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  "body": "\n## Summary\n\nA positive-definite function on a group $G$ is a complex function $\\varphi$ on $G$ with the following property.\nFor all $g_1, \\dots, g_n \\in G$, the matrix $\\big(\\varphi(g_i^{-1}g_j)\\big)_{i,j}$ is [[positive semidefinite|0KX66CZMBB]].\nThe diagonal matrix coefficients $g \\mapsto \\ip{\\pi(g)v}{v}$ of [[unitary representations|74BPYJNZTM]] are positive-definite, and conversely every continuous positive-definite function arises in this way.\nThis correspondence links functions on a group with its representations, and it underlies Bochner's theorem and the Gelfand–Raikov theorem.\n\n## Explanations\n\n### L1 human\n\nOn the real line, the function $\\cos x$ is positive-definite, while $-1$ is not.\nThe idea is that $\\varphi(g^{-1}h)$ behaves like an inner product between the elements $g$ and $h$, so sums $\\sum \\bar c_i c_j \\varphi(g_i^{-1}g_j)$ behave like squared lengths and cannot be negative.\nIndeed $\\cos(y - x) = \\cos x \\cos y + \\sin x \\sin y$ is the dot product of the unit vectors at angles $x$ and $y$.\nEvery positive-definite function is, in this sense, a hidden inner product, and the hidden space is a unitary representation of the group.\n",
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