Definition:Fourier Transform/Real Function

Definition

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

Formulation 1

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

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

Formulation 2

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

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

Formulation 3

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

$\displaystyle \map \FF {\map f t} = \map F s := \dfrac 1 {\sqrt {2 \pi} } \int_{-\infty}^\infty e^{-i s t} \map f t \rd t$

Also known as

The real Fourier transform function is sometimes referred to as the minus-$i$ transform of $\map f t$.

This allows us to distinguish between this and the real inverse Fourier transform function, known in turn as the plus-$i$ transform of $\map F s$.

Ronald N. Bracewell, in his The Fourier Transform and its Applications, 2nd ed. of $1978$, discusses all $3$ of the formulations given in $\mathsf{Pr} \infty \mathsf{fWiki}$, referring to them as System $1$, System $2$ and System $3$.

The numbers assigned to Formulation $1$, Formulation $2$ and Formulation $3$ have been configured so as to correspond to these directly.

Also see

• Results about Fourier transforms can be found here.

Source of Name

This entry was named for Joseph Fourier.