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  "body": "\n## Summary\n\nThe Hilbert transform is the linear operator on functions of a real variable given by convolution with $\\frac{1}{\\pi x}$ in the [[principal value|0SYR10HD4R]] sense.\nLet $u + iv$ be holomorphic on the upper half-plane and decay at infinity.\nThen the transform sends the boundary values of $u$ to those of $v$, so it is the operator of [[harmonic conjugation|050F2FMTKF]] on the line.\nOn the Fourier side it multiplies by $-i \\sgn(\\xi)$.\nIt is the model example of a singular integral operator, and it appears in signal processing and in the [[Kramers–Kronig relations|GFS95PZ5YK]].\n\n## Explanations\n\n### L1 human\n\nThe Hilbert transform shifts the phase of each frequency component of a signal by a quarter turn.\nIt sends $\\cos$ to $\\sin$ and $\\sin$ to $-\\cos$.\nIf $u + iv$ is holomorphic in the upper half-plane and decays at infinity, then on the real line $v$ is the Hilbert transform of $u$.\nFor example, $\\frac{i}{x + i} = \\frac{1}{1 + x^2} + i \\frac{x}{1 + x^2}$, so the transform of $\\frac{1}{1 + x^2}$ is $\\frac{x}{1 + x^2}$.\n",
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