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  "body": "\n## Statement\n\nLet $G$ be a [[locally compact abelian group|ZR09S9R3AW]] with a [[Haar measure|VY0HXD3GC4]] $\\mu$, and let $\\widehat{G}$ be its [[Pontryagin dual|FMW57WNPDB]].\nFor $f \\in L^1(G)$ let $\\widehat{f}$ be its [[Fourier transform|VHN5VW5Q92]].\nThen there is a unique Haar measure $\\nu$ on $\\widehat{G}$ such that:\n\n1. for every $f \\in L^1(G) \\cap L^2(G)$, the function $\\widehat{f}$ lies in $L^2(\\widehat{G}, \\nu)$ and $\\norm{\\widehat{f}}_{L^2(\\nu)} = \\norm{f}_{L^2(\\mu)}$;\n2. the map $f \\mapsto \\widehat{f}$ on $L^1(G) \\cap L^2(G)$ extends uniquely to a unitary isomorphism of [[Hilbert spaces|HM2HHE3J51]] $L^2(G, \\mu) \\to L^2(\\widehat{G}, \\nu)$.\n\n## Remarks\n\nThe measure $\\nu$ is called the dual measure of $\\mu$; scaling $\\mu$ by $c \u003e 0$ scales $\\nu$ by $1/c$.\nOne proof applies [[Bochner's theorem|ZRHAG4NXHC]] to functions $f * \\tilde{f}$, where $\\tilde{f}(x) = \\overline{f(-x)}$, whose transform is $\\abs{\\widehat{f}}^2$.\nThis yields a Haar measure on $\\widehat{G}$ for which the inversion formula holds for such functions.\nEvaluating $f * \\tilde{f}$ at $0$ then gives the isometry, and density of $L^1 \\cap L^2$ in $L^2$ gives the extension.\nExample: for $G = \\mathbb{T}$ with normalised arc length, $\\widehat{G} = \\Z$, $\\nu$ is counting measure, and the theorem is Parseval's identity $\\frac{1}{2\\pi}\\int_0^{2\\pi} \\abs{f(\\theta)}^2 \\, d\\theta = \\sum_{n \\in \\Z} \\abs{\\widehat{f}(n)}^2$.\nFor $G = \\R$ with Lebesgue measure and characters $x \\mapsto e^{2\\pi i \\xi x}$, the dual measure is again Lebesgue measure.\nThe general [[Plancherel theorem for groups|DH3PYQJXH0]] extends the result to some non-abelian groups, where the dual is the unitary dual and $\\nu$ becomes the Plancherel measure.\n\n## Explanations\n\n### L1 human\n\nThe energy of a function, the integral of its squared size, is the same as the total energy of its Fourier transform.\nNo energy is lost or created when a function is split into waves.\nOn the circle this is the familiar fact that the mean square of a periodic function is the sum of the squares of its Fourier coefficients.\n\n### L2 human\n\nThe theorem makes the Fourier transform a unitary operator on $L^2$, so one can transform any square-integrable function, not only integrable ones.\nIt fixes a normalisation of the Haar measure on the dual group, which is the one used in the Fourier inversion formula.\nTogether with [[Pontryagin duality|QTVC6F2P9D]] it shows that $L^2(G)$ is diagonalised by the characters: translation operators become multiplication operators on $L^2(\\widehat{G})$.\n",
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