Definition:Fourier Transform

Definition
The Fourier transform of a Lebesgue integrable function $f: \R^N \to \C$ is the function $\mathcal F \left({f}\right): \R^N \to \C$ given by:
 * $\displaystyle \mathcal F \left({f \left({\xi}\right)}\right) := \int_{\R^N} f \left({\mathbf x}\right) \, e^{-2 \pi i \mathbf x \cdot \xi} \rd \mathbf x$

for $\xi \in \R^N$.

Here, the product $\mathbf x \cdot \xi$ in the exponential is the scalar product of the vectors $\mathbf x$ and $\mathbf \xi$.

Correctness of the definition
The function under the integral in the definition is Lebesgue integrable, as:
 * $\left\vert{f \left({\mathbf x}\right) e^{-2 \pi i \mathbf x \cdot \xi} }\right\vert = \left\vert{f \left({\mathbf x}\right)}\right\vert$

and $f$ is assumed to be integrable.

Also defined as
There exist several slightly different definitions of the Fourier transform which are commonly used; they differ in the choice of the constant $2 \pi$ inside the exponential and/or a multiplicative constant before the integral.

The following definition is also very common:


 * $\displaystyle \mathcal F \left({f \left({\xi}\right)}\right) := \left({2 \pi}\right)^{-\frac N 2} \int_{\R^N} f \left({\mathbf x}\right)\, e^{-i \mathbf x \cdot \xi} \rd \mathbf x$

for $\xi \in \R^N$.

Their properties are essentially the same.

By a simple change of variable one can always translate statements using one of the definitions into statements using another one.

Also denoted as
The Fourier transform of $f$ is also frequently denoted by $\hat f$.

When confusion can arise, one may write $\mathcal F \left[{f}\right]$ instead of $\mathcal F \left({f}\right)$.

Some sources omit the brackets altogether, and deploy it as $\mathcal F f$.

Also see

 * Fourier Transform of Function in Lebesgue Space: the Fourier transform of a function in $L^p \left({\R^N}\right)$ for $1 \le p \le 2$


 * Fourier Transform of Tempered Distribution: more generally on $\R^N$.