domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419801.png" />, having compact support belonging to this domain. More precisely, suppose that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419802.png" /> is defined on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419803.png" />. The support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419804.png" /> is the closure of the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419805.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419806.png" /> is different from zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419807.png" />. Thus one can also say that a function of compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419808.png" /> is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f0419809.png" /> such that its support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198010.png" /> is a closed bounded set located at a distance from the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198012.png" /> by a number greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198014.png" /> is sufficiently small.
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One usually considers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198015.png" />-times continuously-differentiable functions of compact support, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198016.png" /> is a given natural number. Even more often one considers infinitely-differentiable functions of compact support. The function
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 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198017.png" /></td> </tr></table>
+
A function defined in some domain of  $  E  ^ {n} $,
 +
having compact support belonging to this domain. More precisely, suppose that the function  $  f ( x) = f ( x _ {1} \dots x _ {n} ) $
 +
is defined on a domain  $  \Omega \subset  E  ^ {n} $.  
 +
The support of  $  f $
 +
is the closure of the set of points  $  x \in \Omega $
 +
for which  $  f ( x) $
 +
is different from zero  $  ( f ( x) \neq 0) $.  
 +
Thus one can also say that a function of compact support in  $  \Omega $
 +
is a function defined on  $  \Omega $
 +
such that its support  $  \Lambda $
 +
is a closed bounded set located at a distance from the boundary  $  \Gamma $
 +
of  $  \Omega $
 +
by a number greater than  $  \delta > 0 $,
 +
where  $  \delta $
 +
is sufficiently small.
  
can serve as an example of an infinitely-differentiable function of compact support in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198018.png" /> containing the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198019.png" />.
+
One usually considers  $  k $-
 +
times continuously-differentiable functions of compact support, where  $  k $
 +
is a given natural number. Even more often one considers infinitely-differentiable functions of compact support. The function
  
The set of all infinitely-differentiable functions of compact support in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198020.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198021.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198022.png" /> one can define linear functionals (generalized functions, cf. [[Generalized function|Generalized function]]). With the aid of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198023.png" /> one can define generalized solutions (cf. [[Generalized solution|Generalized solution]]) of boundary value problems.
+
$$
 +
\psi ( x) = \
 +
\left \{
  
In theorems concerned with problems on finding generalized solutions, it is often important to know whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198024.png" /> is dense in some concrete space of functions. It is known, for example, that if the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198025.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198026.png" /> is sufficiently smooth, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198027.png" /> is dense in the space of functions
+
\begin{array}{ll}
 +
e ^ {- 1/( | x |  ^ {2} - 1) } , & | x | < 1,
 +
| x |  ^ {2} = \sum _ {j = 1 } ^ { n }  x _ {j}  ^ {2} ,  \\
 +
0 , & | x | \geq  1 , \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198029.png" /></td> </tr></table>
+
\right .$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198030.png" />), that is, in the [[Sobolev space|Sobolev space]] of functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198031.png" /> that vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198032.png" /> along with their normal derivatives of order up to and including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041980/f04198034.png" />).
+
can serve as an example of an infinitely-differentiable function of compact support in a domain  $  \Omega $
 +
containing the sphere  $  | x | \leq  1 $.
 +
 
 +
The set of all infinitely-differentiable functions of compact support in a domain  $  \Omega \subset  E  ^ {n} $
 +
is denoted by  $  D $.
 +
On  $  D $
 +
one can define linear functionals (generalized functions, cf. [[Generalized function|Generalized function]]). With the aid of functions  $  v \in D $
 +
one can define generalized solutions (cf. [[Generalized solution|Generalized solution]]) of boundary value problems.
 +
 
 +
In theorems concerned with problems on finding generalized solutions, it is often important to know whether  $  D $
 +
is dense in some concrete space of functions. It is known, for example, that if the boundary  $  \Gamma $
 +
of a bounded domain  $  \Omega \subset  E  ^ {n} $
 +
is sufficiently smooth, then  $  D $
 +
is dense in the space of functions
 +
 
