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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $f \\colon \\R^n \\to \\C$ be a Lebesgue [[integrable function|X7GTF3D98F]].\nThe *Fourier transform* of $f$ is the function $\\hat f \\colon \\R^n \\to \\C$ given by the [[Lebesgue integral|DWJF7ABPR1]]\n$$\\hat f(\\xi) = \\int_{\\R^n} f(x)\\, e^{-2\\pi i\\, x \\cdot \\xi} \\, dx, \\qquad \\xi \\in \\R^n,$$\nwhere $x \\cdot \\xi = x_1 \\xi_1 + \\dots + x_n \\xi_n$.\nIt is also written $\\mathcal{F} f$.\n\n## Remarks\n\nThe integral converges absolutely because $\\abs{f(x) e^{-2\\pi i x \\cdot \\xi}} = \\abs{f(x)}$.\nSo $\\abs{\\hat f(\\xi)} \\le \\int \\abs{f}$, and $\\hat f$ is continuous by dominated convergence.\nBy the Riemann–Lebesgue lemma, $\\hat f(\\xi) \\to 0$ as $\\abs{\\xi} \\to \\infty$.\nOther common conventions are $\\int f(x) e^{-i x \\cdot \\xi}\\, dx$ and $(2\\pi)^{-n/2} \\int f(x) e^{-i x \\cdot \\xi}\\, dx$; they change the constants in the formulas below.\n\nWith the convention above, if $f$ and $\\partial_j f$ are integrable, then $\\widehat{\\partial_j f}(\\xi) = 2\\pi i \\xi_j \\hat f(\\xi)$.\nIf $\\hat f$ is also integrable, the inversion formula $f(x) = \\int \\hat f(\\xi) e^{2\\pi i x \\cdot \\xi}\\, d\\xi$ holds for almost every $x$.\nPlancherel's theorem extends $\\mathcal F$ to a unitary map of $L^2(\\R^n)$, and duality extends it to [[tempered distributions|AES2T5JPFP]].\n\nExample: $f(x) = e^{-\\pi \\abs{x}^2}$ satisfies $\\hat f = f$.\nFor $n = 1$, completing the square gives $\\hat f(\\xi) = e^{-\\pi \\xi^2} \\int_\\R e^{-\\pi (x + i\\xi)^2}\\, dx$, and the last integral equals $\\int_\\R e^{-\\pi x^2}\\, dx = 1$ by a contour shift.\n\nNon-example: the constant function $f = 1$ on $\\R$ is not integrable, and the integral $\\int_\\R e^{-2\\pi i x \\xi}\\, dx$ diverges for every $\\xi$.\nIts Fourier transform exists only as a tempered distribution, namely the Dirac delta at $0$.\n\n## Explanations\n\n### L1 human\n\nThe transform measures how strongly $f$ oscillates with each frequency $\\xi$.\nIt is the continuous analogue of the list of coefficients of a Fourier series.\n",
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