Definition:Fourier Transform/Real Function/Formulation 2

Definition

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

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

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

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 $\FF f$.

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.

This entry was named for Joseph Fourier.