Definition:Fourier Transform

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

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 $\mathbf \xi$.

Real Function
Let $f: \R \to \R$ be a real function which is Lebesgue integrable.

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

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:


 * $\displaystyle \map \FF {\map f \xi} := \paren {2 \pi}^{-\frac N 2} \int_{\R^N} \map f {\mathbf x} \, e^{-i \mathbf x \cdot \xi} \rd \mathbf x$

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 $\FF \sqbrk f$ instead of $\map \FF f$.

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

Also see

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


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