One of the integral transforms (cf.
Integral transform).
It is a linear operator
acting on a space whose elements are functions
of
real variables. The smallest domain of definition of
is the set
of all infinitely-differentiable functions
of compact support. For such functions
In a certain sense the most natural domain of definition of
is the set
of all infinitely-differentiable functions
that, together with their derivatives, vanish at infinity faster than any power of
.
Formula
(1)
still holds for
,
and
.
Moreover,
is an isomorphism of
onto itself, the inverse mapping
(the
inverse Fourier transform)
is the inverse of the Fourier transform and is given by the formula:
Formula
(1)
also acts on the space
of integrable functions. In order to enlarge the domain of definition of the operator
generalization of
(1)
is necessary. In classical analysis such
a generalization has been constructed for locally integrable
functions with some restriction on their behaviour as
(see
Fourier integral).
In the theory of
generalized functions
the definition of the operator
is free of many requirements of classical analysis.
The basic problems connected with the study of the Fourier transform
are: the investigation of the domain of definition
and the range of values
of
;
as well as studying properties of the mapping
(in particular, conditions for the existence of the inverse operator
and its expression). The inversion formula for the Fourier transform is very simple:
Under the action of the Fourier transform linear operators on the
original space, which are invariant with respect to a shift,
become (under certain conditions) multiplication operators in
the image space. In particular, the
convolution of two functions
and
goes over into the product of the functions
and
:
and differentiation induces multiplication by the independent variable:
In the spaces
,
,
the operator
is defined by the formula
(1)
on the set
and is a bounded operator from
into
,
:
(the
Hausdorff–Young inequality).
admits a continuous extension onto the whole space
which (for
)
is given by
Convergence is understood to be in the norm of
.
If
,
the image of
under the action of
does not coincide with
,
that is, the imbedding
is strict when
(for the case
see
Plancherel theorem).
The inverse operator
is defined on
by
The problem of extending the Fourier transform to a larger class
of functions arises constantly in analysis and its applications. See, for example,
Fourier transform of a generalized function.