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  "body": "\n## Summary\n\nThe Fourier transform on a locally compact abelian group $G$ sends an integrable function $f$ on $G$ to a function $\\widehat{f}$ on the dual group.\nThe value of $\\widehat{f}$ at a character is the integral of $f$ against the conjugate of that character.\nIt unifies Fourier series on the circle, the Fourier transform on $\\R^n$ and the discrete Fourier transform on $\\Z/n\\Z$.\nIt turns convolution into pointwise multiplication, and with the right normalisation it is unitary on $L^2$.\n\n## Explanations\n\n### L1 human\n\nThe Fourier transform measures how much of each wave is present in a function.\nOn a group, the waves are the characters, and the amount of the wave $\\chi$ in $f$ is the average of $f$ times the conjugate of $\\chi$.\nOn the circle this gives the Fourier coefficients $\\frac{1}{2\\pi}\\int_0^{2\\pi} f(\\theta) e^{-in\\theta} \\, d\\theta$.\nOn the group $\\Z/n\\Z$ it gives the discrete Fourier transform used in signal processing.\nOne definition covers all these cases.\n",
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