# 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)