# Difference between revisions of "Generalized derivative"

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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 $\phi$ is the generalized derivative of $f$ with respect to $x_j$ on $\Omega$, and one writes $\phi = | + | 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: | + | \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. | \int\limits_{\Omega}f(x)\frac{\partial \psi}{\partial x_j}(x)\,dx = -\int\limits_{\Omega}\phi(x) \psi(x)\,dx. | ||

\end{equation} | \end{equation} | ||

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

− | \begin{equation}\label{eq: | + | \begin{equation}\label{eq:2} |

− | \int\limits_{\Omega}f(x)D^{\alpha} | + | \int\limits_{\Omega}f(x)D^{\alpha}\psi(x)\,dx = (-1)^{|\alpha|}\int\limits_{\Omega}\phi(x) \psi(x)\,dx, |

\end{equation} | \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 | + | 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 $f$. | ||

− | Another equivalent definition of the generalized derivative $ | + | == 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 $\Omega$. | ||

− | There is the third equivalent definition of a generalized derivative. Suppose that for each closed bounded set $F\subset\Omega$, the functions | + | 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|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 [[#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]]]). | |

− | |||

− | == | + | == 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== | ||

+ | <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=28804