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  "body": "\n## Example\n\nCoiflets are a family of compactly supported orthonormal wavelets constructed by Ingrid Daubechies and named after Ronald Coifman.\nFor an even integer $L \\ge 2$, the coiflet of order $L$ is a real orthonormal wavelet $\\psi$ with [[scaling function|YH4BERB952]] $\\varphi$ such that:\n\n1. $\\psi$ has $L$ [[vanishing moments|7QERXM0C52]]: $\\int_{\\R} x^l \\psi(x) \\, dx = 0$ for $l = 0, 1, \\dots, L-1$.\n2. $\\int_{\\R} \\varphi(x) \\, dx = 1$ and $\\int_{\\R} x^l \\varphi(x) \\, dx = 0$ for $l = 1, \\dots, L-1$.\n\nThe low-pass filter of the coiflet of order $L$ has $3L$ coefficients, so $\\varphi$ and $\\psi$ are supported on intervals of length $3L - 1$.\nFor $L = 2$ the filter has $6$ coefficients.\n\n## Remarks\n\nThe vanishing moments of the scaling function are the distinctive feature.\nFor a smooth function $f$, they make the scaling coefficient $\\ip{f}{\\varphi_{j,k}}$ close to $2^{-j/2} f(2^{-j} k)$, where $\\varphi_{j,k}(x) = 2^{j/2}\\varphi(2^j x - k)$, with an error that is small to high order in $2^{-j}$.\nSo samples of $f$ can be used directly as starting coefficients of the discrete wavelet transform.\nTo get these extra conditions, coiflets need longer filters than the [[Daubechies wavelets|90V8S5CXE6]] with the same number of vanishing moments of $\\psi$.\nCoiflets are also closer to symmetric than Daubechies wavelets.\n\n## Explanations\n\n### L1 human\n\nMost wavelets only ask that the wavelet itself have zero average against low-degree polynomials.\nCoiflets also ask this of the scaling function, apart from its total mass.\nAs a result, the scaling function acts almost like a point sample of the signal.\nThis makes them convenient when the input data are just samples of a smooth function.\n",
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