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 , ), 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 is the generalized derivative of with respect to on , and one writes , if for any infinitely-differentiable function with compact support in (see Function of compact support)
A second, equivalent, definition of the generalized derivative 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  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 ).
|||S.L. Sobolev, "Le problème de Cauchy dans l'espace des fonctionnelles" Dokl. Akad. Nauk SSSR , 3 : 7 (1935) pp. 291–294|
|||S.L. Sobolev, "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales" Mat. Sb. , 1 (1936) pp. 39–72|
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|||S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) Zbl 0307.46024|
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Generalized derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_derivative&oldid=28801