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  "body": "\n## Summary\n\nThe Littlewood–Paley decomposition writes a function $f$ on $\\R^n$ as a sum $f = \\sum_{j \\in \\Z} \\Delta_j f$ of pieces whose [[Fourier transforms|DPVM34ADZV]] are supported in dyadic annuli $\\set{2^{j-1} \u003c \\abs{\\xi} \u003c 2^{j+1}}$.\nThe pieces are obtained by smooth cut-offs in frequency, so each $\\Delta_j f$ is a convolution of $f$ with a rescaled Schwartz function.\nIt is the basic construction of [[Littlewood–Paley theory|AA29VBWBFH]], and it is used to define Besov and Triebel–Lizorkin spaces and paraproducts.\n\n## Explanations\n\n### L1 human\n\nThe decomposition separates a function into frequency bands, each one octave wide.\nThe piece $\\Delta_j f$ contains the oscillations of $f$ at wavelengths about $2^{-j}$, while coarser and finer details are filtered out.\nSmooth filters are used instead of sharp cut-offs, so that each piece is still nicely localized in space.\nFor instance, $\\cos(2\\pi \\cdot 5x)$ on $\\R$ has frequencies $\\pm 5$, and only the pieces with $j = 2$ and $j = 3$ can be nonzero, since $2^{j-1} \u003c 5 \u003c 2^{j+1}$ holds only for these $j$.\n",
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