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  "body": "\n## Summary\n\nThe scaling function $\\varphi$ of a [[multiresolution analysis|1GHQMBR7DE]] is a function whose integer translates form an orthonormal basis of the base space $V_0$.\nSince $V_0 \\subseteq V_1$, it satisfies a refinement equation $\\varphi(x) = \\sqrt2 \\sum_k h_k \\varphi(2x - k)$: it is a combination of half-size copies of itself.\nThe coefficients $h_k$ form the low-pass filter of the [[discrete wavelet transform|P3H4MMC4ZH]], and they also determine the wavelet.\nConversely, one constructs wavelets such as the [[Daubechies wavelets|90V8S5CXE6]] by choosing the coefficients first and solving the refinement equation.\n\n## Explanations\n\n### L1 human\n\nThe box function $1_{[0,1)}$ is made of two boxes of half the width: the one on $[0, \\tfrac12)$ and the one on $[\\tfrac12, 1)$.\nSo $\\varphi(x) = \\varphi(2x) + \\varphi(2x - 1)$.\nAn equation of this kind, where a function equals a combination of squeezed and shifted copies of itself, is a refinement equation.\nMany of its solutions are not simple pictures: they are fractal-like curves fixed by the equation, and the box is only the easiest one.\n",
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