 +
$$
 +
{W ^ { o }  } {} _ {p}  ^ {r} ( \Omega )  = \
 +
\left \{ {f } : {
 +
f \in W _ {p}  ^ {r} ( \Omega ),\
 +
\left .
 +
\frac{\partial  ^ {s} f }{\partial  n  ^ {s} }
 +
\right | _  \Gamma  = 0,\
 +
s = 0 \dots r - 1
 +
} \right \}
 +
$$
 +
 
 +
( $  1 \leq  p \leq  \infty $),  
 +
that is, in the [[Sobolev space|Sobolev space]] of functions of class $  W _ {p}  ^ {r} ( \Omega ) $
 +
that vanish on $  \Gamma $
 +
along with their normal derivatives of order up to and including $  r - 1 $(
 +
$  r = 1, 2 ,\dots $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,  "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,  "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , '''1–2''' , Hermann  (1950–1951)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Lions,  E. Magenes,  "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , '''1–2''' , Hermann  (1950–1951)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Lions,  E. Magenes,  "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5</TD></TR></table>

Latest revision as of 19:40, 5 June 2020


A function defined in some domain of $ E ^ {n} $, having compact support belonging to this domain. More precisely, suppose that the function $ f ( x) = f ( x _ {1} \dots x _ {n} ) $ is defined on a domain $ \Omega \subset E ^ {n} $. The support of $ f $ is the closure of the set of points $ x \in \Omega $ for which $ f ( x) $ is different from zero $ ( f ( x) \neq 0) $. Thus one can also say that a function of compact support in $ \Omega $ is a function defined on $ \Omega $ such that its support $ \Lambda $ is a closed bounded set located at a distance from the boundary $ \Gamma $ of $ \Omega $ by a number greater than $ \delta > 0 $, where $ \delta $ is sufficiently small.

One usually considers $ k $- times continuously-differentiable functions of compact support, where $ k $ is a given natural number. Even more often one considers infinitely-differentiable functions of compact support. The function

$$ \psi ( x) = \ \left \{ \begin{array}{ll} e ^ {- 1/( | x | ^ {2} - 1) } , & | x | < 1, | x | ^ {2} = \sum _ {j = 1 } ^ { n } x _ {j} ^ {2} , \\ 0 , & | x | \geq 1 , \\ \end{array} \right .$$

can serve as an example of an infinitely-differentiable function of compact support in a domain $ \Omega $ containing the sphere $ | x | \leq 1 $.

The set of all infinitely-differentiable functions of compact support in a domain $ \Omega \subset E ^ {n} $ is denoted by $ D $. On $ D $ one can define linear functionals (generalized functions, cf. Generalized function). With the aid of functions $ v \in D $ one can define generalized solutions (cf. Generalized solution) of boundary value problems.

In theorems concerned with problems on finding generalized solutions, it is often important to know whether $ D $ is dense in some concrete space of functions. It is known, for example, that if the boundary $ \Gamma $ of a bounded domain $ \Omega \subset E ^ {n} $ is sufficiently smooth, then $ D $ is dense in the space of functions

$$ {W ^ { o } } {} _ {p} ^ {r} ( \Omega ) = \ \left \{ {f } : { f \in W _ {p} ^ {r} ( \Omega ),\ \left . \frac{\partial ^ {s} f }{\partial n ^ {s} } \right | _ \Gamma = 0,\ s = 0 \dots r - 1 } \right \} $$

( $ 1 \leq p \leq \infty $), that is, in the Sobolev space of functions of class $ W _ {p} ^ {r} ( \Omega ) $ that vanish on $ \Gamma $ along with their normal derivatives of order up to and including $ r - 1 $( $ r = 1, 2 ,\dots $).

References

[1] V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)
[2] S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)

Comments

References

[a1] L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1950–1951)
[a2] J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)
[a3] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
How to Cite This Entry:
Function of compact support. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_of_compact_support&oldid=47009
This article was adapted from an original article by S.M. Nikol'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article