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  "body": "\n## Definition\n\nLet $\\psi \\colon \\R \\to \\C$ be a [[wavelet|4PZ35C0GV3]] and let $p \\ge 1$ be an integer.\nSuppose that $x \\mapsto x^k \\psi(x)$ is [[Lebesgue integrable|DWJF7ABPR1]] on $\\R$ for $k = 0, \\dots, p - 1$.\nThe wavelet $\\psi$ has *$p$ vanishing moments* if its [[moments|01XAHPADBA]] of order less than $p$ vanish:\n$$\\int_\\R x^k \\, \\psi(x) \\, dx = 0 \\qquad \\text{for } k = 0, 1, \\dots, p - 1.$$\n\n## Remarks\n\nEquivalently, $\\int_\\R P(x) \\psi(x) \\, dx = 0$ for every polynomial $P$ of degree less than $p$.\nIn terms of the Fourier transform, under the same integrability, this says that $\\hat\\psi$ has a zero of order at least $p$ at $\\xi = 0$.\nThe *number* of vanishing moments of $\\psi$ is the largest such $p$.\n\nExample: the [[Haar wavelet|JW5PA9XA7F]] $\\psi = 1_{[0, 1/2)} - 1_{[1/2, 1)}$ has $\\int \\psi = \\tfrac12 - \\tfrac12 = 0$, so it has $1$ vanishing moment.\n\nNon-example: the Haar wavelet does not have $2$ vanishing moments, since $\\int_\\R x \\psi(x) \\, dx = \\tfrac18 - \\tfrac38 = -\\tfrac14 \\ne 0$.\n",
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