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  "body": "\n## Definition\n\nLet $n \\ge 1$, and let $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi} \\, dx$ be the [[Fourier transform|DPVM34ADZV]], extended to [[tempered distributions|AES2T5JPFP]].\nFix a smooth function $\\chi \\colon \\R^n \\to [0, 1]$ with $\\chi(\\xi) = 1$ for $\\abs{\\xi} \\le 1$ and $\\chi(\\xi) = 0$ for $\\abs{\\xi} \\ge 2$.\nLet $\\psi(\\xi) = \\chi(\\xi) - \\chi(2\\xi)$.\nFor $j \\in \\Z$ and $f$ a tempered distribution, the *Littlewood–Paley piece* $\\Delta_j f$ is defined by\n$$\\widehat{\\Delta_j f}(\\xi) = \\psi(2^{-j}\\xi) \\, \\hat f(\\xi).$$\nThe *Littlewood–Paley decomposition* of $f$ is the series $\\sum_{j \\in \\Z} \\Delta_j f$.\n\n## Remarks\n\nThe function $\\psi$ vanishes for $\\abs{\\xi} \\le 1/2$ and for $\\abs{\\xi} \\ge 2$, so $\\widehat{\\Delta_j f}$ is supported in $\\set{2^{j-1} \\le \\abs{\\xi} \\le 2^{j+1}}$.\nFor $\\xi \\ne 0$ the sum telescopes: $\\sum_{j = -N}^{M} \\psi(2^{-j}\\xi) = \\chi(2^{-M}\\xi) - \\chi(2^{N+1}\\xi)$, which tends to $1$ as $M, N \\to \\infty$.\nHence $f = \\sum_{j \\in \\Z} \\Delta_j f$ with convergence in $L^2(\\R^n)$ for $f \\in L^2(\\R^n)$, and in the [[Lp space|SASP14MA1Z]] $L^p(\\R^n)$ for $1 \u003c p \u003c \\infty$.\nEach $\\Delta_j f = f * \\check\\psi_j$ is a convolution with the Schwartz function $\\check\\psi_j(x) = 2^{jn} \\check\\psi(2^j x)$, where $\\check\\psi$ is the inverse Fourier transform of $\\psi$.\nAn inhomogeneous version keeps $\\Delta_j$ for $j \\ge 1$ and replaces all $j \\le 0$ by the single low-frequency piece with multiplier $\\chi$.\n\nExample: if $\\hat f$ vanishes outside $\\set{2^{k-1} \\le \\abs{\\xi} \\le 2^{k+1}}$ for some $k \\in \\Z$, then $\\Delta_j f = 0$ unless $j \\in \\set{k - 1, k, k + 1}$.\nIndeed $\\psi(2^{-j}\\xi) \\ne 0$ needs $2^{j-1} \u003c \\abs{\\xi} \u003c 2^{j+1}$, and this meets the support of $\\hat f$ only when $k - 2 \u003c j \u003c k + 2$.\n\nNon-example: the cut-offs $\\mathbf{1}_{\\set{2^j \\le \\abs{\\xi} \u003c 2^{j+1}}}$ also sum to $1$ on $\\xi \\ne 0$, but they are not smooth, so they do not fit this definition.\nFor $n \\ge 2$ such sharp cut-offs are badly behaved on $L^p(\\R^n)$ for $p \\ne 2$, as Fefferman's ball multiplier theorem shows for the indicator of a ball.\n",
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    "name": "Littlewood–Paley decomposition",
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