# 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 | + | Let $f$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g0437902.png" /> be locally integrable functions on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g0437903.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g0437904.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g0437905.png" />, that is, Lebesgue integrable on any closed bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g0437906.png" />. 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]]) |

<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> | <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> |

## Revision as of 12:27, 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 be locally integrable functions on an open set in the -dimensional space , that is, Lebesgue integrable on any closed bounded set . 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)

(1) |

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 [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 |

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Generalized derivative.

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