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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $f \\colon \\R^n \\to \\C$ be [[integrable|X7GTF3D98F]].\nThe *FBI transform* of $f$ is the function on $\\R^n \\times (\\R^n \\setminus \\set{0})$ given by\n$$Tf(x, \\xi) = \\int_{\\R^n} e^{i \\xi \\cdot (x - y) - \\frac{1}{2} \\abs{\\xi} \\abs{x - y}^2}\\, f(y)\\, dy.$$\nThe same formula, read as a pairing, defines $Tf$ for a distribution $f$ with compact support.\n\n## Remarks\n\nThe factor $e^{-\\frac12 \\abs{\\xi} \\abs{x - y}^2}$ is a Gaussian window centred at $x$ of width of order $\\abs{\\xi}^{-1/2}$, and $e^{i\\xi \\cdot (x - y)}$ selects the frequency $\\xi$.\nAuthors use different normalizations of the window and of the constant in front; the qualitative properties do not depend on these choices.\nThe main property concerns the analytic [[wave front set|3T5Q9WVQX8]].\nCut $f$ off near a point $x_0$, and let $\\xi_0 \\ne 0$.\nThen $(x_0, \\xi_0)$ lies outside the analytic wave front set of $f$ exactly when $Tf(x, \\lambda \\xi)$ decays like $e^{-c\\lambda}$ as $\\lambda \\to \\infty$.\nHere $c \u003e 0$ is a constant and the bound is uniform for $(x, \\xi)$ near $(x_0, \\xi_0)$.\n\nExample: $n = 1$ and $f(y) = e^{-y^2/2}$.\nFor $\\xi \u003e 0$ the exponent is a quadratic polynomial in $y$.\nThe Gaussian integral gives\n$$Tf(x, \\xi) = \\sqrt{\\frac{2\\pi}{1 + \\xi}}\\, \\exp\\Big( i\\xi x - \\frac{\\xi x^2}{2} + \\frac{(\\xi x - i\\xi)^2}{2(1 + \\xi)} \\Big),$$\nwhose modulus is $\\sqrt{2\\pi/(1+\\xi)}\\, \\exp\\big(-\\frac{\\xi (x^2 + \\xi)}{2(1 + \\xi)}\\big)$; it decays like $e^{-\\xi/2}$ as $\\xi \\to \\infty$, as it must for a real analytic function.\n\nNon-example: $f(y) = e^{\\abs{y}^3}$ is not integrable, and the integral defining $Tf(x, \\xi)$ diverges for every $(x, \\xi)$, because $e^{\\abs{y}^3}$ grows faster than any Gaussian decays.\n",
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