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  "body": "\n## Example\n\nThe *rectangular function* is $\\operatorname{rect} \\colon \\R \\to \\R$ with $\\operatorname{rect}(x) = 1$ for $\\abs{x} \u003c \\frac12$, $\\operatorname{rect}(x) = \\frac12$ for $\\abs{x} = \\frac12$, and $\\operatorname{rect}(x) = 0$ for $\\abs{x} \u003e \\frac12$.\nIt is integrable, so its [[Fourier transform|DPVM34ADZV]] is defined by the integral.\nFor $\\xi \\ne 0$,\n$$\\widehat{\\operatorname{rect}}(\\xi) = \\int_{-1/2}^{1/2} e^{-2\\pi i x \\xi}\\, dx = \\frac{e^{\\pi i \\xi} - e^{-\\pi i \\xi}}{2\\pi i \\xi} = \\frac{\\sin(\\pi \\xi)}{\\pi \\xi},$$\nand $\\widehat{\\operatorname{rect}}(0) = 1$.\nThis function is the normalized sinc function, $\\operatorname{sinc}(\\xi) = \\sin(\\pi\\xi)/(\\pi\\xi)$.\n\n## Remarks\n\nThe transform is continuous and tends to $0$ at infinity, as for every integrable function.\nIt decays only like $1/\\abs{\\xi}$ and is not integrable, which reflects the jumps of $\\operatorname{rect}$.\nSo the inversion formula does not hold as an absolutely convergent integral.\nIt holds in $L^2(\\R)$.\nAs a symmetric improper integral it gives $\\frac12$ at $x = \\pm\\frac12$, the average of the two one-sided limits, which is why the value $\\frac12$ is used there.\n\nBy the [[convolution theorem|QY8ETVDA1Z]], the triangle function $\\operatorname{rect} * \\operatorname{rect}(x) = \\max(1 - \\abs{x}, 0)$ has transform $\\operatorname{sinc}^2$, which is integrable.\nRead in the other direction, $\\operatorname{sinc}$ is a [[band-limited function|P3T104G564]] whose transform is $\\operatorname{rect}$.\n\n## Explanations\n\n### L1 human\n\nThe rectangular function is a box: it switches on at $-\\frac12$ and off at $\\frac12$.\nIts frequency content is the sinc wave $\\sin(\\pi\\xi)/(\\pi\\xi)$, a central peak with slowly shrinking ripples.\nSharp edges in time need many high frequencies, which is why the ripples die out so slowly.\n",
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