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  "body": "\n## Example\n\nFix a width $\\sigma \u003e 0$.\nThe *Ricker wavelet*, also called the Mexican hat wavelet, is\n$$\\psi(t) = \\frac{2}{\\sqrt{3\\sigma} \\, \\pi^{1/4}} \\left(1 - \\frac{t^2}{\\sigma^2}\\right) e^{-t^2/(2\\sigma^2)}, \\qquad t \\in \\R.$$\nUp to a negative constant factor, it is the second derivative of the Gaussian $e^{-t^2/(2\\sigma^2)}$.\n\nCheck of the zero mean, with $\\sigma = 1$.\nSince $\\int_{\\R} e^{-t^2/2} \\, dt = \\sqrt{2\\pi}$ and $\\int_{\\R} t^2 e^{-t^2/2} \\, dt = \\sqrt{2\\pi}$, the integral of $(1 - t^2) e^{-t^2/2}$ is $0$.\nCheck of the norm, with $\\sigma = 1$.\nThe integral of $(1 - t^2)^2 e^{-t^2}$ is $\\sqrt{\\pi}\\,(1 - 1 + 3/4) = 3\\sqrt{\\pi}/4$, and multiplying by $4/(3\\sqrt{\\pi})$ gives $\\norm{\\psi}_2^2 = 1$.\nThe general case follows by the substitution $t = \\sigma s$.\n\n## Remarks\n\nThe Fourier transform of $\\psi$ is a constant times $\\omega^2 e^{-\\sigma^2\\omega^2/2}$.\nIt vanishes to second order at $\\omega = 0$, so $\\psi$ satisfies the [[admissibility condition|2BBB8NY5ZA]] and has two [[vanishing moments|7QERXM0C52]]: $\\int \\psi = 0$ and $\\int t \\psi(t) \\, dt = 0$, while $\\int t^2 \\psi(t) \\, dt \\ne 0$.\nIt does not generate an orthonormal basis; it is used in the [[continuous wavelet transform|57FR21NDGV]].\nIn two variables, the analogous function is the negative Laplacian of a Gaussian, which detects blobs and edges in images.\nThe Ricker wavelet is also a standard model of a seismic source pulse.\n\n## Explanations\n\n### L1 human\n\nDraw a bell curve, push its sides down below zero, and keep the middle as a bump: that is the Mexican hat shape.\nThe positive bump and the two negative dips balance, so the total area is zero.\nSliding this shape along a signal gives a strong response where the signal has a peak of matching width.\n",
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