Fourier transform of a generalized function

An extension of the Fourier transform from test functions to generalized functions (cf. Generalized function). Let be a space of test functions on which the Fourier transformation ,

is defined and on which is an isomorphism of onto a space of test functions . Then the Fourier transformation is defined on the space of generalized functions by

and this is an isomorphism of onto the space of generalized functions .

Examples.

1) , . Here the inverse of is the operation

and the basic formulas for are

2) Let , , , where is the set of all functions for which , and where , .

3) , , where is the set of all entire functions satisfying the growth condition: There is a number such that for any one can find a such that

Fourier series of generalized functions.

If a generalized function is periodic with -period , , then and it can be expanded in a trigonometric series,

converging to in ; here

Examples.

4) , in particular .

5) , in particular .

6) , where is the Heaviside function.

The Fourier transform of the convolution of generalized functions.

Let the direct product of two generalized functions and in admit an extension to functions of the form , for all . Namely, suppose that for any sequence , , in with the properties: , , , , (uniformly on any compact set), the sequence

has a limit, denoted by , which does not depend on the sequence from the class indicated. In this case the functional that acts according to the formula , , is called the convolution of the generalized functions and , . The convolution does not exist for all pairs of generalized functions and . It automatically exists if for any the set

is bounded in (in particular if or has compact support). If the convolution exists, then it is commutative: ; and it commutes with shifts and with derivatives: ; the Dirac -function plays the role of "identity" : . Convolution is a non-associative operation. However, there are associative (and commutative) convolution algebras. The Dirac delta-function serves as the identity in them. For example, the set consisting of generalized functions from with support in a convex, acute, closed cone with vertex at is a convolution algebra. The set forms a convolution subalgebra of . Notation: , (when ). The formula for the Fourier transform of the convolution

is valid in the following cases:

a) , has compact support;

b) ;

c) , has compact support;

d) . In this case the product of the generalized functions and is understood to be the limit in of the product , , as , , where and denote the Laplace transforms of and (see Generalized functions, product of).

References

[1]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)
[2]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1 , Acad. Press (1964) (Translated from Russian)
[3]	L. Schwartz, "Théorie des distributions" , 2 , Hermann (1951)
[4]	P. Antosik, J. Mikusiński, R. Sikorski, "Theory of distributions. The sequential approach" , Elsevier (1973)
[5]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)

Comments

For other normalizations used in defining Fourier transforms, cf. Fourier transform.

The Heaviside function on is defined by if and if .

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
[a2]	D.S. Jones, "The theory of generalized functions" , Cambridge Univ. Press (1982)