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  "body": "\n## Example\n\nUse the [[Fourier transform|DPVM34ADZV]] $\\hat f(\\omega) = \\int_{\\R} f(x) e^{-i\\omega x} \\, dx$.\nFix an infinitely differentiable function $\\nu \\colon \\R \\to [0,1]$ with $\\nu(x) = 0$ for $x \\le 0$, $\\nu(x) = 1$ for $x \\ge 1$, and $\\nu(x) + \\nu(1-x) = 1$ for all $x$.\nThe *Meyer wavelet* $\\psi$ is the function in $L^2(\\R)$ whose Fourier transform is\n$$\\hat\\psi(\\omega) = \\begin{cases} e^{i\\omega/2} \\sin\\left(\\frac{\\pi}{2} \\nu\\left(\\frac{3\\abs{\\omega}}{2\\pi} - 1\\right)\\right) \u0026 \\text{if } \\frac{2\\pi}{3} \\le \\abs{\\omega} \\le \\frac{4\\pi}{3}, \\\\ e^{i\\omega/2} \\cos\\left(\\frac{\\pi}{2} \\nu\\left(\\frac{3\\abs{\\omega}}{4\\pi} - 1\\right)\\right) \u0026 \\text{if } \\frac{4\\pi}{3} \\le \\abs{\\omega} \\le \\frac{8\\pi}{3}, \\\\ 0 \u0026 \\text{otherwise.} \\end{cases}$$\nThe functions $\\psi_{j,k}(x) = 2^{j/2} \\psi(2^j x - k)$, for $j, k \\in \\Z$, form an orthonormal basis of $L^2(\\R)$.\nSo $\\psi$ is an orthonormal wavelet.\nIt comes from a [[multiresolution analysis|1GHQMBR7DE]] whose scaling function $\\varphi$ has $\\hat\\varphi(\\omega) = 1$ for $\\abs{\\omega} \\le 2\\pi/3$ and $\\hat\\varphi(\\omega) = 0$ for $\\abs{\\omega} \\ge 4\\pi/3$.\n\n## Remarks\n\nThe two formulas agree at $\\abs{\\omega} = 4\\pi/3$, since there the sine has argument $\\pi/2$ and the cosine has argument $0$.\nThe condition $\\nu(x) + \\nu(1-x) = 1$ makes the values of $\\abs{\\hat\\psi}^2$ on overlapping bands add up to $1$, which is the key to orthonormality.\nSince $\\hat\\psi$ vanishes near $\\omega = 0$, every moment $\\int x^n \\psi(x) \\, dx$ is zero.\nSince $\\hat\\psi$ has compact support, $\\psi$ is band-limited and infinitely differentiable.\nSince $\\nu$ is $C^\\infty$, $\\hat\\psi$ is $C^\\infty$ with compact support, so $\\psi$ decays faster than any power of $\\abs{x}$.\nThe support of $\\psi$ itself is not compact.\nThe Meyer wavelet can be seen as a smoothed version of the [[Shannon wavelet|6C1S8HD38Z]], whose Fourier transform is a sharp indicator function.\n\n## Explanations\n\n### L1 human\n\nThe Meyer wavelet is built in frequency space rather than in time.\nIts spectrum lives in two bands, one on each side of the origin, and the edges of the bands are rounded off smoothly instead of being cut sharply.\nBecause the edges are smooth, the wavelet is concentrated in time as well: it is a smooth wave that dies out quickly away from the origin.\nShifting and dyadically stretching it gives an orthonormal basis for all finite-energy signals on the line.\n",
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