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

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

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