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  "body": "\n## Statement\n\nLet $G$ be a [[locally compact abelian group|ZR09S9R3AW]] and let $\\widehat{G}$ be its [[Pontryagin dual|FMW57WNPDB]].\nLet $\\varphi \\colon G \\to \\C$ be a continuous function.\nCall $\\varphi$ [[positive-definite|3PRZ02APFA]] if $\\sum_{j,k=1}^{n} c_j \\overline{c_k}\\, \\varphi(x_j - x_k) \\ge 0$ for all $n \\ge 1$, $x_1, \\dots, x_n \\in G$ and $c_1, \\dots, c_n \\in \\C$.\nThen $\\varphi$ is positive-definite if and only if there is a finite positive [[Radon measure|HZNEQQBCW4]] $\\mu$ on $\\widehat{G}$ with\n$$\\varphi(x) = \\int_{\\widehat{G}} \\chi(x) \\, d\\mu(\\chi) \\quad \\text{for all } x \\in G.$$\nThe measure $\\mu$ is then unique, and $\\mu(\\widehat{G}) = \\varphi(0)$.\n\n## Remarks\n\nThe easy direction: if $\\varphi$ has this form, then $\\sum_{j,k} c_j \\overline{c_k} \\varphi(x_j - x_k) = \\int_{\\widehat{G}} \\abs{\\sum_j c_j \\chi(x_j)}^2 \\, d\\mu(\\chi) \\ge 0$.\nFor the converse, $\\varphi$ defines a positive linear functional on $L^1(G)$, hence on the commutative group C*-algebra of $G$.\nThe spectrum of this algebra is $\\widehat{G}$, and the Riesz representation theorem then gives $\\mu$.\nExample: for $G = \\R$, the function $\\varphi(x) = e^{-x^2/2}$ is positive-definite, and $\\mu$ is the standard normal distribution, since $\\int_\\R e^{i\\xi x} \\frac{1}{\\sqrt{2\\pi}} e^{-\\xi^2/2} \\, d\\xi = e^{-x^2/2}$.\nFor $G = \\R$, the theorem says that the continuous positive-definite functions with $\\varphi(0) = 1$ are exactly the [[characteristic functions|0V5B6D6BQH]] of probability distributions.\nContinuity is needed: on $\\R$ the indicator function of $\\set{0}$ is positive-definite but is not the transform of any measure, since such transforms are continuous.\n\n## Explanations\n\n### L1 human\n\nA positive-definite function is one that behaves like a correlation: certain sums built from it are never negative.\nBochner's theorem says that such functions are exactly the averages of waves with nonnegative weights.\nIn probability, it says which functions are characteristic functions of random variables.\n\n### L2 human\n\nThe theorem is a key tool in the proof of the [[Plancherel theorem for abelian groups|V9K9TG0K3J]] and of [[Pontryagin duality|QTVC6F2P9D]].\nFor $G = \\Z$ it is Herglotz's theorem on positive-definite sequences, with measures on the circle.\nIn the theory of stationary processes it gives the spectral measure of a stationary covariance function.\n",
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