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  "body": "\n## Summary\n\nMany [[integral transforms|YCK709QB06]] extend from functions to [[tempered distributions|8TK1JNG87X]] by duality.\nOne moves the transform onto the [[Schwartz test functions|CG9JEATW3Q]].\nThe transform $Tu$ of a distribution $u$ acts on a test function $\\varphi$ as $u$ acts on $T^t\\varphi$.\nHere $T^t$ is the transform with the transposed kernel.\nThis works when $T^t$ maps the Schwartz space continuously into itself.\nThe model case is the [[Fourier transform of tempered distributions|5PVSQXCTCT]]; the [[Laplace transform of distributions|6Z3XJVYT2Q]] uses a related construction.\n\n## Explanations\n\n### L1 human\n\nObjects like the Dirac delta or the constant function $1$ on $\\R$ have no Fourier transform as an ordinary integral.\nThe trick is never to integrate them directly.\nInstead one asks how the transform of the object would act on nice test functions, and uses the rule $\\int (Tf)\\varphi = \\int f\\, (T^t\\varphi)$, which holds for ordinary functions.\nThe right-hand side makes sense for distributions, so it is taken as the definition.\nIn this way the delta at $0$ gets the Fourier transform $1$.\n",
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