Namespaces
Variants
Actions

Difference between revisions of "Generalized derivative"

From Encyclopedia of Mathematics
Jump to: navigation, search
 
(13 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
{{TEX|done}}
 +
 
''of function type''
 
''of function type''
  
 
An extension of the idea of a derivative to some classes of non-differentiable functions. The first definition is due to S.L. Sobolev (see [[#References|[1]]], [[#References|[2]]]), who arrived at a definition of a generalized derivative from the point of view of his concept of a [[Generalized function|generalized function]].
 
An extension of the idea of a derivative to some classes of non-differentiable functions. The first definition is due to S.L. Sobolev (see [[#References|[1]]], [[#References|[2]]]), who arrived at a definition of a generalized derivative from the point of view of his concept of a [[Generalized function|generalized function]].
  
Let $f$ and $\phi$ be locally integrable functions on an open set $\Omega\subset \mathbb R^n$, that is, Lebesgue integrable on any closed bounded set $F\subset\Omega$. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g0437907.png" /> is the generalized derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g0437908.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g0437909.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379010.png" />, and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379011.png" />, if for any infinitely-differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379012.png" /> with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379013.png" /> (see [[Function of compact support|Function of compact support]])
+
Let $f$ and $\phi$ be locally integrable functions on an open set $\Omega\subset \mathbb R^n$, that is, Lebesgue integrable on any closed bounded set $F\subset\Omega$. Then $\phi$ is the generalized derivative of $f$ with respect to $x_j$ on $\Omega$, and one writes $\phi = \partial f / \partial x_j$ (or $\phi = D_jf$), if for any infinitely-differentiable function $\psi$ with compact support in $\Omega$ (see [[Function of compact support|Function of compact support]])
 +
 
 +
\begin{equation}\label{eq:1}
 +
\int\limits_{\Omega}f(x)\frac{\partial \psi}{\partial x_j}(x)\,dx = -\int\limits_{\Omega}\phi(x) \psi(x)\,dx.
 +
\end{equation}
 +
 
 +
Generalized derivatives of a higher order $D^{\alpha}f$ are defined as follows.
 +
 
 +
\begin{equation}\label{eq:2}
 +
\int\limits_{\Omega}f(x)D^{\alpha}\psi(x)\,dx = (-1)^{|\alpha|}\int\limits_{\Omega}\phi(x) \psi(x)\,dx,
 +
\end{equation}
 +
 
 +
where multiindex $\alpha = (\alpha_1,\dots,\alpha_n)$, $x=(x_1,\dots,x_n)$, $|\alpha| = \alpha_1+\dots+\alpha_n$ and differential operator
 +
\begin{equation*}
 +
D^{\alpha}  = \frac{\partial^{\alpha_1+\dots+\alpha_n}}{\partial x_1^{\alpha_1}\dots\partial x_n^{\alpha_n}}
 +
\end{equation*}
 +
is just short notation.  In this case $\phi = D^{\alpha}f$ is $\alpha$-th generalized derivatives of function $f$.
 +
 +
== Alternative Definitions ==
 +
 
 +
Another equivalent definition of the generalized derivative $\partial f / \partial x_j$ is the following. If $f$ can be modified on a set of $n$-dimensional measure zero so that the modified function (which will again be denoted by $f$) is locally [[Absolutely_continuous_function#Absolute_continuity_of_a_function|absolutely continuous]] with respect to $x_j$ for almost-all (in the sense of the $(n-1)$-dimensional Lebesgue measure) $x^j=(x_1,\dots,x_{j-1},x_{j+1},\dots,x_n)$ belonging to the projection $\Omega^j$ of $\Omega$ onto the plane $x_j=0$, then $f$ has partial derivative (in the usual sense of the word) $\partial f / \partial x_j$  [[Almost-everywhere|almost-everywhere]] on $\Omega$. If a function $\phi = \partial f / \partial x_j$ almost-everywhere on $\Omega$, then $\phi$ is a generalized derivative of $f$ with respect to $x_j$ on $\Omega$. Thus, a generalized derivative is defined almost-everywhere on $\Omega$ if $f$ is continuous and the ordinary derivative $\partial f / \partial x_j$ is continuous on $\Omega$, then it is also a generalized derivative of $f$ with respect to $x_j$ on $\Omega$.
  
<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/g043790/g04379014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
  
A second, equivalent, definition of the generalized derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379015.png" /> is the following. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379016.png" /> can be modified on a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379017.png" />-dimensional measure zero so that the modified function (which will again be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379018.png" />) is locally absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379019.png" /> for almost-all (in the sense of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379020.png" />-dimensional Lebesgue measure) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379021.png" /> belonging to the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379023.png" /> onto the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379024.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379025.png" /> has partial derivative (in the usual sense of the word) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379026.png" /> [[Almost-everywhere|almost-everywhere]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379027.png" />. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379028.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379030.png" /> is a generalized derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379031.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379033.png" />. Thus, a generalized derivative is defined almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379034.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379035.png" /> is continuous and the ordinary derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379036.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379037.png" />, then it is also a generalized derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379038.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379040.png" />.
+
There is the third equivalent definition of a generalized derivative. Suppose that there is sequence of functions $f_{\nu}\in C^1(\Omega)$, $\nu=1,2,\dots$ such that for each closed bounded set $F\subset\Omega$, the functions $f$ and $\phi$, defined on $\Omega$, have the properties:
  
