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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f, g \\in L^1(\\R^n)$.\nLet $\\hat h(\\xi) = \\int_{\\R^n} h(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ denote the [[Fourier transform|DPVM34ADZV]] of $h \\in L^1(\\R^n)$.\nThen\n$$\\int_{\\R^n} \\hat f(y)\\, g(y)\\, dy = \\int_{\\R^n} f(y)\\, \\hat g(y)\\, dy,$$\nand both integrals converge absolutely.\n\n## Remarks\n\nBoth sides equal the double integral $\\int\\int f(x) g(y) e^{-2\\pi i x \\cdot y}\\, dx\\, dy$.\nThis double integral converges absolutely, with bound $\\norm{f}_1 \\norm{g}_1$, so [[Fubini's theorem|0R0T1VWH5X]] allows either order of integration.\nAbsolute convergence of each side also follows from $\\abs{\\hat f} \\le \\norm{f}_1$ and $g \\in$ [[$L^1$|SASP14MA1Z]].\n\nExample: take $n = 1$, $f = \\mathbf{1}_{[-1/2, 1/2]}$ and $g(y) = e^{-\\pi y^2}$, so that $\\hat g = g$.\nThe right side is $\\int_{-1/2}^{1/2} e^{-\\pi y^2}\\, dy$.\nThe left side is $\\int_\\R \\frac{\\sin \\pi y}{\\pi y} e^{-\\pi y^2}\\, dy$, and the formula says the two numbers are equal.\n\nThe formula is the basis of the definition of the Fourier transform of a tempered distribution $T$, namely $\\hat T(\\varphi) = T(\\hat \\varphi)$.\nBy the formula, this agrees with the transform of $f$ when $T$ is integration against $f$.\n\n## Explanations\n\n### L1 human\n\nThe transform can be moved from one factor to the other inside an integral of a product.\nIt is a symmetry of the Fourier transform with respect to the pairing $\\int f g$.\n\n### L2 human\n\nThe formula is the main tool in proofs of the Fourier inversion theorem: one pairs $\\hat f$ with a Gaussian of shrinking width and moves the transform onto the Gaussian.\nIt is also what makes the extension of the transform to tempered distributions consistent with the integral definition.\nWith complex conjugates it leads to the Parseval form $\\int f \\bar g = \\int \\hat f \\overline{\\hat g}$ on $L^1 \\cap L^2$.\n",
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