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  "body": "\n## Summary\n\nRecovering a function $f$ from the values of its [[Laplace transform|0V49JEZW06]] $F(s)$ at real points $s \u003e 0$ is an ill-posed problem in the sense of Hadamard.\nThe solution is unique, but it does not depend continuously on the data.\nTransforms that are very close can come from functions that are far apart.\nSo small errors in measured values of $F$ can produce large errors in the computed $f$, and numerical inversion needs regularisation.\nThe problem fails the stability condition in the definition of a [[well-posed problem|W09B9NZ9B5]].\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform smooths a function a great deal: it averages $f$ against the slowly varying weights $e^{-s t}$.\nFast wiggles in $f$ almost cancel in these averages, so they leave nearly no trace in $F$.\nFor example, $e^{-t} \\sin(n t)$ has size about $1/2$ for every $n$, but its transform $n/((s+1)^2 + n^2)$ shrinks to $0$ as $n$ grows.\nSo from a slightly noisy $F$ one cannot tell whether such wiggles are present, and inversion becomes unstable.\n",
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