Editing Fourier transform (section)

=== Angular frequency (''&omega;'') ===
When the independent variable (<math>x</math>) represents ''time'' (often denoted by <math>t</math>), the transform variable (<math>\xi</math>) represents [[frequency]] (often denoted by <math>f</math>). For example, if time is measured in [[second]]s, then frequency is in [[hertz]]. The Fourier transform can also be written in terms of [[angular frequency]], <math>\omega = 2\pi \xi,</math> whose units are [[radian]]s per second.

The substitution <math>\xi = \tfrac{\omega}{2 \pi}</math> into {{EquationNote|Eq.1}}  produces this convention, where function <math>\widehat f</math> is relabeled <math>\widehat {f_1}:</math>
<math display="block">\begin{align}
\widehat {f_3}(\omega) &\triangleq  \int_{-\infty}^{\infty} f(x)\cdot e^{-i\omega x}\, dx = \widehat{f_1}\left(\tfrac{\omega}{2\pi}\right),\\
f(x) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} \widehat{f_3}(\omega)\cdot e^{i\omega x}\, d\omega.
\end{align}
</math>
Unlike the {{EquationNote|Eq.1}} definition, the Fourier transform is no longer a [[unitary transformation]], and there is less symmetry between the formulas for the transform and its inverse.  Those properties are restored by splitting the <math>2 \pi</math> factor evenly between the transform and its inverse, which leads to another convention:
<math display="block">\begin{align}
\widehat{f_2}(\omega) &\triangleq \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)\cdot e^{- i\omega x}\, dx = \frac{1}{\sqrt{2\pi}}\  \ \widehat{f_1}\left(\tfrac{\omega}{2\pi}\right), \\
f(x) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \widehat{f_2}(\omega)\cdot e^{ i\omega x}\, d\omega.
\end{align}</math>
Variations of all three conventions can be created by conjugating the complex-exponential [[integral kernel|kernel]] of both the forward and the reverse transform. The signs must be opposites.

{| class="wikitable"
|+ Summary of popular forms of the Fourier transform, one-dimensional
|-
! ordinary frequency {{mvar|ξ}} (Hz)
! unitary
| <math>\begin{align}
\widehat{f_1}(\xi)\ &\triangleq\ \int_{-\infty}^{\infty} f(x)\, e^{-i 2\pi \xi x}\, dx = \sqrt{2\pi}\ \ \widehat{f_2}(2 \pi \xi) = \widehat{f_3}(2 \pi \xi) \\
f(x) &= \int_{-\infty}^{\infty} \widehat{f_1}(\xi)\, e^{i 2\pi x \xi}\, d\xi \end{align}</math>
|-
! rowspan="2" | angular frequency {{mvar|ω}} (rad/s)
! unitary
| <math>\begin{align}
\widehat{f_2}(\omega)\ &\triangleq\ \frac{1}{\sqrt{2\pi}}\ \int_{-\infty}^{\infty} f(x)\,  e^{-i \omega x}\, dx = \frac{1}{\sqrt{2\pi}}\ \ \widehat{f_1} \! \left(\frac{\omega}{2 \pi}\right) = \frac{1}{\sqrt{2\pi}}\ \ \widehat{f_3}(\omega) \\
f(x) &= \frac{1}{\sqrt{2\pi}}\ \int_{-\infty}^{\infty} \widehat{f_2}(\omega)\, e^{i \omega x}\, d\omega \end{align}</math>
|-
! non-unitary
| <math>\begin{align}
\widehat{f_3}(\omega) \ &\triangleq\ \int_{-\infty}^{\infty} f(x)\,  e^{-i\omega x}\, dx = \widehat{f_1} \left(\frac{\omega}{2 \pi}\right) = \sqrt{2\pi}\ \ \widehat{f_2}(\omega) \\
f(x) &= \frac{1}{2 \pi} \int_{-\infty}^{\infty} \widehat{f_3}(\omega)\, e^{i \omega x}\, d\omega \end{align}</math>
|}

{| class="wikitable"
|+ Generalization for {{math|''n''}}-dimensional functions
|-
! ordinary frequency {{mvar|ξ}} (Hz)
! unitary
| <math>\begin{align}
\widehat{f_1}(\xi)\ &\triangleq\ \int_{\mathbb{R}^n} f(x) e^{-i 2\pi \xi\cdot x}\, dx = (2 \pi)^\frac{n}{2}\widehat{f_2}(2\pi \xi) = \widehat{f_3}(2\pi \xi) \\
f(x) &= \int_{\mathbb{R}^n} \widehat{f_1}(\xi) e^{i 2\pi \xi\cdot x}\, d\xi \end{align}</math>
|-
! rowspan="2" | angular frequency {{mvar|ω}} (rad/s)
! unitary
| <math>\begin{align}
\widehat{f_2}(\omega)\ &\triangleq\ \frac{1}{(2 \pi)^\frac{n}{2}} \int_{\mathbb{R}^n} f(x) e^{-i \omega\cdot x}\, dx = \frac{1}{(2 \pi)^\frac{n}{2}} \widehat{f_1} \! \left(\frac{\omega}{2 \pi}\right) = \frac{1}{(2 \pi)^\frac{n}{2}} \widehat{f_3}(\omega) \\
f(x) &= \frac{1}{(2 \pi)^\frac{n}{2}} \int_{\mathbb{R}^n} \widehat{f_2}(\omega)e^{i \omega\cdot x}\, d\omega \end{align}</math>
|-
! non-unitary
| <math>\begin{align}
\widehat{f_3}(\omega) \ &\triangleq\ \int_{\mathbb{R}^n} f(x) e^{-i\omega\cdot x}\, dx = \widehat{f_1} \left(\frac{\omega}{2 \pi}\right) = (2 \pi)^\frac{n}{2} \widehat{f_2}(\omega) \\
f(x) &= \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \widehat{f_3}(\omega) e^{i \omega\cdot x}\, d\omega \end{align}</math>
|}