Editing Fourier transform (section)

=== Functional relationships, one-dimensional ===
The Fourier transforms in this table may be found in {{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} &\widehat{f}(\xi) \triangleq \widehat {f_1}(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>
|<math>\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_2}(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|<math>\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_3}(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|Definitions
|-
| 101
|<math> a\, f(x) + b\, g(x)\,</math>
|<math> a\, \widehat{f}(\xi) + b\, \widehat{g}(\xi)\,</math>
|<math> a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\,</math>
|<math> a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\,</math>
|Linearity
|-
| 102
|<math> f(x - a)\,</math>
|<math> e^{-i 2\pi \xi a} \widehat{f}(\xi)\,</math>
|<math> e^{- i a \omega} \widehat{f}(\omega)\,</math>
|<math> e^{- i a \omega} \widehat{f}(\omega)\,</math>
|Shift in time domain
|-
| 103
|<math> f(x)e^{iax}\,</math>
|<math> \widehat{f} \left(\xi - \frac{a}{2\pi}\right)\,</math>
|<math> \widehat{f}(\omega - a)\,</math>
|<math> \widehat{f}(\omega - a)\,</math>
|Shift in frequency domain, dual of 102
|-
| 104
|<math> f(a x)\,</math>
|<math> \frac{1}{|a|} \widehat{f}\left( \frac{\xi}{a} \right)\,</math>
|<math> \frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)\,</math>
|<math> \frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)\,</math>
|Scaling in the time domain. If {{math|{{abs|''a''}}}} is large, then {{math|''f''(''ax'')}} is concentrated around 0 and{{br}}<math> \frac{1}{|a|}\hat{f} \left( \frac{\omega}{a} \right)\,</math>{{br}}spreads out and flattens.
|-
| 105
|<math> \widehat {f_n}(x)\,</math>
|<math> \widehat {f_1}(x) \ \stackrel{\mathcal{F}_1}{\longleftrightarrow}\  f(-\xi)\,</math>
|<math> \widehat {f_2}(x) \ \stackrel{\mathcal{F}_2}{\longleftrightarrow}\  f(-\omega)\,</math>
|<math> \widehat {f_3}(x) \ \stackrel{\mathcal{F}_3}{\longleftrightarrow}\ 2\pi f(-\omega)\,</math>
|The same transform is applied twice, but ''x'' replaces the frequency variable (''ξ'' or ''ω'') after the first transform.
|-
| 106
|<math> \frac{d^n f(x)}{dx^n}\,</math>
|<math> (i 2\pi \xi)^n \widehat{f}(\xi)\,</math>
|<math> (i\omega)^n \widehat{f}(\omega)\,</math>
|<math> (i\omega)^n \widehat{f}(\omega)\,</math>
|n{{superscript|th}}-order derivative.
As {{math|''f''}} is a [[Schwartz space|Schwartz function]]
|-
|106.5
|<math>\int_{-\infty}^{x} f(\tau) d \tau</math>
|<math>\frac{\widehat{f}(\xi)}{i 2 \pi \xi} + C \, \delta(\xi)</math>
|<math>\frac{\widehat{f} (\omega)}{i\omega} + \sqrt{2 \pi} C \delta(\omega)</math>
|<math>\frac{\widehat{f} (\omega)}{i\omega} + 2 \pi C \delta(\omega)</math>
|Integration.<ref>{{Cite web |date=2015 |orig-date=2010 |title=The Integration Property of the Fourier Transform |url=https://www.thefouriertransform.com/transform/integration.php |url-status=live |archive-url=https://web.archive.org/web/20220126171340/https://www.thefouriertransform.com/transform/integration.php |archive-date=2022-01-26 |access-date=2023-08-20 |website=The Fourier Transform .com}}</ref> Note: <math>\delta</math> is the [[Dirac delta function]] and <math>C</math> is the average ([[DC component|DC]]) value of <math>f(x)</math> such that <math>\int_{-\infty}^\infty (f(x) - C) \, dx = 0</math>
|-
| 107
|<math> x^n f(x)\,</math>
|<math> \left (\frac{i}{2\pi}\right)^n \frac{d^n \widehat{f}(\xi)}{d\xi^n}\,</math>
|<math> i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}</math>
|<math> i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}</math>
|This is the dual of 106
|-
| 108
|<math> (f * g)(x)\,</math>
|<math> \widehat{f}(\xi) \widehat{g}(\xi)\,</math>
|<math> \sqrt{2\pi}\ \widehat{f}(\omega) \widehat{g}(\omega)\,</math>
|<math> \widehat{f}(\omega) \widehat{g}(\omega)\,</math>
|The notation {{math|''f'' ∗ ''g''}} denotes the [[convolution]] of {{mvar|f}} and {{mvar|g}} — this rule is the [[convolution theorem]]
|-
| 109
|<math> f(x) g(x)\,</math>
|<math> \left(\widehat{f} * \widehat{g}\right)(\xi)\,</math>
|<math> \frac{1}\sqrt{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\,</math>
|<math> \frac{1}{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\,</math>
|This is the dual of 108
|-
| 110
|For {{math|''f''(''x'')}} purely real
|<math> \widehat{f}(-\xi) = \overline{\widehat{f}(\xi)}\,</math>
|<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\,</math>
|<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\,</math>
|Hermitian symmetry. {{math|{{overline|''z''}}}} indicates the [[complex conjugate]].
|-
<!-- A Symmetry section has been added instead of this.
| 111
|For {{math|''f''(''x'')}} purely real and [[even function|even]]
| colspan=3 align=center |<math>\widehat f </math> is a purely real and [[even function]].
|
|-
| 112
|For {{math|''f''(''x'')}} purely real and [[odd function|odd]]
| colspan=3 align=center |<math>\widehat f </math> is a purely [[imaginary number|imaginary]] and [[odd function]].
|
|-->
| 113
|For {{math|''f''(''x'')}} purely imaginary
|<math> \widehat{f}(-\xi) = -\overline{\widehat{f}(\xi)}\,</math>
|<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\,</math>
|<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\,</math>
|{{math|{{overline|''z''}}}} indicates the [[complex conjugate]].
|-
| 114
| <math> \overline{f(x)}</math>|| <math> \overline{\widehat{f}(-\xi)}</math> || <math> \overline{\widehat{f}(-\omega)}</math> || <math> \overline{\widehat{f}(-\omega)}</math> || [[Complex conjugate|Complex conjugation]], generalization of 110 and 113
|-
|115
|<math> f(x) \cos (a x)</math>
|<math> \frac{ \widehat{f}\left(\xi - \frac{a}{2\pi}\right)+\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2}</math>
|<math> \frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}\,</math>
|<math> \frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}</math>
|This follows from rules 101 and 103 using [[Euler's formula]]:{{br}}<math>\cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}.</math>
|-
|116
|<math> f(x)\sin( ax)</math>
|<math> \frac{\widehat{f}\left(\xi-\frac{a}{2\pi}\right)-\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2i}</math>
|<math> \frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}</math>
|<math> \frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}</math>
|This follows from 101 and 103 using [[Euler's formula]]:{{br}}<math>\sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}.</math>
|}