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)