View source for Generalized function, derivative of a

A weak extension of the operation of ordinary [[Differentiation|differentiation]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438201.png" /> be a [[Generalized function|generalized function]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438202.png" />. The generalized (weak) derivative

<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/g043820/g0438203.png" /></td> </tr></table>

of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438204.png" /> is defined by the equation

<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/g043820/g0438205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

Since the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438206.png" /> is linear and continuous from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438207.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438208.png" />, the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g0438209.png" /> defined by the right-hand side of (*) is a generalized function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382014.png" />.

The following properties hold for the derivatives of a generalized function: the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382015.png" /> is linear and continuous from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382016.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382017.png" />; any generalized function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382018.png" /> is infinitely differentiable (in the generalized sense); the result of differentiation does not depend on the order; the [[Leibniz formula|Leibniz formula]] is valid for the differentiation of a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382019.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382020.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382021.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382022.png" />. It may happen that a certain generalized derivative can be identified with some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382023.png" />-function. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382024.png" /> is a [[Generalized derivative|generalized derivative]] of function type.

===Examples.===


1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382026.png" /> is the Heaviside function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382027.png" /> is the Dirac function (cf. [[Delta-function|Delta-function]] for both).

2) The general solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382028.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382029.png" /> is an arbitrary constant.

3) The [[Trigonometric series|trigonometric series]]

<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/g043820/g04382030.png" /></td> </tr></table>

converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382031.png" /> and it can be differentiated term-by-term in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043820/g04382032.png" /> infinitely many times.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1''' , Hermann  (1950)  {{MR|0035918}} {{ZBL|0037.07301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,   "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)  {{MR|0165337}} {{ZBL|0123.09003}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5  {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR></table>