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  "body": "\n## Summary\n\nThe Fourier transform of a tempered distribution $T$ is again a tempered distribution.\nIt acts on a Schwartz function $\\varphi$ as $T$ acts on the Fourier transform of $\\varphi$.\nIt extends the Fourier transform from integrable and square-integrable functions to all tempered distributions.\nIt is a linear bijection of the space of tempered distributions onto itself.\nIt turns derivatives into multiplication by polynomials, which makes it a main tool for linear partial differential equations.\n\n## Explanations\n\n### L1 human\n\nMany useful objects, such as constant functions, polynomials and the Dirac delta, have no Fourier transform as an integral.\nThe trick is to move the transform onto smooth, fast-decaying test functions, where the integral always makes sense.\nIn this way the delta at $0$ has Fourier transform the constant function $1$, and the constant $1$ has Fourier transform the delta.\nThis matches the idea that a spike contains all frequencies equally.\n",
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