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A weak extension of the operation of ordinary [[Differentiation|differentiation]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438201.png" /> be a [[Generalized function|generalized function]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438202.png" />. The generalized (weak) derivative
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<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/g/g043/g043820/g0438203.png" /></td> </tr></table>
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of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438204.png" /> is defined by the equation
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A weak extension of the operation of ordinary [[Differentiation|differentiation]]. Let  $  f $
 +
be a [[Generalized function|generalized function]],  $  f \in D  ^  \prime  ( O) $.  
 +
The generalized (weak) derivative
  
<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/g/g043/g043820/g0438205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$
 +
D  ^  \alpha  f  = \
  
Since the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438206.png" /> is linear and continuous from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438207.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438208.png" />, the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438209.png" /> defined by the right-hand side of (*) is a generalized function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382014.png" />.
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\frac{\partial  ^ {| \alpha | } f }{\partial  x _ {1} ^ {\alpha _ {1} } \dots
 +
\partial  x _ {n} ^ {\alpha _ {n} } }
 +
,\ \
 +
| \alpha | = \alpha _ {1} + \dots + \alpha _ {n} ,
 +
$$
  
The following properties hold for the derivatives of a generalized function: the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382015.png" /> is linear and continuous from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382016.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382017.png" />; any generalized function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382018.png" /> is infinitely differentiable (in the generalized sense); the result of differentiation does not depend on the order; the [[Leibniz formula|Leibniz formula]] is valid for the differentiation of a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382019.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382020.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382021.png" />.
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of order  $  \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $
 +
is defined by the equation
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382022.png" />. It may happen that a certain generalized derivative can be identified with some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382023.png" />-function. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382024.png" /> is a [[Generalized derivative|generalized derivative]] of function type.
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$$ \tag{* }
 +
( D  ^  \alpha  f , \phi )  = \
 +
( - 1 ) ^ {| \alpha | }
 +
( f , D  ^  \alpha  \phi ) ,\ \
 +
\phi \in D ( O) .
 +
$$
 +
 
 +
Since the operation  $  \phi \mapsto (- 1) ^ {| \alpha | } D  ^  \alpha  \phi $
 +
is linear and continuous from  $  D ( O) $
 +
into  $  D ( O) $,
 +
the functional  $  D  ^  \alpha  f $
 +
defined by the right-hand side of (*) is a generalized function in  $  D  ^  \prime  ( O) $.  
 +
If  $  f \in C  ^ {p} ( O) $,
 +
then  $  D  ^  \alpha  f \in C ^ {p - | \alpha | } ( O) $
 +
for all  $  \alpha $
 +
with  $  | \alpha | \leq  p $.
 +
 
 +
The following properties hold for the derivatives of a generalized function: the operation  $  f \mapsto D  ^  \alpha  f $
 +
is linear and continuous from  $  D  ^  \prime  ( O) $
 +
into  $  D  ^  \prime  ( O) $;
 +
any generalized function in  $  D  ^  \prime  ( O) $
 +
is infinitely differentiable (in the generalized sense); the result of differentiation does not depend on the order; the [[Leibniz formula|Leibniz formula]] is valid for the differentiation of a product  $  af $,
 +
when  $  a \in C  ^  \infty  ( O) $;
 +
and  $  \supp  D  ^  \alpha  f \subset  \supp  f $.
 +
 
 +
Let  $  f \in L _ { \mathop{\rm loc}  }  ^ {1} ( O) $.
 +
It may happen that a certain generalized derivative can be identified with some $  L _ { \mathop{\rm loc}  }  ^ {1} ( O) $-
 +
function. In this case $  D  ^  \alpha  f ( x) $
 +
is a [[Generalized derivative|generalized derivative]] of function type.
  
 
===Examples.===
 
===Examples.===
  
 +
1)  $  \theta  ^  \prime  = \delta $,
 +
where  $  \theta $
 +
is the Heaviside function and  $  \delta $
 +
is the Dirac function (cf. [[Delta-function|Delta-function]] for both).
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382026.png" /> is the Heaviside function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382027.png" /> is the Dirac function (cf. [[Delta-function|Delta-function]] for both).
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2) The general solution of the equation $  u  ^  \prime  = 0 $
 
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in the class $  D  ^  \prime  $
2) The general solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382028.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382029.png" /> is an arbitrary constant.
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is an arbitrary constant.
  
