Generalized function, derivative of a
A weak extension of the operation of ordinary differentiation. Let be a generalized function,
. The generalized (weak) derivative
![]() |
of order is defined by the equation
![]() | (*) |
Since the operation is linear and continuous from
into
, the functional
defined by the right-hand side of (*) is a generalized function in
. If
, then
for all
with
.
The following properties hold for the derivatives of a generalized function: the operation is linear and continuous from
into
; any generalized function in
is infinitely differentiable (in the generalized sense); the result of differentiation does not depend on the order; the Leibniz formula is valid for the differentiation of a product
, when
; and
.
Let . It may happen that a certain generalized derivative can be identified with some
-function. In this case
is a generalized derivative of function type.
Examples.
1) , where
is the Heaviside function and
is the Dirac function (cf. Delta-function for both).
2) The general solution of the equation in the class
is an arbitrary constant.
3) The trigonometric series
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converges in and it can be differentiated term-by-term in
infinitely many times.
References
[1] | L. Schwartz, "Théorie des distributions" , 1 , Hermann (1950) MR0035918 Zbl 0037.07301 |
[2] | S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) MR0165337 Zbl 0123.09003 |
Comments
References
[a1] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 MR0617913 Zbl 0435.46002 |
[a2] | L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001 |
Generalized function, derivative of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_function,_derivative_of_a&oldid=17641