Definition:Inverse Fourier Transform/Real Function

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Definition

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


Formulation 1

The inverse Fourier transform of $F$ is defined and denoted as:

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


Formulation 2

The inverse Fourier transform of $F$ is defined and denoted as:

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


Formulation 3

The inverse Fourier transform of $F$ is defined and denoted as:

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


Also known as

The real inverse Fourier transform function is sometimes referred to as the plus-$i$ transform of $\map F s$.

This allows us to distinguish between this and the real Fourier transform function, known in turn as the minus-$i$ transform of $\map f t$.


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 inverse Fourier transforms can be found here.


Source of Name

This entry was named for Joseph Fourier.


Sources