 
3) The [[Trigonometric series|trigonometric series]]
 
3) The [[Trigonometric series|trigonometric series]]
  
<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/g/g043/g043820/g04382030.png" /></td> </tr></table>
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$$
 +
\sum _ {k = - \infty } ^  \infty  a _ {k} e  ^ {ikx} ,\ \
 +
| a _ {k} |  \leq  A ( 1 + | k | )  ^ {m} ,
 +
$$
  
converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382031.png" /> and it can be differentiated term-by-term in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382032.png" /> infinitely many times.
+
converges in $  D  ^  \prime  $
 +
and it can be differentiated term-by-term in $  D  ^  \prime  $
 +
infinitely many times.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , '''1''' , Hermann  (1950)  {{MR|0035918}} {{ZBL|0037.07301}} </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)  {{MR|0165337}} {{ZBL|0123.09003}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , '''1''' , Hermann  (1950)  {{MR|0035918}} {{ZBL|0037.07301}} </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)  {{MR|0165337}} {{ZBL|0123.09003}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5  {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5  {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR></table>

Revision as of 19:41, 5 June 2020


A weak extension of the operation of ordinary differentiation. Let $ f $ be a generalized function, $ f \in D ^ \prime ( O) $. The generalized (weak) derivative

$$ D ^ \alpha f = \ \frac{\partial ^ {| \alpha | } f }{\partial x _ {1} ^ {\alpha _ {1} } \dots \partial x _ {n} ^ {\alpha _ {n} } } ,\ \ | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} , $$

of order $ \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $ is defined by the equation

$$ \tag{* } ( D ^ \alpha f , \phi ) = \ ( - 1 ) ^ {| \alpha | } ( f , D ^ \alpha \phi ) ,\ \ \phi \in D ( O) . $$

Since the operation $ \phi \mapsto (- 1) ^ {| \alpha | } D ^ \alpha \phi $ is linear and continuous from $ D ( O) $ into $ D ( O) $, the functional $ D ^ \alpha f $ defined by the right-hand side of (*) is a generalized function in $ D ^ \prime ( O) $. If $ f \in C ^ {p} ( O) $, then $ D ^ \alpha f \in C ^ {p - | \alpha | } ( O) $ for all $ \alpha $ with $ | \alpha | \leq p $.

The following properties hold for the derivatives of a generalized function: the operation $ f \mapsto D ^ \alpha f $ is linear and continuous from $ D ^ \prime ( O) $ into $ D ^ \prime ( O) $; any generalized function in $ D ^ \prime ( O) $ is infinitely differentiable (in the generalized sense); the result of differentiation does not depend on the order; the Leibniz formula is valid for the differentiation of a product $ af $, when $ a \in C ^ \infty ( O) $; and $ \supp D ^ \alpha f \subset \supp f $.

Let $ f \in L _ { \mathop{\rm loc} } ^ {1} ( O) $. It may happen that a certain generalized derivative can be identified with some $ L _ { \mathop{\rm loc} } ^ {1} ( O) $- function. In this case $ D ^ \alpha f ( x) $ is a generalized derivative of function type.

Examples.

1) $ \theta ^ \prime = \delta $, where $ \theta $ is the Heaviside function and $ \delta $ is the Dirac function (cf. Delta-function for both).

2) The general solution of the equation $ u ^ \prime = 0 $ in the class $ D ^ \prime $ is an arbitrary constant.

3) The trigonometric series

$$ \sum _ {k = - \infty } ^ \infty a _ {k} e ^ {ikx} ,\ \ | a _ {k} | \leq A ( 1 + | k | ) ^ {m} , $$

converges in $ D ^ \prime $ and it can be differentiated term-by-term in $ D ^ \prime $ infinitely many times.

References

[1] L. Schwartz, "Théorie des distributions" , 1 , Hermann (1950) MR0035918 Zbl 0037.07301
[2] S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) MR0165337 Zbl 0123.09003

Comments

References

[a1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 MR0617913 Zbl 0435.46002
[a2] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001
How to Cite This Entry:
Generalized function, derivative of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_function,_derivative_of_a&oldid=47071
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article