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''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]])
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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 $\phi = \frac{\partial f}{\partial x_j}$, if for any infinitely-differentiable function $\psi$ with compact support in $\Omega$ (see [[Function of compact support|Function of compact support]])
  
<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>
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\begin{equation}\label{eq:1}
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\int\limits_{\Omega}f(x)\frac{\partial \psi}{\partial x_j}(x)\,dx = -\int\limits_{\Omega}\phi(x) \psi(x)\,dx.
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\end{equation}
  
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" />.
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Another equivalent definition of the generalized derivative $\frac{\partial f}{\partial x_j}$ 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_function#Absolute_continuity_of_a_function|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="absmi
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ddle" 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" />.
  
 
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.
 
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.

Revision as of 14:02, 21 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 $\phi = \frac{\partial f}{\partial x_j}$, 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}

Another equivalent definition of the generalized derivative $\frac{\partial f}{\partial x_j}$ is the following. If can be modified on a set of -dimensional measure zero so that the modified function (which will again be denoted by ) is locally absolutely continuous with respect to for almost-all (in the sense of the -dimensional Lebesgue measure) belonging to the projection of onto the plane , then has partial derivative (in the usual sense of the word) almost-everywhere on . If a function almost-everywhere on , then is a generalized derivative of with respect to on . Thus, a generalized derivative is defined almost-everywhere on ; if is continuous and the ordinary derivative is continuous on , then it is also a generalized derivative of with respect to on .

Generalized derivatives of a higher order are defined by induction. They are independent (almost-everywhere) of the order of differentiation.

There is a third equivalent definition of a generalized derivative. Suppose that for each closed bounded set , the functions and , defined on , have the properties:

and suppose that the functions , and their partial derivatives are continuous on . Then is the generalized partial derivative of with respect to on () (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 that is locally summable on , considered as a generalized function, and let be the partial derivative in the sense of the theory of generalized functions. If represents a function that is locally summable on , then 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 are considered). Subsequently, many investigators arrived at this concept independently of their predecessors (on this question see [4]).

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=28802
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