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  "body": "\n## Summary\n\nThe Weierstrass transform of a function $f$ on the real line is its average against a Gaussian weight of variance $2$ centred at each point.\nIt is the solution at time $t = 1$ of the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$ with initial value $f$.\nIt smooths every function it is applied to, and on the Fourier side it multiplies by a Gaussian.\nWeierstrass used this smoothing in his proof of the approximation theorem that bears his name.\n\n## Explanations\n\n### L1 human\n\nImagine $f$ as a temperature profile along a long thin rod.\nLet the heat flow for one unit of time: sharp peaks spread out and the profile becomes smooth.\nThe new profile is the Weierstrass transform of $f$.\nEach new value is a weighted mean of old values, with the largest weights on nearby points.\nFor example, the transform of $x^2$ is $x^2 + 2$, because averaging adds the variance $2$ of the Gaussian weight.\n",
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