User:Maximilian Janisch/Sandbox

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 $\Phi \rightarrow \Psi$ (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:

\begin{equation} F ( D ^ { \alpha } f ) = ( i x ) ^ { \alpha } F f \end{equation}

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

\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). admits a continuous extension onto the whole space which (for $1 < p \leq 2$) is given by

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(3)

Convergence is understood to be in the norm of . If $p \neq 2$, 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 $F L y$ 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

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

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

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(a1)

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(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

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[a1]	W. Rudin, "Functional analysis" , McGraw-Hill (1973)