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  "body": "\n## Summary\n\nThe admissibility condition asks that a function $\\psi$ in $L^2(\\R)$ have finite constant $C_\\psi = \\int \\abs{\\hat\\psi(\\xi)}^2 / \\abs{\\xi} \\, d\\xi$.\nIt makes the [[continuous wavelet transform|57FR21NDGV]] with respect to $\\psi$ preserve energy up to the factor $C_\\psi$.\nSo the transform can be inverted.\nFor integrable $\\psi$ it forces $\\int \\psi = 0$: an admissible function must oscillate.\n\n## Explanations\n\n### L1 human\n\nA wavelet transform measures a signal against stretched and shifted copies of one shape.\nFor this to lose no information, the shape must not contain a constant part, because very wide copies would then just measure the average of the signal again and again.\nThe admissibility condition makes this precise in terms of the Fourier transform near frequency zero.\nThe Mexican hat curve, which dips below zero on both sides, passes the test.\nA plain bell curve, which is positive everywhere, fails it.\n",
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