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  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $L^p(\\R^n)$ denote the [[Lebesgue spaces|SASP14MA1Z]] with respect to Lebesgue measure.\nFor $f \\in L^1(\\R^n)$ let $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi} \\, dx$ be its [[Fourier transform|DPVM34ADZV]].\n\n1. If $f \\in L^1(\\R^n) \\cap L^2(\\R^n)$, then $\\hat f \\in L^2(\\R^n)$ and\n$$\\int_{\\R^n} \\abs{\\hat f(\\xi)}^2 \\, d\\xi = \\int_{\\R^n} \\abs{f(x)}^2 \\, dx.$$\n2. The map $f \\mapsto \\hat f$ on $L^1(\\R^n) \\cap L^2(\\R^n)$ extends uniquely to a [[unitary operator|MM0EJ2GDE4]] $\\mathcal F$ of the [[Hilbert space|HM2HHE3J51]] $L^2(\\R^n)$ onto itself.\n\n## Remarks\n\nProof idea: for Schwartz functions the identity follows from the multiplication formula $\\int \\hat f \\, \\overline{\\hat g} = \\int f \\, \\overline{g}$ and Fourier inversion.\nSchwartz functions are dense in $L^1 \\cap L^2$ for the norm $\\norm{f}_1 + \\norm{f}_2$, which gives part 1.\nSince $L^1 \\cap L^2$ is dense in $L^2$, the isometry extends uniquely to $L^2$.\nThe extension is onto because its range is closed and contains the Schwartz space.\n\nFor $f \\in L^2$ the transform is computed as the $L^2$ limit $\\mathcal F f = \\lim_{R \\to \\infty} \\int_{\\abs{x} \\le R} f(x) e^{-2\\pi i x \\cdot \\xi} \\, dx$, because the integral itself may diverge.\nThe inner product is also preserved: $\\int \\mathcal F f \\, \\overline{\\mathcal F g} = \\int f \\, \\overline{g}$ for $f, g \\in L^2(\\R^n)$.\n\nExample: for $f = \\mathbf{1}_{[-1/2, 1/2]}$ on $\\R$, $\\hat f(\\xi) = \\frac{\\sin \\pi \\xi}{\\pi \\xi}$ and $\\int \\abs{f}^2 = 1$.\nSo the theorem gives $\\int_\\R \\left( \\frac{\\sin \\pi \\xi}{\\pi \\xi} \\right)^2 d\\xi = 1$.\n\nThe analogue for Fourier series is [[Parseval's identity|047RFCNVSK]].\nThe theorem generalizes to [[locally compact abelian groups|V9K9TG0K3J]] and, with representations in place of characters, to [[more general groups|DH3PYQJXH0]].\n\n## Explanations\n\n### L1 human\n\nThe energy of a signal, the integral of its squared size, is the same whether one computes it from the signal or from its frequency content.\nSo the Fourier transform preserves lengths and angles, like a rotation in infinite dimensions.\nThis lets one define the Fourier transform of every square-integrable function, even when the defining integral does not converge.\n\n### L2 human\n\nPlancherel's theorem makes $L^2$ the natural home of the Fourier transform.\nIt is the starting point of the $L^p$ theory, through interpolation with the trivial bound from $L^1$ to $L^\\infty$.\nIn quantum mechanics it says that the position and momentum representations of a wave function carry the same total probability.\nIt also underlies the definition of Sobolev spaces $H^s$ by weights on the Fourier side.\n",
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  "name": "Plancherel theorem",
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