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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f \\colon \\R^n \\to \\C$ be [[integrable|X7GTF3D98F]] for [[Lebesgue measure|2Z2QVWZ262]].\nFor $\\xi \\in \\R^n$ let\n$$\\hat{f}(\\xi) = \\int_{\\R^n} f(x) \\, e^{-2 \\pi i \\, x \\cdot \\xi} \\, dx,$$\na [[Lebesgue integral|DWJF7ABPR1]], where $x \\cdot \\xi$ is the dot product.\nThen $\\hat{f}(\\xi) \\to 0$ as $\\abs{\\xi} \\to \\infty$.\n\n## Remarks\n\nProof idea: for the indicator function of a box, $\\hat{f}$ is a product of terms like $(e^{-2\\pi i b \\xi_k} - e^{-2\\pi i a \\xi_k}) / (-2\\pi i \\xi_k)$, which tends to $0$.\nLinear combinations of such indicators, the step functions, are dense in $L^1(\\R^n)$.\nSince $\\abs{\\hat{f}(\\xi) - \\hat{g}(\\xi)} \\le \\int \\abs{f - g}$ for every $\\xi$, the property passes to all integrable $f$.\n\nExample: for $f = \\mathbf{1}_{[-1/2, 1/2]}$ on $\\R$, $\\hat{f}(\\xi) = \\sin(\\pi \\xi) / (\\pi \\xi)$ for $\\xi \\ne 0$, which tends to $0$.\n\nThe same proof gives the version for Fourier coefficients.\nIf $f$ is integrable on $[0, 2\\pi]$, then $\\int_0^{2\\pi} f(x) e^{-ikx} \\, dx \\to 0$ as $\\abs{k} \\to \\infty$ over the integers.\nThe lemma gives no rate of decay in general.\n\n## Explanations\n\n### L1 human\n\nMultiplying a function by a fast oscillating wave and integrating gives almost complete cancellation.\nThe positive and negative parts of the wave nearly balance on each small piece where the function is almost constant.\nSo the Fourier transform of an integrable function becomes small at high frequencies.\n\n### L2 human\n\nThe lemma says that the Fourier transform maps $L^1(\\R^n)$ into the space $C_0(\\R^n)$ of continuous functions vanishing at infinity.\nIn Fourier series it is a first step in the study of pointwise convergence, through the Dini and Jordan criteria.\nThe image of $L^1$ is a proper subset of $C_0$, so the decay at infinity alone does not characterise Fourier transforms of integrable functions.\n",
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