Editing Fourier transform (section)

=== Square-integrable functions, one-dimensional ===
The Fourier transforms in this table may be found in {{harvtxt|Campbell|Foster|1948}}, {{harvtxt|Erdélyi|1954}}, or {{harvtxt|Kammler|2000|loc=appendix}}.
{| class="wikitable"
! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks
|-
|
|<math> f(x)\,</math>
|<math>\begin{align} &\hat{f}(\xi) \triangleq \hat f_1(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>
|<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_2(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_3(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|Definitions
|-
|{{anchor|rect}} 201
|<math> \operatorname{rect}(a x) \,</math>
|<math> \frac{1}{|a|}\, \operatorname{sinc}\left(\frac{\xi}{a}\right)</math>
|<math> \frac{1}{\sqrt{2 \pi a^2}}\, \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>
|<math> \frac{1}{|a|}\, \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>
|The [[rectangular function|rectangular pulse]] and the ''normalized'' [[sinc function]], here defined as {{math|1=sinc(''x'') = {{sfrac|sin(π''x'')|π''x''}}}}
|-
| 202
|<math> \operatorname{sinc}(a x)\,</math>
|<math> \frac{1}{|a|}\, \operatorname{rect}\left(\frac{\xi}{a} \right)\,</math>
|<math> \frac{1}{\sqrt{2\pi a^2}}\, \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)</math>
|<math> \frac{1}{|a|}\, \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)</math>
|Dual of rule 201. The [[rectangular function]] is an ideal [[low-pass filter]], and the [[sinc function]] is the [[Anticausal system|non-causal]] impulse response of such a filter. The [[sinc function]] is defined here as {{math|1=sinc(''x'') = {{sfrac|sin(π''x'')|π''x''}}}}
|-
| 203
|<math> \operatorname{sinc}^2 (a x)</math>
|<math> \frac{1}{|a|}\, \operatorname{tri} \left( \frac{\xi}{a} \right) </math>
|<math> \frac{1}{\sqrt{2\pi a^2}}\, \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>
|<math> \frac{1}{|a|}\, \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>
| The function {{math|tri(''x'')}} is the [[triangular function]]
|-
| 204
|<math> \operatorname{tri} (a x)</math>
|<math> \frac{1}{|a|}\, \operatorname{sinc}^2 \left( \frac{\xi}{a} \right) \,</math>
|<math> \frac{1}{\sqrt{2\pi a^2}} \, \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) </math>
|<math> \frac{1}{|a|} \, \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) </math>
| Dual of rule 203.
|-
| 205
|<math> e^{- a x} u(x) \,</math>
|<math> \frac{1}{a + i 2\pi \xi}</math>
|<math> \frac{1}{\sqrt{2 \pi} (a + i \omega)}</math>
|<math> \frac{1}{a + i \omega}</math>
|The function {{math|''u''(''x'')}} is the [[Heaviside step function|Heaviside unit step function]] and {{math|''a'' > 0}}.
|-
| 206
|<math> e^{-\alpha x^2}\,</math>
|<math> \sqrt{\frac{\pi}{\alpha}}\, e^{-\frac{(\pi \xi)^2}{\alpha}}</math>
|<math> \frac{1}{\sqrt{2 \alpha}}\, e^{-\frac{\omega^2}{4 \alpha}}</math>
|<math> \sqrt{\frac{\pi}{\alpha}}\, e^{-\frac{\omega^2}{4 \alpha}}</math>
|This shows that, for the unitary Fourier transforms, the [[Gaussian function]] {{math|''e''<sup>−''αx''<sup>2</sup></sup>}} is its own Fourier transform for some choice of {{mvar|α}}. For this to be integrable we must have {{math|Re(''α'') > 0}}.
|-
| 208
|<math> e^{-a|x|} \,</math>
|<math> \frac{2 a}{a^2 + 4 \pi^2 \xi^2} </math>
|<math> \sqrt{\frac{2}{\pi}} \, \frac{a}{a^2 + \omega^2} </math>
|<math> \frac{2a}{a^2 + \omega^{2}} </math>
|For {{math|Re(''a'') > 0}}. That is, the Fourier transform of a [[Laplace distribution|two-sided decaying exponential function]] is a [[Lorentzian function]].
|-
| 209
|<math> \operatorname{sech}(a x) \,</math>
|<math> \frac{\pi}{a} \operatorname{sech} \left( \frac{\pi^2}{ a} \xi \right)</math>
|<math> \frac{1}{a}\sqrt{\frac{\pi}{2}} \operatorname{sech}\left( \frac{\pi}{2 a} \omega \right)</math>
|<math> \frac{\pi}{a}\operatorname{sech}\left( \frac{\pi}{2 a} \omega \right)</math>
|[[Hyperbolic function|Hyperbolic secant]] is its own Fourier transform
|-
| 210
|<math> e^{-\frac{a^2 x^2}2} H_n(a x)\,</math>
|<math> \frac{\sqrt{2\pi}(-i)^n}{a} e^{-\frac{2\pi^2\xi^2}{a^2}} H_n\left(\frac{2\pi\xi}a\right)</math>
|<math> \frac{(-i)^n}{a} e^{-\frac{\omega^2}{2 a^2}} H_n\left(\frac \omega a\right)</math>
|<math> \frac{(-i)^n \sqrt{2\pi}}{a} e^{-\frac{\omega^2}{2 a^2}} H_n\left(\frac \omega a \right)</math>
|{{math|''H<sub>n</sub>''}} is the {{mvar|n}}th-order [[Hermite polynomial]]. If {{math|''a'' {{=}} 1}} then the Gauss–Hermite functions are [[eigenfunction]]s of the Fourier transform operator. For a derivation, see [[Hermite polynomials#Hermite functions as eigenfunctions of the Fourier transform|Hermite polynomial]]. The formula reduces to 206 for {{math|''n'' {{=}} 0}}.
|}