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
,
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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
.
Contents
Examples.
1) ,
. Here the inverse of
is the operation
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and the basic formulas for are
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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
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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
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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
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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
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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
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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) |
Fourier transform of a generalized function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_transform_of_a_generalized_function&oldid=12995