Definition:Fourier Transform

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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:

$\ds \map {\map \FF f} {\mathbf s} := \int_{\R^N} \map f {\mathbf x} e^{-2 \pi i \mathbf x \cdot \mathbf s} \rd \mathbf x$

for $\mathbf s \in \R^N$.

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

In this context $\map \FF f$ is to be considered the operator.

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:

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

Tempered Distribution

Let $T \in \map {\SS'} \R$ be a tempered distribution.

Let $\map \SS \R$ be the Schwartz space.

The Fourier transform $\hat T$ of (the tempered distribution) $T$:

$\hat T \in \map {\SS'} \R$

is defined as:

$\forall \phi \in \map \SS \R: \map {\hat T} \phi := \map T {\hat \phi}$

Correctness of the definition

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The function under the integral in the definition is Lebesgue integrable, as:

$\size {\map f {\mathbf x} e^{-2 \pi i \mathbf x \cdot \mathbf s} } = \size {\map f {\mathbf x} }$

and $f$ is assumed to be integrable.

Also defined as

There exist a number of 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:

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

for $\mathbf s \in \R^N$.

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 often seen denoted by $\hat f$, as this notation has considerable usefulness.

The style of $\FF$ can vary across different sources. $\mathsf{Pr} \infty \mathsf{fWiki}$ uses $\FF$ as standard.

Some sources write $\FF \sqbrk f$ instead of $\map \FF f$.

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

Also see

• Properties of Fourier Transform

• 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.

Sources

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