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  "body": "\n## Summary\n\nThe $n$-th Fourier coefficient $\\hat f(n)$ of a $2\\pi$-periodic integrable function $f$ measures how much of the wave $e^{inx}$ is present in $f$.\nIt is the average of $f(x) e^{-inx}$ over one period, and it is the inner product of $f$ with the [[Fourier monomial|TZYSW0975T]] $e_n$.\nThe sequence $(\\hat f(n))_{n \\in \\Z}$ determines $f$ up to a null set, and the [[Fourier series|G12KXDHPPS]] of $f$ is $\\sum_n \\hat f(n) e^{inx}$.\nThe size of the coefficients for large $\\abs{n}$ reflects the smoothness of $f$.\n\n## Explanations\n\n### L1 human\n\nTo find how much of the frequency $n$ is in a signal, multiply the signal by the wave $e^{-inx}$ and average over one period.\nThe other frequencies average out to zero, and only the part with frequency $n$ survives.\nFor example, for $f(x) = 3 + 5 \\cos 2x$ the coefficient $\\hat f(0)$ is $3$, the coefficients $\\hat f(2)$ and $\\hat f(-2)$ are $\\frac{5}{2}$, and all the others are $0$.\n",
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