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==History==
The ''rect'' function has been introduced 1953 by [[Philip Woodward|Woodward]]<ref>{{Cite journal |last=Klauder |first=John R |title=The Theory and Design of Chirp Radars |pages=745–808 |journal=Bell System Technical Journal |year=1960 |volume=39 |issue=4 |doi=10.1002/j.1538-7305.1960.tb03942.x |url=https://ieeexplore.ieee.org/document/6773600 |url-access=subscription }}</ref> in "Probability and Information Theory, with Applications to Radar"<ref>{{Cite book |last=Woodward |first=Philipp M |title=Probability and Information Theory, with Applications to Radar |publisher=Pergamon Press |pages=29 |year=1953 }}</ref> as an ideal [[Window function#Rectangular window|cutout operator]], together with the [[Sinc function|''sinc'' function]]<ref>{{Cite book |last=Higgins |first=John Rowland |title=Sampling Theory in Fourier and Signal Analysis: Foundations |pages=4 |publisher=Oxford University Press Inc. |year=1996 |isbn=0198596995 }}</ref><ref>{{Cite book |last=Zayed |first=Ahmed I |title=Handbook of Function and Generalized Function Transformations |pages=507 |publisher=CRC Press |year=1996 |isbn=9780849380761 }}</ref> as an ideal [[Whittaker–Shannon interpolation formula|interpolation operator]], and their counter operations which are [[Sampling (signal processing)|sampling]] ([[Dirac comb#Dirac-comb identity|''comb'' operator]]) and [[Periodic summation|replicating]] ([[Dirac comb#Dirac-comb identity|''rep'' operator]]), respectively.
==Relation to the boxcar function==
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The [[Fourier transform#Tables of important Fourier transforms|unitary Fourier transforms]] of the rectangular function are<ref name="wolfram"/>
<math display="block">\int_{-\infty}^\infty \operatorname{rect}(t)\cdot e^{-i 2\pi f t} \, dt
=\frac{\sin(\pi f)}{\pi f} = \operatorname{sinc}(\pi f) =\operatorname{sinc}_\pi(f),</math>
using ordinary frequency {{mvar|f}}, where [[sinc function|<math>\operatorname{sinc}_\pi</math>]] is the normalized form<ref>Wolfram MathWorld, https://mathworld.wolfram.com/SincFunction.html</ref> of the [[sinc function]] and
<math display="block">\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \operatorname{rect}(t)\cdot e^{-i \omega t} \, dt
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==See also==
*[[Fourier transform]]
*[[Square wave (waveform)|Square wave]]
*[[Step function]]
*[[Top-hat filter]]
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