Generalized derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379041.png" /> of a higher order are defined by induction. They are independent (almost-everywhere) of the order of differentiation.
+
\begin{equation*}
 +
\lim\limits_{\nu\to\infty}\int\limits_{F}|f_{\nu}-f|\,dx=0,
 +
\end{equation*}
  
There is a third equivalent definition of a generalized derivative. Suppose that for each closed bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379042.png" />, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379044.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379045.png" />, have the properties:
+
\begin{equation*}
 +
\lim\limits_{\nu\to\infty}\int\limits_{F}\left|\frac{\partial f_{\nu}}{\partial x_j}-\phi\right|\,dx=0.
 +
\end{equation*}
  
<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/g043790/g04379046.png" /></td> </tr></table>
+
Then $\phi$ is the generalized partial derivative of $f$ with respect to $x_j$ on $\Omega$ ($\phi = \partial f / \partial x_j$) (see also [[Sobolev space|Sobolev space]]).
  
<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/g043790/g04379047.png" /></td> </tr></table>
 
  
and suppose that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379049.png" /> and their partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379050.png" /> are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379051.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379052.png" /> is the generalized partial derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379053.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379054.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379055.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379056.png" />) (see also [[Sobolev space|Sobolev space]]).
+
From the point of view of the theory of generalized functions, a generalized derivative can be defined as follows: Suppose one is given a function $f$ that is locally summable on $\Omega$, considered as a generalized function, and let $\partial f / \partial x_j = \phi$ be the partial derivative in the sense of the theory of generalized functions. If $\phi$ represents a function that is locally summable on $\Omega$, then $\phi$ is a generalized derivative (in the first (original) sense).
  
From the point of view of the theory of generalized functions, a generalized derivative can be defined as follows: Suppose one is given a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379057.png" /> that is locally summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379058.png" />, considered as a generalized function, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379059.png" /> be the partial derivative in the sense of the theory of generalized functions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379060.png" /> represents a function that is locally summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379061.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379062.png" /> is a generalized derivative (in the first (original) sense).
+
The concept of a generalized derivative had been considered even earlier (see [[#References|[3]]] for example, where generalized derivatives with integrable square on $\Omega$ are considered). Subsequently, many investigators arrived at this concept independently of their predecessors (on this question see [[#References|[4]]]).
  
The concept of a generalized derivative had been considered even earlier (see [[#References|[3]]] for example, where generalized derivatives with integrable square on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379063.png" /> are considered). Subsequently, many investigators arrived at this concept independently of their predecessors (on this question see [[#References|[4]]]).
 
  
====References====
+
== Examples ==
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.L. Sobolev,   "Le problème de Cauchy dans l'espace des fonctionnelles"  ''Dokl. Akad. Nauk SSSR'' , '''3''' :  7  (1935) pp. 291–294  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,  "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales"  ''Mat. Sb.'' , '''1'''  (1936) pp. 39–72  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Levi,  "Sul principio di Dirichlet"  ''Rend. Circ. Mat. Palermo'' , '''22'''  (1906)  pp. 293–359  {{MR|}}  {{ZBL|37.0414.06}}  {{ZBL|37.0414.04}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975) (Translated from Russian)  {{MR|}} {{ZBL|0307.46024}} </TD></TR></table>
+
Let $\Omega$ is interval $(-1,1)\subset \mathbb R$. Then the function $f(x) = |x|$ has the generalized derivative $Du(x)$ which is $\operatorname{sgn} x$.
  
 +
At the same conditions the function $f(x) = \operatorname{sgn} x$ does not have generalized derivative.
  
 +
==References==
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.L. Sobolev,  "Le problème de Cauchy dans l'espace des fonctionnelles"  ''Dokl. Akad. Nauk SSSR'' , '''3''' :  7  (1935)  pp. 291–294  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,  "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales"  ''Mat. Sb.'' , '''1'''  (1936)  pp. 39–72  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Levi,  "Sul principio di Dirichlet"  ''Rend. Circ. Mat. Palermo'' , '''22'''  (1906)  pp. 293–359  {{MR|}}  {{ZBL|37.0414.06}}  {{ZBL|37.0414.04}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)  {{MR|}} {{ZBL|0307.46024}} </TD></TR></table>
  
 
====Comments====
 
====Comments====

Latest revision as of 10:49, 30 November 2012


of function type

An extension of the idea of a derivative to some classes of non-differentiable functions. The first definition is due to S.L. Sobolev (see [1], [2]), who arrived at a definition of a generalized derivative from the point of view of his concept of a generalized function.

