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 $\Phi$ and the range of values $F \Phi = \Psi$ of $F$; as well as studying properties of the mapping $F: \ \Phi \rightarrow \Psi$( in particular, conditions for the existence of the inverse operator $F ^ {\ -1}$ and its expression). The inversion formula for the Fourier transform is very simple:

$$F ^ {\ -1} [g (x)] \ = \ F [g (-x)].$$

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 $f$ and $g$ goes over into the product of the functions $Ff$ and $Fg$:

$$F (f * g) \ = \ Ff \cdot Fg;$$

and differentiation induces multiplication by the independent variable:

$$F (D^ \alpha f \ ) \ = \ (ix)^ \alpha Ff.$$

In the spaces $L _{p} ( \mathbf R^{n} )$, $1 \leq p \leq 2$, the operator $F$ is defined by the formula (1) on the set $D _{F} = (L _{1} \cap L _{p} ) ( \mathbf R^{n} )$ and is a bounded operator from $L _{p} ( \mathbf R^{n} )$ into $L _{q} ( \mathbf R^{n} )$, $p^{-1} + q^{-1} = 1$:

$$\begin{equation} \left\{\frac{1}{(2 \pi)^{n / 2}} \int_{\mathbf{R}^{n}}|(F f)(x)|^{q} d x\right\}^{1 / q} \leq\left\{\frac{1}{(2 \pi)^{n / 2}} \int_{\mathbf{R}^{n}}|f(x)|^{p} d x\right\}^{1 / p} \end{equation}$$

(the Hausdorff–Young inequality). $F$ admits a continuous extension onto the whole space $L _{p} ( \mathbf R^{n} )$ which (for $1 < p \leq 2$) is given by

$$\tag{3} (Ff \ ) (x) \ = \ \lim\limits _ {R \rightarrow \infty} {}^{q} \ { \frac{1}{(2 \pi ) ^ n/2} } \int\limits _ {| \xi | < R} f ( \xi ) e ^ {-i \xi x} \ d \xi \ = \ \widetilde{f} (x).$$

Convergence is understood to be in the norm of $L _{q} ( \mathbf R^{n} )$. If $p \neq 2$, the image of $L _{p}$ under the action of $F$ does not coincide with $L _{q}$, that is, the imbedding $FL _{p} \subset L _{q}$ is strict when $1 \leq p < 2$( for the case $p = 2$ see Plancherel theorem). The inverse operator $F ^ {\ -1}$ is defined on $FL _{p}$ by

$$(F ^ {\ -1} \widetilde{f} \ ) \ = \ \lim\limits _ {R \rightarrow \infty} {}^{p} \ { \frac{1}{(2 \pi ) ^ n/2} } \int\limits _ {| \xi | < R} \widetilde{f} ( \xi ) e ^ {i \xi x} \ d \xi ,\ \ 1 < p \leq 2.$$

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 $x = (x _{1} \dots x _{n} )$ and $\xi = ( \xi _{1} \dots \xi _{n} )$ then $x \cdot \xi$ denotes the scalar product $\sum _{ {i = 1}^{n}} x _{i} \xi _{i}$.

If in (1) the "normalizing factor" $(1/ {2 \pi} )^{n/2}$ is replaced by some constant $\alpha$, then in (2) it must be replaced by $\beta$ with $\alpha \beta = (1/ {2 \pi} )^{n}$.

At least two other conventions for the "normalization factor" are in common use:

$$\tag{a1} (F \phi ) (x) \ = \ \int\limits _ {\mathbf R ^ n} \phi ( \xi ) e ^ {- ix \cdot \xi} \ d \xi ,$$

$$(F ^ {\ -1} \phi ) (x) \ = \ \frac{1}{(2 \pi ) ^ n} \int\limits _ {\mathbf R ^ n} \phi ( \xi ) e ^ {ix \cdot \xi} \ d \xi ,$$

$$\tag{a2} (F \phi ) (x) \ = \ \int\limits _ {\mathbf R ^ n} \phi ( \xi ) e ^ {- 2 \pi ix \cdot \xi} \ d \xi ,$$

$$(F ^ {\ -1} \phi ) (x) \ = \ \int\limits _ {\mathbf R^{n} } \phi ( \xi ) e ^ {2 \pi ix \cdot \xi} \ d \xi .$$

The convention of the article leads to the Fourier transform as a unitary operator from $L _{2} ( \mathbf R^{n} )$ 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)