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  "body": "\n## Definition\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]].\nLet $f \\in L^1(G, \\mu)$, the [[group algebra|6GD25S53ND]] of $\\mu$-integrable functions.\nThe *Fourier transform* of $f$ is the function $\\widehat{f} \\colon \\widehat{G} \\to \\C$ given by\n$$\\widehat{f}(\\chi) = \\int_G f(x) \\overline{\\chi(x)} \\, d\\mu(x).$$\n\n## Remarks\n\nExample: for $G = \\R$ with Lebesgue measure, the continuous characters are $\\chi_\\xi(x) = e^{2\\pi i \\xi x}$ for $\\xi \\in \\R$, and the formula gives $\\widehat{f}(\\chi_\\xi) = \\int_\\R f(x) e^{-2\\pi i \\xi x} \\, dx$, the classical Fourier transform.\nFor $G = \\mathbb{T}$ with normalised arc length and $\\chi_n(z) = z^n$, it gives the Fourier coefficients of $f$.\nNon-example: for $f(x) = 1$ on $\\R$, the integral does not converge because $f \\notin L^1(\\R)$; the transform of such $f$ only makes sense as a distribution.\nThe integral converges for $f \\in L^1$ because $\\abs{\\chi(x)} = 1$, and $\\abs{\\widehat{f}(\\chi)} \\le \\norm{f}_1$.\nThe function $\\widehat{f}$ is continuous on $\\widehat{G}$ and vanishes at infinity, and $\\widehat{f * g} = \\widehat{f}\\,\\widehat{g}$ for the convolution $f * g$.\nSome authors use $\\chi(x)$ instead of $\\overline{\\chi(x)}$; this changes $\\widehat{f}(\\chi)$ into $\\widehat{f}(\\overline{\\chi})$.\n",
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