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  "body": "\n## Definition\n\nLet $f \\in L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]).\nThe *Wigner distribution* of $f$ is the function $Wf \\colon \\R^2 \\to \\C$,\n$$Wf(x, \\omega) = \\int_{\\R} f\\!\\left(x + \\tfrac{t}{2}\\right) \\overline{f\\!\\left(x - \\tfrac{t}{2}\\right)} \\, e^{-2\\pi i t \\omega} \\, dt,$$\nfor $x \\in \\R$ (time) and $\\omega \\in \\R$ (frequency).\n\n## Remarks\n\nThe integral converges absolutely for each $x$, by the Cauchy–Schwarz inequality.\n$Wf$ is real-valued, since replacing $t$ by $-t$ turns the integral into its complex conjugate.\nFor $f$ in the Schwartz space, $\\int_\\R Wf(x, \\omega) \\, d\\omega = \\abs{f(x)}^2$ and $\\int_\\R Wf(x, \\omega) \\, dx = \\abs{\\hat f(\\omega)}^2$, where $\\hat f$ is the [[Fourier transform|DPVM34ADZV]].\nIn quantum mechanics the Wigner function of a wave function is usually written with Planck's constant and a normalizing factor.\n\nExample: let $f(t) = 2^{1/4} e^{-\\pi t^2}$, so that $\\norm{f}_2 = 1$.\nThen $f(x + t/2) \\overline{f(x - t/2)} = \\sqrt2 \\, e^{-2\\pi x^2} e^{-\\pi t^2/2}$, and the Fourier transform of $e^{-\\pi t^2/2}$ is $\\sqrt2 \\, e^{-2\\pi \\omega^2}$.\nSo $Wf(x, \\omega) = 2 e^{-2\\pi (x^2 + \\omega^2)}$, and integrating in $\\omega$ gives $\\sqrt2 e^{-2\\pi x^2} = \\abs{f(x)}^2$.\n\nNon-example: the Wigner distribution is quadratic in $f$, not linear.\nFor instance $W(2f) = 4 Wf \\ne 2 Wf$ when $f \\ne 0$.\n",
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