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  "body": "\n## Summary\n\nA Wiener–Hopf equation is a [[convolution integral equation|EY75YD988Z]] on the half-line.\nThe kernel depends only on $x - t$, but the unknown lives on $[0, \\infty)$ and the integral runs over $[0, \\infty)$.\nThe cut at $0$ breaks the translation symmetry, so the [[Fourier transform|DPVM34ADZV]] alone does not solve it.\nIt is solved by the [[Wiener–Hopf method|H4NXEY6QTZ]].\nThis method factors the symbol of the equation into pieces that extend holomorphically to the two half-planes.\nSuch equations model radiative transfer in a half-space, diffraction by a half-plane, and optimal causal filtering.\n\n## Explanations\n\n### L1 human\n\nImagine a medium that fills half of space, such as a star's atmosphere seen as a slab of infinite depth.\nEach layer interacts with the others through a rule that depends only on their distance, but there are no layers beyond the surface.\nThe equation for the state of the layers is then a convolution, cut off at the surface.\nA small example: the function $\\varphi(x) = 1 + x$ satisfies $\\varphi(x) = \\int_0^\\infty \\frac{1}{2} e^{-\\abs{x - t}} \\varphi(t)\\, dt$ for all $x \\ge 0$.\n",
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