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  "body": "\n## Summary\n\nA positive-definite function on $\\R^n$ is a complex function $f$ such that every matrix $\\big(f(x_i - x_j)\\big)_{i,j}$ built from finitely many points is positive semidefinite.\nIt is the case $G = \\R^n$ of a [[positive-definite function on a group|3PRZ02APFA]].\nBy [[Bochner's theorem|ZRHAG4NXHC]], the continuous positive-definite functions on $\\R^n$ are exactly the Fourier transforms of finite positive measures.\nThey appear as characteristic functions in probability, as covariance functions of stationary processes, and as kernels in interpolation and machine learning.\n\n## Explanations\n\n### L1 human\n\nPick some points on the line and form the table of values $f(x_i - x_j)$.\nFor a positive-definite function, this table always behaves like a table of inner products, so it never produces a negative \"squared length\".\nThe cosine is the simplest example, and so is the Gaussian $e^{-x^2}$.\nBeing positive-definite forces $f(0)$ to be the largest value of $\\abs{f}$.\n",
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