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  "body": "\n## Summary\n\nThe FBI transform, named after Fourier, Bros and Iagolnitzer, is a phase-space transform.\nIt looks at a function near a point $x$ at frequency $\\xi$ through a Gaussian window of width about $\\abs{\\xi}^{-1/2}$.\nIt is a tool of microlocal analysis.\nSmoothness and real analyticity of a function near a point, in given directions, can be read off from the decay of its FBI transform.\nIn particular, it characterizes the analytic [[wave front set|3T5Q9WVQX8]] through exponential decay.\nIt is a close relative of the [[Bargmann transform|QPH28QH036]] and of the short-time Fourier transform with a Gaussian window.\n\n## Explanations\n\n### L1 human\n\nTo see whether a function is smooth near a point, one can zoom in there and look at its high-frequency content.\nThe FBI transform does this with a Gaussian magnifying glass placed at $x$ and tuned to frequency $\\xi$.\nThe higher the frequency, the narrower the glass.\nIf the result dies off very fast as the frequency grows, the function is real analytic near $x$ in that direction of frequency.\n",
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