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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\C$ be a function in the [[Schwartz space|CG9JEATW3Q]].\nLet $\\widehat{f}(\\xi) = \\int_\\R f(x) e^{-2\\pi i x \\xi} \\, dx$ be its [[Fourier transform|DPVM34ADZV]].\nThen\n$$\\sum_{n \\in \\Z} f(n) = \\sum_{k \\in \\Z} \\widehat{f}(k),$$\nand both series converge absolutely.\n\n## Remarks\n\nThe proof periodises $f$: the function $F(x) = \\sum_{n \\in \\Z} f(x + n)$ is smooth and has period $1$.\nIts [[Fourier coefficients|G12KXDHPPS]] are $\\int_0^1 F(x) e^{-2\\pi i k x} \\, dx = \\widehat{f}(k)$, so its Fourier series $\\sum_k \\widehat{f}(k) e^{2\\pi i k x}$ converges to $F(x)$, and $x = 0$ gives the formula.\nExample: for $t \u003e 0$ and $f(x) = e^{-\\pi t x^2}$ one has $\\widehat{f}(\\xi) = t^{-1/2} e^{-\\pi \\xi^2 / t}$, and the formula gives $\\sum_{n} e^{-\\pi t n^2} = t^{-1/2} \\sum_{k} e^{-\\pi k^2/t}$, the transformation law of the Jacobi theta function.\nThe Schwartz hypothesis can be weakened, but some decay and regularity of $f$ and $\\widehat{f}$ is needed for both sides to converge to the same value.\nLet $G$ be a locally compact abelian group and $H$ a discrete subgroup with $G/H$ compact.\nA similar formula then holds, with the sum on the right over the characters of $G$ trivial on $H$.\nIt needs suitable conditions on $f$ and a suitable normalisation of the Haar measures.\n\n## Explanations\n\n### L1 human\n\nSample a function at the integers and add up the values.\nThe formula says that one gets the same total by sampling the Fourier transform at the integers and adding those values.\nA function that is spread out has a transform that is concentrated, so one side can be hard to compute while the other side is easy.\n\n### L2 human\n\nThe formula is the basic link between a lattice and its dual lattice; in $\\R^n$ it reads $\\sum_{v \\in L} f(v) = \\frac{1}{\\operatorname{vol}(\\R^n/L)} \\sum_{w \\in L^*} \\widehat{f}(w)$ for a lattice $L$ with dual lattice $L^*$.\nApplied to Gaussians it gives the functional equation of theta functions, which leads to the functional equation of the Riemann zeta function.\nIn signal processing it underlies the sampling theorem, and in analytic number theory it is used to count lattice points.\n",
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