# 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 dot product of the vectors $\mathbf x$ and $\mathbf \xi$.

### Real Function

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

The **Fourier transform** of $f$ is defined and denoted as:

- $\displaystyle \map \FF {\map f t} = \map F s := \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$

## 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 $\R^N$.

- Results about
**Fourier transforms**can be found here.

## Source of Name

This entry was named for Joseph Fourier.