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  "body": "\n## Definition\n\nLet $(h_n)_{n \\in \\Z}$ be a finite sequence of real numbers with $\\sum_n h_n = \\sqrt{2}$.\nLet $\\varphi_0 \\in L^2(\\R)$ be a starting function, usually the indicator function of $[0, 1)$.\nThe *cascade algorithm* with filter $(h_n)$ is the iteration\n$$\\varphi_{i+1}(x) = \\sqrt{2} \\sum_{n} h_n \\, \\varphi_i(2x - n), \\qquad i = 0, 1, 2, \\dots$$\nSuppose the sequence $(\\varphi_i)$ converges in $L^2(\\R)$ to a limit $\\varphi$.\nThen $\\varphi$ satisfies the refinement equation $\\varphi(x) = \\sqrt{2}\\sum_n h_n \\varphi(2x - n)$.\nSo $\\varphi$ is the [[scaling function|YH4BERB952]] of the filter.\n\n## Remarks\n\nThe condition $\\sum_n h_n = \\sqrt{2}$ keeps the integral fixed: $\\int \\varphi_{i+1} = \\sqrt{2}\\sum_n h_n \\cdot \\frac{1}{2}\\int\\varphi_i = \\int \\varphi_i$.\nThe values of $\\varphi_i$ at the points $k 2^{-i}$ can be computed by repeated upsampling by $2$ and convolution with $\\sqrt{2}\\,h$.\nThis is the inverse fast wavelet transform applied to a single coefficient.\n\nExample: for the Haar filter $h_0 = h_1 = 1/\\sqrt{2}$ and $\\varphi_0 = \\mathbf{1}_{[0,1)}$, one step gives $\\varphi_1(x) = \\mathbf{1}_{[0,1)}(2x) + \\mathbf{1}_{[0,1)}(2x - 1) = \\mathbf{1}_{[0,1)}(x)$.\nSo the iteration is constant and converges at once to the Haar scaling function.\n\nNon-example: for the filter $h_0 = h_1 = 1$, the sum is $2 \\ne \\sqrt{2}$.\nFrom $\\varphi_0 = \\mathbf{1}_{[0,1)}$ one gets $\\varphi_i = 2^{i/2}\\mathbf{1}_{[0,1)}$, which does not converge.\n\nFor the filters of the Daubechies wavelets, the iterates converge to the scaling function in $L^2(\\R)$.\nFor some filters that satisfy the sum condition, the iteration still fails to converge, so convergence depends on further properties of the filter.\n\n## Explanations\n\n### L1 human\n\nThe cascade algorithm turns the self-similarity rule of a scaling function into a repeated computation.\nEach round replaces the current shape by a weighted sum of half-size, shifted copies of it.\n",
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