Let $f$ and $\phi$ be locally integrable functions on an open set $\Omega\subset \mathbb R^n$, that is, Lebesgue integrable on any closed bounded set $F\subset\Omega$. Then $\phi$ is the generalized derivative of $f$ with respect to $x_j$ on $\Omega$, and one writes $\phi = \partial f / \partial x_j$ (or $\phi = D_jf$), if for any infinitely-differentiable function $\psi$ with compact support in $\Omega$ (see Function of compact support)

\begin{equation}\label{eq:1} \int\limits_{\Omega}f(x)\frac{\partial \psi}{\partial x_j}(x)\,dx = -\int\limits_{\Omega}\phi(x) \psi(x)\,dx. \end{equation}

Generalized derivatives of a higher order $D^{\alpha}f$ are defined as follows.

\begin{equation}\label{eq:2} \int\limits_{\Omega}f(x)D^{\alpha}\psi(x)\,dx = (-1)^{|\alpha|}\int\limits_{\Omega}\phi(x) \psi(x)\,dx, \end{equation}

where multiindex $\alpha = (\alpha_1,\dots,\alpha_n)$, $x=(x_1,\dots,x_n)$, $|\alpha| = \alpha_1+\dots+\alpha_n$ and differential operator \begin{equation*} D^{\alpha} = \frac{\partial^{\alpha_1+\dots+\alpha_n}}{\partial x_1^{\alpha_1}\dots\partial x_n^{\alpha_n}} \end{equation*} is just short notation. In this case $\phi = D^{\alpha}f$ is $\alpha$-th generalized derivatives of function $f$.

Alternative Definitions

Another equivalent definition of the generalized derivative $\partial f / \partial x_j$ is the following. If $f$ can be modified on a set of $n$-dimensional measure zero so that the modified function (which will again be denoted by $f$) is locally absolutely continuous with respect to $x_j$ for almost-all (in the sense of the $(n-1)$-dimensional Lebesgue measure) $x^j=(x_1,\dots,x_{j-1},x_{j+1},\dots,x_n)$ belonging to the projection $\Omega^j$ of $\Omega$ onto the plane $x_j=0$, then $f$ has partial derivative (in the usual sense of the word) $\partial f / \partial x_j$ almost-everywhere on $\Omega$. If a function $\phi = \partial f / \partial x_j$ almost-everywhere on $\Omega$, then $\phi$ is a generalized derivative of $f$ with respect to $x_j$ on $\Omega$. Thus, a generalized derivative is defined almost-everywhere on $\Omega$ if $f$ is continuous and the ordinary derivative $\partial f / \partial x_j$ is continuous on $\Omega$, then it is also a generalized derivative of $f$ with respect to $x_j$ on $\Omega$.


There is the third equivalent definition of a generalized derivative. Suppose that there is sequence of functions $f_{\nu}\in C^1(\Omega)$, $\nu=1,2,\dots$ such that for each closed bounded set $F\subset\Omega$, the functions $f$ and $\phi$, defined on $\Omega$, have the properties:

\begin{equation*} \lim\limits_{\nu\to\infty}\int\limits_{F}|f_{\nu}-f|\,dx=0, \end{equation*}

\begin{equation*} \lim\limits_{\nu\to\infty}\int\limits_{F}\left|\frac{\partial f_{\nu}}{\partial x_j}-\phi\right|\,dx=0. \end{equation*}

Then $\phi$ is the generalized partial derivative of $f$ with respect to $x_j$ on $\Omega$ ($\phi = \partial f / \partial x_j$) (see also Sobolev space).


From the point of view of the theory of generalized functions, a generalized derivative can be defined as follows: Suppose one is given a function $f$ that is locally summable on $\Omega$, considered as a generalized function, and let $\partial f / \partial x_j = \phi$ be the partial derivative in the sense of the theory of generalized functions. If $\phi$ represents a function that is locally summable on $\Omega$, then $\phi$ is a generalized derivative (in the first (original) sense).

The concept of a generalized derivative had been considered even earlier (see [3] for example, where generalized derivatives with integrable square on $\Omega$ are considered). Subsequently, many investigators arrived at this concept independently of their predecessors (on this question see [4]).


Examples

Let $\Omega$ is interval $(-1,1)\subset \mathbb R$. Then the function $f(x) = |x|$ has the generalized derivative $Du(x)$ which is $\operatorname{sgn} x$.

At the same conditions the function $f(x) = \operatorname{sgn} x$ does not have generalized derivative.

References

[1] S.L. Sobolev, "Le problème de Cauchy dans l'espace des fonctionnelles" Dokl. Akad. Nauk SSSR , 3 : 7 (1935) pp. 291–294
[2] S.L. Sobolev, "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales" Mat. Sb. , 1 (1936) pp. 39–72
[3] B. Levi, "Sul principio di Dirichlet" Rend. Circ. Mat. Palermo , 22 (1906) pp. 293–359 Zbl 37.0414.06 Zbl 37.0414.04
[4] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) Zbl 0307.46024

Comments

References

[a1] S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965) MR0178246 Zbl 0142.37401
How to Cite This Entry:
Generalized derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_derivative&oldid=28801
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