{| class="wikitable"
! !! Funktion !! Fouriertransformation <br /> unitær, ordinær frekvens !! Fouriertransformation <br /> unitær, angulær frekvens !! Fouriertransformation <br /> ikke-unitær, angulær frekvens !! Bemærkninger
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| style="text-align:center;"|<math>\displaystyle f(x)</math>
| style="text-align:center;"|<math>\displaystyle \hat{f}(\xi)=</math> <br />
<math>\displaystyle \int_{-\infty}^\infty f(x) e^{-2\pi ix\xi}\,dx</math>
| style="text-align:center;"|<math>\displaystyle \hat{f}(\omega)=</math> <br />
<math>\displaystyle \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx</math>
| style="text-align:center;"|<math>\displaystyle \hat{f}(\nu)=</math> <br />
<math>\displaystyle \int_{-\infty}^\infty f(x) e^{-i\nu x}\, dx</math>
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|{{anchor|rect}} 201
|<math>\displaystyle \operatorname{rect}(a x) \,</math>
|<math>\displaystyle \frac{1}{|a|}\cdot \operatorname{sinc}\left(\frac{\xi}{a}\right)</math>
|<math>\displaystyle \frac{1}{\sqrt{2 \pi a^2}}\cdot \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>
|<math>\displaystyle \frac{1}{|a|}\cdot \operatorname{sinc}\left(\frac{\nu}{2\pi a}\right)</math>
| <!-- The [[rectangular function|rectangular pulse]] and the ''normalized'' [[sinc function]], here defined as sinc(''x'') = sin(''{{pi}}x'')/(''{{pi}}x'') -->
|-
| 202
|<math>\displaystyle \operatorname{sinc}(a x)\,</math>
|<math>\displaystyle \frac{1}{|a|}\cdot \operatorname{rect}\left(\frac{\xi}{a} \right)\,</math>
|<math>\displaystyle \frac{1}{\sqrt{2\pi a^2}}\cdot \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)</math>
|<math>\displaystyle \frac{1}{|a|}\cdot \operatorname{rect}\left(\frac{\nu}{2 \pi a}\right)</math>
|Duale af regel 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 sinc(''x'') = sin(''{{pi}}x'')/(''{{pi}}x'') -->
|-
| 203
|<math>\displaystyle \operatorname{sinc}^2 (a x)</math>
|<math>\displaystyle \frac{1}{|a|}\cdot \operatorname{tri} \left( \frac{\xi}{a} \right) </math>
|<math>\displaystyle \frac{1}{\sqrt{2\pi a^2}}\cdot \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>
|<math>\displaystyle \frac{1}{|a|}\cdot \operatorname{tri} \left( \frac{\nu}{2\pi a} \right) </math>
| <!-- The function tri(''x'') is the [[triangular function]] -->
|-
| 204
|<math>\displaystyle \operatorname{tri} (a x)</math>
|<math>\displaystyle \frac{1}{|a|}\cdot \operatorname{sinc}^2 \left( \frac{\xi}{a} \right) \,</math>
|<math>\displaystyle \frac{1}{\sqrt{2\pi a^2}} \cdot \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) </math>
|<math>\displaystyle \frac{1}{|a|} \cdot \operatorname{sinc}^2 \left( \frac{\nu}{2\pi a} \right) </math>
| Duale af regel 203.
|-
| 205
|<math>\displaystyle e^{- a x} u(x) \,</math>
|<math>\displaystyle \frac{1}{a + 2 \pi i \xi}</math>
|<math>\displaystyle \frac{1}{\sqrt{2 \pi} (a + i \omega)}</math>
|<math>\displaystyle \frac{1}{a + i \nu}</math>
| <!-- The function ''u''(''x'') is the [[Heaviside step function|Heaviside unit step function]] and ''a''&nbsp;>&nbsp;0. -->
|-
| 206
|<math>\displaystyle e^{-\alpha x^2}\,</math>
|<math>\displaystyle \sqrt{\frac{\pi}{\alpha}}\cdot e^{-\frac{(\pi \xi)^2}{\alpha}}</math>
|<math>\displaystyle \frac{1}{\sqrt{2 \alpha}}\cdot e^{-\frac{\omega^2}{4 \alpha}}</math>
|<math>\displaystyle \sqrt{\frac{\pi}{\alpha}}\cdot e^{-\frac{\nu^2}{4 \alpha}}</math>
| <!-- This shows that, for the unitary Fourier transforms, the [[Gaussian function]] exp(−''αx''<sup>2</sup>) is its own Fourier transform for some choice of ''α''. For this to be integrable we must have Re(''α'')&gt;0. -->
|-
| 207
|<math>\displaystyle \operatorname{e}^{-a|x|} \,</math>
|<math>\displaystyle \frac{2 a}{a^2 + 4 \pi^2 \xi^2} </math>
|<math>\displaystyle \sqrt{\frac{2}{\pi}} \cdot \frac{a}{a^2 + \omega^2} </math>
|<math>\displaystyle \frac{2a}{a^2 + \nu^2} </math>
| <!-- For ''a''&nbsp;>&nbsp;0. That is, the Fourier transform of a [[Laplace distribution|two-sided decaying exponential function]] is a [[Lorentzian function]]. -->
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| 208
|<math>\displaystyle \operatorname{sech}(a x) \,</math>
|<math>\displaystyle \frac{\pi}{a} \operatorname{sech} \left( \frac{\pi^2}{ a} \xi \right)</math>
|<math>\displaystyle \frac{1}{a}\sqrt{\frac{\pi}{2}}\operatorname{sech}\left( \frac{\pi}{2 a} \omega \right)</math>
|<math>\displaystyle \frac{\pi}{a}\operatorname{sech}\left( \frac{\pi}{2 a} \nu \right)</math>
| <!-- [[Hyperbolic function|Hyperbolic secant]] is its own Fourier transform -->
|-
| 209
|<math>\displaystyle e^{-\frac{a^2 x^2}2} H_n(a x)\,</math>
|<math>\displaystyle \frac{\sqrt{2\pi}(-i)^n}{a}</math><br>
&nbsp; <math>\cdot e^{-\frac{2\pi^2\xi^2}{a^2}} H_n\left(\frac{2\pi\xi}a\right)</math>
|<math>\displaystyle \frac{(-i)^n}{a}</math><br>
&nbsp; <math>\cdot e^{-\frac{\omega^2}{2 a^2}} H_n\left(\frac \omega a\right)</math>
|<math>\displaystyle \frac{(-i)^n \sqrt{2\pi}}{a}</math><br>
&nbsp; <math>\cdot e^{-\frac{\nu^2}{2 a^2}} H_n\left(\frac \nu a \right)</math>
| <!-- <math>H_n</math> is the ''n''th-order [[Hermite polynomial]]. If ''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 ''n'' = 0. -->
|}
Samfund Datalogi Videnskab Matematik