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  "body": "\n## Summary\n\nA [[wavelet|4PZ35C0GV3]] $\\psi$ has $p$ vanishing moments if it is orthogonal to all polynomials of degree less than $p$, that is, $\\int x^k \\psi(x) \\, dx = 0$ for $k = 0, \\dots, p - 1$.\nThen the wavelet coefficients of a function that is smooth near a point are very small at fine scales near that point.\nThis is why wavelets with many vanishing moments compress smooth signals well.\nThe [[Daubechies wavelets|90V8S5CXE6]] have the shortest support among orthonormal wavelets with a given number of vanishing moments.\n\n## Explanations\n\n### L1 human\n\nA wavelet with vanishing moments cannot see polynomials of low degree: its inner product with them is zero.\nA smooth function looks like a polynomial on a small interval, by Taylor's formula.\nSo a narrow wavelet with many vanishing moments sees almost nothing of a smooth function, and its coefficient is close to zero.\nLarge coefficients appear only near jumps and other rough spots, which is what makes compression possible.\n",
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