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  "body": "\n## Example\n\nWrite $\\operatorname{sinc}(t) = \\sin(\\pi t)/(\\pi t)$ for $t \\ne 0$ and $\\operatorname{sinc}(0) = 1$.\nUse the [[Fourier transform|DPVM34ADZV]] $\\hat f(\\omega) = \\int_{\\R} f(t) e^{-i\\omega t} \\, dt$.\nThe real *Shannon wavelet* is\n$$\\psi(t) = 2\\operatorname{sinc}(2t) - \\operatorname{sinc}(t) = \\operatorname{sinc}\\left(\\frac{t}{2}\\right) \\cos\\left(\\frac{3\\pi t}{2}\\right).$$\nIts Fourier transform is the indicator function of the set $\\set{\\omega : \\pi \u003c \\abs{\\omega} \\le 2\\pi}$, up to a set of measure zero.\n\nCheck.\nThe transform of $\\operatorname{sinc}$ is the indicator of $[-\\pi, \\pi]$.\nThe transform of $2\\operatorname{sinc}(2t)$ is the indicator of $[-2\\pi, 2\\pi]$.\nTheir difference is the indicator of the two bands.\nThe second formula follows from $\\sin(2\\pi t) - \\sin(\\pi t) = 2\\cos(3\\pi t/2)\\sin(\\pi t/2)$.\n\n## Remarks\n\nThe functions $\\psi_{j,k}(t) = 2^{j/2}\\psi(2^j t - k)$, $j, k \\in \\Z$, form an orthonormal basis of $L^2(\\R)$.\nThe reason is that the two bands, shifted by multiples of $2\\pi$, tile the line, and their dyadic dilates tile $\\R \\setminus \\set{0}$.\nThe scaling function of the underlying [[multiresolution analysis|1GHQMBR7DE]] is $\\varphi = \\operatorname{sinc}$, and its approximation spaces are spaces of band-limited functions.\nThe Shannon wavelet is perfectly localized in frequency but poorly localized in time: $\\abs{\\psi(t)}$ decays only like $1/\\abs{t}$, so $\\psi$ is not integrable.\nThe [[Meyer wavelet|Q5RN2FNX6Z]] smooths the sharp edges of $\\hat\\psi$ to gain fast decay in time.\n\n## Explanations\n\n### L1 human\n\nThe Shannon wavelet is an ideal band-pass filter turned into a function.\nIt keeps exactly the frequencies between $\\pi$ and $2\\pi$ in absolute value and nothing else.\nIts stretched copies split the frequency line into octaves with no overlap.\nThe price for these sharp cuts is a function that ripples on for a long time before it fades.\n",
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