Fourier transform
One of the integral transforms (cf. Integral transform). It is a linear operator $F$ acting on a space whose elements are functions $f$ of $n$ real variables. The smallest domain of definition of $F$ is the set $D=C_0^\infty$ of all infinitely-differentiable functions $\phi$ of compact support. For such functions
\begin{equation} (F\phi)(x) = \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot \int_{\mathbf R^n} \phi(\xi) e^{-i x \xi} \, \mathrm d\xi. \end{equation}
In a certain sense the most natural domain of definition of $F$ is the set $S$ of all infinitely-differentiable functions $\phi$ that, together with their derivatives, vanish at infinity faster than any power of $\frac{1}{|x|}$. Formula (1) still holds for $\phi\in S$, and $(F \phi)(x) \equiv \psi(x)\in S$. Moreover, $F$ is an isomorphism of $S$ onto itself, the inverse mapping $F^{-1}$ (the inverse Fourier transform) is the inverse of the Fourier transform and is given by the formula:
\begin{equation} \phi(x) = (F^{-1} \psi)(x) = \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot \int_{\mathbf R^n} \psi(\xi) e^{i x \xi} \, \mathrm d\xi. \end{equation}
Formula (1) also acts on the space $L_{1}\left(\mathbf{R}^{n}\right)$ of integrable functions. In order to enlarge the domain of definition of the operator $F$ 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 $|x|\to\infty$ (see Fourier integral). In the theory of generalized functions the definition of the operator $F$ is free of many requirements of classical analysis.
The basic problems connected with the study of the Fourier transform $F$ 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:
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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
:
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and differentiation induces multiplication by the independent variable:
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In the spaces ,
, the operator
is defined by the formula (1) on the set
and is a bounded operator from
into
,
:
\begin{equation} \left\{\frac{1}{(2 \pi)^{n / 2}} \int_{\mathbf{R}^{n}}|(F f)(x)|^{q} d x\right\}^{1 / q} \leq\left\{\frac{1}{(2 \pi)^{n / 2}} \int_{\mathbf{R}^{n}}|f(x)|^{p} d x\right\}^{1 / p} \end{equation}
(the Hausdorff–Young inequality). admits a continuous extension onto the whole space
which (for
) is given by
![]() | (3) |
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
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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.
References
[1] | E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) |
[2] | A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) |
[3] | E.M. Stein, G. Weiss, "Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971) |
Comments
Instead of "generalized function" the term "distributiondistribution" is often used.
If and
then
denotes the scalar product
.
If in (1) the "normalizing factor" is replaced by some constant
, then in (2) it must be replaced by
with
.
At least two other conventions for the "normalization factor" are in common use:
![]() | (a1) |
![]() |
![]() | (a2) |
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The convention of the article leads to the Fourier transform as a unitary operator from into itself, and so does the convention (a2). Convention (a1) is more in line with harmonic analysis.
References
[a1] | W. Rudin, "Functional analysis" , McGraw-Hill (1973) |
Fourier transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_transform&oldid=44378