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  "body": "\n## Example\n\nFor each integer $N \\ge 1$, Ingrid Daubechies constructed an orthonormal [[wavelet|4PZ35C0GV3]] $\\psi$ from a [[multiresolution analysis|1GHQMBR7DE]].\nIts [[scaling function|YH4BERB952]] $\\varphi$ satisfies a refinement equation $\\varphi(x) = \\sqrt2 \\sum_{k=0}^{2N-1} h_k \\varphi(2x - k)$ with $2N$ real coefficients.\nBoth $\\varphi$ and $\\psi$ have compact support of length $2N - 1$, and $\\psi$ has $N$ [[vanishing moments|7QERXM0C52]].\nFor $N = 1$ this is the Haar wavelet.\n\nThe case $N = 2$ has the filter\n$$h_0 = \\frac{1 + \\sqrt3}{4\\sqrt2}, \\quad h_1 = \\frac{3 + \\sqrt3}{4\\sqrt2}, \\quad h_2 = \\frac{3 - \\sqrt3}{4\\sqrt2}, \\quad h_3 = \\frac{1 - \\sqrt3}{4\\sqrt2}.$$\nCheck of the filter conditions.\nThe sum is $\\frac{8}{4\\sqrt2} = \\sqrt2$, as needed for $\\int \\varphi = 1$.\nThe sum of squares is $\\frac{(4 + 2\\sqrt3) + (12 + 6\\sqrt3) + (12 - 6\\sqrt3) + (4 - 2\\sqrt3)}{32} = 1$.\nAnd $h_0 h_2 + h_1 h_3 = \\frac{(1 + \\sqrt3)(3 - \\sqrt3) + (3 + \\sqrt3)(1 - \\sqrt3)}{32} = \\frac{2\\sqrt3 - 2\\sqrt3}{32} = 0$.\nSo the filter is orthogonal to its shift by $2$, which reflects the orthogonality of the translates $\\varphi(\\cdot - k)$.\n\nThe wavelet is $\\psi(x) = \\sqrt2 \\sum_k g_k \\varphi(2x - k)$ with $g_k = (-1)^k h_{1-k}$, so its coefficients are $g_{-2} = h_3$, $g_{-1} = -h_2$, $g_0 = h_1$, $g_1 = -h_0$.\nCheck of two vanishing moments: $\\sum_k g_k = h_3 - h_2 + h_1 - h_0 = 0$ and $\\sum_k k g_k = -2h_3 + h_2 - h_0 = \\frac{-2 + 2\\sqrt3 + 3 - \\sqrt3 - 1 - \\sqrt3}{4\\sqrt2} = 0$.\nThese two discrete conditions give $\\int \\psi = 0$ and $\\int x \\psi(x) \\, dx = 0$.\nThe scaling function of this case is continuous but not smooth, and it has no closed formula; it is computed from the refinement equation.\n\n## Explanations\n\n### L1 human\n\nDaubechies wavelets are a family of wavelets that live on short intervals and are smoother as the index grows.\nEach one is fixed by a short list of numbers, its filter, and the discrete wavelet transform uses only these numbers.\nThe four numbers above give the simplest one after the Haar wavelet.\nTheir shape looks jagged, almost like a fractal, but they are widely used in signal and image compression.\n",
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