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  "body": "\n## Definition\n\nLet $(V_j)_{j \\in \\Z}$ be a [[multiresolution analysis|1GHQMBR7DE]] of $L^2(\\R)$.\nA *scaling function* of it is a function $\\varphi \\in V_0$ whose integer translates $(\\varphi(\\cdot - k))_{k \\in \\Z}$ form an orthonormal basis of $V_0$.\nThe functions $\\sqrt2 \\, \\varphi(2 \\cdot - k)$, $k \\in \\Z$, form an orthonormal basis of $V_1$, and $\\varphi \\in V_1$.\nHence there is a unique sequence $(h_k)_{k \\in \\Z} \\in \\ell^2(\\Z)$ with\n$$\\varphi(x) = \\sqrt2 \\sum_{k \\in \\Z} h_k \\, \\varphi(2x - k),$$\nwith convergence in $L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]).\nThis identity is the *refinement equation* (or two-scale equation) of $\\varphi$, and $(h_k)$ is its *low-pass filter*; explicitly $h_k = \\ip{\\varphi}{\\sqrt2 \\, \\varphi(2 \\cdot - k)}$.\n\n## Remarks\n\nOrthonormality of the translates of $\\varphi$ gives $\\sum_k h_k \\overline{h_{k + 2m}} = \\delta_{m,0}$ for all $m \\in \\Z$.\nIf $\\varphi$ is integrable with $\\int \\varphi \\ne 0$, integrating the refinement equation gives $\\sum_k h_k = \\sqrt2$.\nThe function $\\psi(x) = \\sqrt2 \\sum_k (-1)^k \\overline{h_{1-k}} \\, \\varphi(2x - k)$ is then an orthonormal wavelet.\n\nExample: in the Haar multiresolution analysis, $\\varphi = 1_{[0, 1)}$ and $\\varphi(x) = \\varphi(2x) + \\varphi(2x - 1)$.\nSo $h_0 = h_1 = 1/\\sqrt2$ and $h_k = 0$ otherwise.\nThe formula for $\\psi$ gives $\\psi(x) = \\varphi(2x) - \\varphi(2x - 1) = 1_{[0, 1/2)}(x) - 1_{[1/2, 1)}(x)$, the Haar wavelet.\n\nNon-example: $\\varphi = 1_{[0, 2)}/\\sqrt2$ satisfies the refinement equation $\\varphi(x) = \\varphi(2x) + \\varphi(2x - 2)$, but its integer translates overlap, so $\\ip{\\varphi}{\\varphi(\\cdot - 1)} = \\tfrac12 \\ne 0$ and it is not the scaling function of a multiresolution analysis in the sense above.\n",
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