Generalized derivative
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 $f$ 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) 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:
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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 |
Generalized derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_derivative&oldid=28804