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  "body": "\n## Statement\n\nLet $f \\in L^1(\\R)$, the Lebesgue space of integrable functions, and let $\\widehat{f}(\\xi) = \\int_\\R f(x) e^{-2\\pi i x\\xi} \\, dx$ be its [[Fourier transform|DPVM34ADZV]].\nFor $a \\in \\R$ let $f_a(x) = f(x - a)$.\nThen the linear span of the translates $\\set{f_a : a \\in \\R}$ is dense in $L^1(\\R)$ if and only if $\\widehat{f}(\\xi) \\ne 0$ for every $\\xi \\in \\R$.\n\n## Remarks\n\nThe \"only if\" part is easy: if $\\widehat{f}(\\xi_0) = 0$, then every $g$ in the closed span has $\\widehat{g}(\\xi_0) = 0$, because $\\widehat{f_a}(\\xi) = e^{-2\\pi i a\\xi}\\widehat{f}(\\xi)$ and $g \\mapsto \\widehat{g}(\\xi_0)$ is continuous on $L^1$; but some $g \\in L^1$ has $\\widehat{g}(\\xi_0) \\ne 0$.\nThe \"if\" part uses [[Wiener's lemma|NC768TWZ3D]] type arguments: closed translation-invariant subspaces of $L^1(\\R)$ are ideals for convolution, and an ideal whose transforms have no common zero is dense.\nExample: the Gaussian $f(x) = e^{-\\pi x^2}$ has $\\widehat{f}(\\xi) = e^{-\\pi\\xi^2} \u003e 0$, so linear combinations of its translates approximate every integrable function in $L^1$.\nThe same statement holds for every [[locally compact abelian group|ZR09S9R3AW]] $G$, with $f \\in L^1(G)$ and the [[Fourier transform|VHN5VW5Q92]] on the dual group.\nThere is also an $L^2$ version: the translates of $f \\in L^2(\\R)$ span a dense subspace of $L^2(\\R)$ if and only if the zero set of $\\widehat{f}$ has Lebesgue measure $0$.\n\n## Explanations\n\n### L1 human\n\nTake one integrable function and all its shifted copies.\nThe theorem tells when sums of these copies can approximate every integrable function.\nThe answer is: exactly when the function contains every frequency, that is, its Fourier transform never vanishes.\n\n### L2 human\n\nWiener used the theorem to give general Tauberian theorems: if $\\phi$ is bounded and $f * \\phi(x) \\to c \\int f$ as $x \\to \\infty$ for one $f$ with nonvanishing transform, then the same holds for all $g \\in L^1$.\nThis gives a short route to Tauberian results for summation methods and, through the Wiener–Ikehara theorem, to the prime number theorem.\nIn Banach algebra language, the theorem says that a closed ideal of $L^1(\\R)$ whose Fourier transforms have no common zero is all of $L^1(\\R)$.\n",
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