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  "body": "\n## Example\n\nLet $T \u003e 0$.\nThe *Dirac comb* of period $T$ is the sum of [[Dirac deltas|0KPHENZ5S7]] at the points $kT$, $k \\in \\Z$:\n$$\\operatorname{III}_T(\\varphi) = \\sum_{k \\in \\Z} \\varphi(kT) \\quad \\text{for } \\varphi \\in \\mathcal{S}(\\R),$$\nwhere $\\mathcal{S}(\\R)$ is the [[Schwartz space|CG9JEATW3Q]].\nIt is a [[tempered distribution|AES2T5JPFP]].\nIndeed, $\\abs{\\varphi(kT)} \\le (1 + k^2 T^2)^{-1} \\sup_x (1 + x^2) \\abs{\\varphi(x)}$, so\n$$\\abs{\\operatorname{III}_T(\\varphi)} \\le \\Big( \\sum_{k \\in \\Z} \\frac{1}{1 + k^2 T^2} \\Big) \\big( p_{0,0}(\\varphi) + p_{2,0}(\\varphi) \\big),$$\nwhere $p_{a,b}(\\varphi) = \\sup_x \\abs{x^a \\varphi^{(b)}(x)}$.\nThis is a bound by finitely many Schwartz seminorms.\n\n## Remarks\n\nIts [[Fourier transform|5PVSQXCTCT]], with the convention $\\hat\\varphi(\\xi) = \\int \\varphi(x) e^{-2\\pi i x \\xi}\\, dx$, is again a Dirac comb:\n$$\\widehat{\\operatorname{III}_T} = \\frac{1}{T} \\operatorname{III}_{1/T}.$$\nThis identity is a restatement of the [[Poisson summation formula|GARGJYSGFG]] $\\sum_k \\varphi(kT) = \\frac1T \\sum_k \\hat\\varphi(k/T)$, applied to $\\hat\\varphi$ and using $\\hat{\\hat\\varphi}(x) = \\varphi(-x)$.\nFor $T = 1$ the comb is its own Fourier transform.\n\nMultiplying a function by $\\operatorname{III}_T$ models sampling at spacing $T$.\nBy the identity above, sampling makes the spectrum periodic with period $1/T$, which is the mechanism behind aliasing and the sampling theorem.\n\n## Explanations\n\n### L1 human\n\nThe Dirac comb is an infinite row of unit spikes placed at equal distance $T$.\nIts spectrum is again a row of spikes, at distance $1/T$.\nSpikes close together in time give spikes far apart in frequency, and the reverse.\n",
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