Generalized derivative

from Encyclopedia of Mathematics.

An extension of the idea of a derivative to some classes of non-differentiable functions. The first definition is due to S.L. Sobolev, 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 $\phi = \frac{\partial f}{\partial x_j}$ (or $\phi = D_jf$), if for any infinitely-differentiable function $\psi$ with compact support in $\Omega$.
$$\begin{equation}\label{eq:1} \int\limits_{\Omega}f(x)\frac{\partial \psi}{\partial x_j}(x)\,dx = -\int\limits_{\Omega}\phi(x) \psi(x)\,dx. \end{equation}$$
Generalized derivatives of a higher order $D^{\alpha}_xf$ are defined as follows
$$\begin{equation}\label{eq:2} \int\limits_{\Omega}f(x)D^{\alpha}_x\psi(x)\,dx = (-1)^{|\alpha|}\int\limits_{\Omega}\phi(x) \psi(x)\,dx, \end{equation}$$
where multiindex $\alpha = (\alpha_1,\dots,\alpha_n)$, $x=(x_1,\dots,x_n)$, $|\alpha| = \alpha_1+\dots+\alpha_n$ and differential operator $D^{\alpha}_x$ is just short notation for $\frac{\partial^{\alpha_1+\dots+\alpha_n}}{\partial x_1^{\alpha_1}\dots\partial x_n^{\alpha_n}}$. In this case $\phi = D^{\alpha}_xf$ is $\alpha$-th generalized derivatives of function $f$.

Another equivalent definition of the generalized derivative $\frac{\partial f}{\partial x_j}$ 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) $\frac{\partial f}{\partial x_j}$ almost-everywhere on $\Omega$. If a function $\phi = \frac{\partial f}{\partial x_j}$ 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 $\frac{\partial f}{\partial x_j}$ 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).