Definition:Fourier Transform

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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

  • Results about Fourier transforms can be found here.

Source of Name

This entry was named for Joseph Fourier.